❆◆❚Ô◆■❖ ❋❆❇■❆◆❖ P❆■❱❆

❱❖▲❯▼❊ ❊ ➪❘❊❆ ❉❊ ❙Ó▲■❉❖❙ ●❊❖▼➱❚❘■❈❖❙ ❯❙❆◆❉❖ ❖ P❘■◆❈❮P■❖❉❊ ❈❆❱❆▲■❊❘■

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦ ♣❛rt❡ ❞❛s❡①✐❣ê♥❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦❞❡ ▼❛❣✐st❡r ❙❝✐❡♥t✐❛❡✳

❱■➬❖❙❆▼■◆❆❙ ●❊❘❆■❙ ✲ ❇❘❆❙■▲

✷✵✶✺

11/6/2015 FichaCatalografica :: Fichacatalografica

https://www3.dti.ufv.br/bbt/ficha/Fichacatalografica/aprovada/201505280328029483 2/3

Ficha catalográfica preparada pela Biblioteca Central da Universidade Federal de

Viçosa - Campus Viçosa

T

Paiva, Antônio Fabiano, 1971-

P149v

2015

Volume e área de sólidos geométricos usando o Princípio de Cavalieri

/ Antônio Fabiano Paiva. - Viçosa, MG, 2015.

x, 67f. : il. (algumas color.) ; 29 cm.

Orientador: Mércio Botelho Faria.

Dissertação (mestrado) - Universidade Federal de Viçosa.

Referências bibliográficas: f.66-67.

1. Geometria sólida. 2. Princípio de Cavalieri (Matemática). 3.

Cálculo de volume. I. Universidade Federal de Viçosa. Departamento de

Matemática. Programa de Pós-graduação em Matemática. II. Título.

CDD 22. ed. 516.156

❆◆❚Ô◆■❖ ❋❆❇■❆◆❖ P❆■❱❆

❱❖▲❯▼❊ ❊ ➪❘❊❆ ❉❊ ❙Ó▲■❉❖❙ ●❊❖▼➱❚❘■❈❖❙ ❯❙❆◆❉❖ ❖ P❘■◆❈❮P■❖❉❊ ❈❆❱❆▲■❊❘■

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ à ❯♥✐✈❡rs✐❞❛❞❡❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❝♦♠♦ ♣❛rt❡ ❞❛s❡①✐❣ê♥❝✐❛s ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧✱ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦❞❡ ▼❛❣✐st❡r ❙❝✐❡♥t✐❛❡✳

❆P❘❖❱❆❉❆✿ ✼ ❞❡ ❛❜r✐❧ ❞❡ ✷✵✶✺✳

▼❛r✐♦ ❏♦sé ❞❡ ❙♦✉③❛ ❲❛❧t❡r ❍✉❛r❛❝❛ ❱❛r❣❛s

▼❡r❝✐♦ ❇♦t❡❧❤♦ ❋❛r✐❛✭❖r✐❡♥t❛❞♦r✮

❆❣r❛❞❡❝✐♠❡♥t♦s

❆ ❉❡✉s ♣♦r t♦❞♦s ♦s ♠❡✉s ❝♦♥❤❡❝✐♠❡♥t♦s ❡ ❢♦rç❛ ♣❛r❛ ❝♦♥s❡❣✉✐r ❧✉t❛r✦

➚ ▲❡♥✐✱ ♠✐♥❤❛ q✉❡r✐❞❛ ❡s♣♦s❛✱ ♣❡❧♦ ❛♣♦✐♦✱ ❝♦♠♣r❡❡♥sã♦✱ ❢♦rç❛ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧♦ ❛♠♦r❝❡❞✐❞♦ ❛ ♠✐♠ ❞✉r❛♥t❡ t♦❞♦ ❡st❡ t❡♠♣♦✳

❆♦ ♠❡✉ ✜❧❤♦✱ ●❛❜r✐❡❧✱ ♣❡❧♦ ❝❛r✐♥❤♦ ✐♥❝♦♥❞✐❝✐♦♥❛❧ ❞❛❞♦ ❛ ♠✐♠ ❞✉r❛♥t❡ ♦s ♠❡✉s ❡st✉❞♦s✳

❆♦s ♠❡✉s ♣❛✐s✱ ■✈❛♥ ❡ ❊❞♥❛✱ ♣❡❧❛s ♦r❛çõ❡s✱ ❝♦♥s❡❧❤♦s ❞✉r❛♥t❡ ❛s ♠✐♥❤❛s ✈✐❛❣❡♥s ❡ ❛♠✐♥❤❛ ❡st❛❞✐❛ ❡♠ ❱✐ç♦s❛✳

❆ ♠✐♥❤❛ ✐r♠ã✱ ❋❛❜✐❡❧❡✱ q✉❡ ♠❡s♠♦ ❞✐st❛♥t❡ s❡✐ q✉❡ t♦r❝❡✉ ❡ ♠❡ ❛❥✉❞♦✉ ♠❡♥t❛❧♠❡♥t❡ ❡❞♦ s❡✉ ❥❡✐t♦✳

❆♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦✱ ♣❡❧♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦✱ ❢♦rç❛ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ✉♠❛ ❞✐✈❡rsã♦s❡♠ t❛♠❛♥❤♦ ❡♥q✉❛♥t♦ ❡st✉❞á✈❛♠♦s✳

❆♦s ♠❡✉s ♥♦✈♦s ❛♠✐❣♦s ❆♥❞ré✱ ❉❛♥✐❡❧✱ ❊❧✐❛s✱ ●✐❧❜❡rt♦ ❡ ❱✐✈✐❛♥❡✱ ♦s q✉❛✐s ❢♦r♠❛♠♦s ❛♦♥♦ss♦ ❥❡✐t♦ ✉♠❛ ✈❡r❞❛❞❡✐r❛ ✐r♠❛♥❞❛❞❡✳

❆s ❡s❝♦❧❛s ❛s q✉❛✐s tr❛❜❛❧❤❡✐ ❞✉r❛♥t❡ ♦s ❞♦✐s ❛♥♦s ❞❡ ❡st✉❞♦ ♣♦r ❝♦♠♣r❡❡♥❞❡r❡♠ ❛s ♠✐♥❤❛s♥❡❝❡ss✐❞❛❞❡s ❡ ♦r❣❛♥✐③❛çõ❡s ❞❡ ❤♦rár✐♦s✳

❆♦ ♣r♦❢❡ss♦r✱ ❉♦✉t♦r ▼ér❝✐♦ ❇♦t❡❧❤♦✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛✱ ❡♥s✐♥❛♠❡♥t♦s ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧❛♣❛❝✐ê♥❝✐❛✳

➚ ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳

✐✐

❙✉♠ár✐♦

❘❡s✉♠♦ ✈✐

❆❜str❛❝t ✈✐✐

■♥tr♦❞✉çã♦ ✶

✶ ❈♦♥t❛♥❞♦ ✉♠ ♣♦✉❝♦ ❞❡ ❍✐stór✐❛ ✸

✷ ❖ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✱ ♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ❡ ♦ ❊◆❊▼ ✾

✷✳✶ P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

✷✳✷ ❊①❛♠❡ ◆❛❝✐♦♥❛❧ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ✲ ❊◆❊▼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

✷✳✷✳✶ ❊■❳❖❙ ❈❖●◆■❚■❱❖❙ ✭❝♦♠✉♥s ❛ t♦❞❛s ❛s ár❡❛s ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦✮ ✳ ✳ ✶✷

✷✳✷✳✷ ❈♦♠♣❡tê♥❝✐❛ ❞❡ ár❡❛ ✷ ✲ ❯t✐❧✐③❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❣❡♦♠étr✐❝♦ ♣❛r❛ r❡✲❛❧✐③❛r ❛ ❧❡✐t✉r❛ ❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❛ r❡❛❧✐❞❛❞❡ ❡ ❛❣✐r s♦❜r❡ ❡❧❛✳ ✳ ✳ ✳ ✳ ✶✸

✷✳✷✳✸ ❈♦♠♣❡tê♥❝✐❛ ❞❡ ár❡❛ ✸ ✲ ❈♦♥str✉✐r ♥♦çõ❡s ❞❡ ❣r❛♥❞❡③❛s ❡ ♠❡❞✐❞❛s ♣❛r❛❛ ❝♦♠♣r❡❡♥sã♦ ❞❛ r❡❛❧✐❞❛❞❡ ❡ ❛ s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❞♦ ❝♦t✐❞✐❛♥♦✳ ✳ ✶✸

✸ ◆♦çõ❡s ■♥✐❝✐❛✐s ❛ r❡s♣❡✐t♦ ❞♦s ♣♦❧✐❡❞r♦s ❞❡ P❧❛tã♦ ❡ ❛❧❣✉♥s só❧✐❞♦s ✶✹

✸✳✶ P♦❧✐❡❞r♦s ❊s♣❡❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺

✸✳✶✳✶ Pr✐s♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺

✸✳✶✳✷ P✐râ♠✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺

✸✳✷ ❙ó❧✐❞♦s ❢♦r♠❛❞♦s ♣♦r s✉♣❡r❢í❝✐❡s ♥ã♦ ♣❧❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✐✐✐

❙❯▼➪❘■❖ ✐✈

✸✳✷✳✶ ❈✐❧✐♥❞r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✸✳✷✳✷ ❈♦♥❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✸✳✷✳✸ ❊s❢❡r❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✸✳✸ ➪r❡❛s ❡ ✈♦❧✉♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✸✳✹ ❊①❡♠♣❧♦s ❞❡ P♦❧✐❡❞r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✹ ❆ ♥♦çã♦ ✐♥t✉✐t✐✈❛ ❛ r❡s♣❡✐t♦ ❞❡ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦ ❡ ✈♦❧✉♠❡ ❞❡ ✉♠♣❛r❛❧❡❧❡♣í♣❡❞♦ ✷✵

✺ ❖ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ✷✹

✺✳✶ Pr♦❝❡❞✐♠❡♥t♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✻ ❯s♦ ❞❡ r❡❝✉rs♦s ❡❧❡trô♥✐❝♦s ❡ s♦❢t✇❛r❡s ♥♦ ❡♠♣r❡❣♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✲✈❛❧✐❡r✐ ✷✽

✼ ❯s♦ ❞❡ ♠❛t❡r✐❛❧ ✏❝♦♥❝r❡t♦✑ ♥♦ ❡st✉❞♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ✸✺

✽ ❉❡♠♦♥str❛çõ❡s ❡ ❡①❡♠♣❧✐✜❝❛çõ❡s ✐♠♣♦rt❛♥t❡s ✸✽

✽✳✶ ❉❡♠♦♥str❛çã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡s✱ ✉s❛♥❞♦r❡❝✉rs♦s ❞❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧ ❝♦♠♦ ❥✉st✐✜❝❛t✐✈❛ ♣❛r❛ ❞♦❝❡♥t❡s✳ ✳ ✸✽

✽✳✷ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ✜❣✉r❛s ♣❧❛♥❛s ✲ ❉❡♠♦♥str❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✽✳✸ ➪r❡❛ ❞❡ ✉♠❛ r❡❣✐ã♦ ❞❡t❡r♠✐♥❛❞❛ ♣♦r ✉♠❛ ❝✉r✈❛ y = x2 ❡ ♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s✳ ✹✵

✽✳✸✳✶ ❈♦♥❝❡✐t♦s ❡ ✐♥❢♦r♠❛çõ❡s ♣r❡❧✐♠✐♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

✽✳✹ ❱♦❧✉♠❡ ❞❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ q✉❛❧q✉❡r ✲ ❉❡♠♦♥str❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷

✽✳✺ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ ❞❡ ❜❛s❡ tr✐❛♥❣✉❧❛r ✲ ❉❡♠♦♥str❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✽✳✻ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ q✉❛❧q✉❡r ✭♥ã♦ s❡♥❞♦ tr✐❛♥❣✉❧❛r✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✽✳✼ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✾ Pr♦♣♦st❛ ♣❛r❛ ♦ tr❛❜❛❧❤♦ ❞✐❞át✐❝♦ ❝♦♠ ♣r✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♥♦ ❊♥s✐♥♦▼é❞✐♦ ✹✼

✈

✾✳✶ ❆t✐✈✐❞❛❞❡s ❛❧t❡r♥❛t✐✈❛s q✉❡ ❡♥✈♦❧✈❡♠ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

✾✳✶✳✶ ❆t✐✈✐❞❛❞❡ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

✾✳✶✳✷ ❆t✐✈✐❞❛❞❡ ✲ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

✶✵ ❘❡s♦❧✉çã♦ ❞♦s ❡①❡r❝í❝✐♦s t❡ór✐❝♦s ❞♦ ❝❛♣ít✉❧♦ ✾✳ ✺✻

❈♦♥❝❧✉sã♦ ✻✺

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✻✻

❘❡s✉♠♦

P❆■❱❆✱ ❆♥tô♥✐♦ ❋❛❜✐❛♥♦✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❛❜r✐❧ ❞❡ ✷✵✶✺✳ ❱♦❧✉♠❡ ❡➪r❡❛ ❞❡ ❙ó❧✐❞♦s ●❡♦♠étr✐❝♦s ❯s❛♥❞♦ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳ ❖r✐❡♥t❛❞♦r✿ ▼ér❝✐♦❇♦t❡❧❤♦ ❋❛r✐❛✳

◆❡st❛ ❞✐ss❡rt❛çã♦ ❡st✉❞❛♠♦s ❡ ❡❧❛❜♦r❛♠♦s ❛❧❣✉♠❛s ♣r♦♣♦st❛s ❞❡ ❛t✐✈✐❞❛❞❡s ❡❞✉❝❛❝✐♦♥❛✐s

q✉❡♠ ❡♥✈♦❧✈❡♠ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✱ ❜❡♠ ❝♦♠♦ ♦ s❡✉ ✉s♦ ❡ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❢❡✐t♦ ❝♦♠

r❡❝✉rs♦s ❡❧❡trô♥✐❝♦s✱ t❡♥❞♦ ❝♦♠♦ ♣r✐♥❝✐♣❛❧ ❢❡rr❛♠❡♥t❛ ♦ s♦❢t✇❛r❡ ●❡♦❣❡❜r❛ ✸❉ ❡ ♠❛t❡r✐❛❧

❝♦♥❝r❡t♦✳ ❖ ♥♦ss♦ tr❛❜❛❧❤♦ ❝♦♥st❛ ❞❡ ✉♠❛ ❛♣r❡s❡♥t❛çã♦ t❡ór✐❝❛✱ ❜❡♠ ❝♦♠♦ ✉♠❛ r❡✈✐sã♦

❜✐❜❧✐♦❣rá✜❝❛ ❞❡ ❝♦♥❝❡✐t♦s q✉❡ sã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ ♥♦✈♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❡

r❡s♦❧✉çã♦ ❞❡ ❡①❡r❝í❝✐♦s ❡ ♣r♦❜❧❡♠❛s ♣rát✐❝♦s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ❡ ♥♦ ❝♦t✐❞✐❛♥♦✱ ♣♦❞❡♥❞♦ s❡r

❡❧❛❜♦r❛❞♦s ♣❛r❛ q✉❛❧q✉❡r ♥❡❝❡ss✐❞❛❞❡ q✉❡ ♦s ❡❞✉❝❛♥❞♦s ♣♦ss❛♠ t❡r ❡♠ s✉❛ ✈✐❞❛ ❡st✉❞❛♥✲

t✐❧✳ ❆♣r❡s❡♥t❛♠♦s✱ ❛ss✐♠✱ ❛t✐✈✐❞❛❞❡s q✉❡ ♣♦❞❡♠ s❡r✈✐r ❞❡ ❡①❡♠♣❧♦s ❡ ✐♥❝❡♥t✐✈♦ ❛ ❢✉t✉r♦s

tr❛❜❛❧❤♦s ❝♦♠ ♦✉tr❛s ♣❛rt❡s ❞❛ ❣❡♦♠❡tr✐❛ ❡ ❡♠ ♦✉tr♦s r❛♠♦s ❞❛ ▼❛t❡♠át✐❝❛✳

✈✐

❆❜str❛❝t

P❆■❱❆✱ ❆♥tô♥✐♦ ❋❛❜✐❛♥♦✱ ▼✳❙❝✳✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❱✐ç♦s❛✱ ❆♣r✐❧ ✷✵✶✺✳ ❱♦❧✉♠❡ ❛♥❞❆r❡❛ ♦❢ ●❡♦♠❡tr✐❝s ❙♦❧✐❞s ❯s✐♥❣ t❤❡ ❈❛✈❛❧✐❡r✐➫ s Pr✐♥❝✐♣❧❡ ❆❞✈✐s♦r✿ ▼ér❝✐♦ ❇♦t❡❧❤♦❋❛r✐❛✳

■♥ t❤✐s ❞✐ss❡rt❛t✐♦♥ ✇❡ st✉❞✐❡❞ ❛♥❞ ❝r❡❛t❡❞ s♦♠❡ ♣r♦♣♦s❛❧s ❢♦r ❡❞✉❝❛t✐♦♥❛❧ ❛❝t✐✈✐t✐❡s ✇❤✐❝❤

✐♥✈♦❧✈❡ t❤❡ Pr✐♥❝✐♣❧❡ ♦❢ ❈❛✈❛❧✐❡r✐✱ ❛s ✇❡❧❧ ❛s ✐ts ✉s❡ ❛♥❞ ❛♣♣❧✐❝❛❜✐❧✐t② ❞♦♥❡ ✇✐t❤ ❡❧❡❝tr♦♥✐❝

r❡s♦✉r❝❡s✱ ❤❛✈✐♥❣ ❛s t❤❡ ♠❛✐♥ t♦♦❧ ●❡♦❣❡❜r❛ ✸❉ s♦❢t✇❛r❡ ❛♥❞ ❝♦♥❝r❡t❡ ♠❛t❡r✐❛❧s✳ ❖✉r ✇♦r❦

❝♦♥s✐sts ♦❢ ❛ t❤❡♦r❡t✐❝❛❧ ♣r❡s❡♥t❛t✐♦♥✱ ❛♥❞ ❛ ❧✐t❡r❛t✉r❡ r❡✈✐❡✇ ♦❢ ❝♦♥❝❡♣ts t❤❛t ❛r❡ ✐♠♣♦rt❛♥t

❢♦r ♦❜t❛✐♥✐♥❣ ♥❡✇ ❦♥♦✇❧❡❞❣❡ ❛♥❞ s♦❧✈✐♥❣ ♣r❛❝t✐❝❛❧ ❡①❡r❝✐s❡s ❛♥❞ ♣r♦❜❧❡♠s ✐♥ t❤❡ ❝❧❛ssr♦♦♠

❛♥❞ ✐♥ ❞❛✐❧② ❧✐❢❡✳ ❚❤❡s❡ ❡①❡r❝✐s❡s ❝❛♥ ❜❡ ♣r❡♣❛r❡❞ ❢♦r ❛♥② ♥❡❡❞ t❤❛t t❤❡ st✉❞❡♥ts ❝❛♥ ❤❛✈❡

✐♥ t❤❡✐r st✉❞❡♥t ❧✐❢❡✳ ❚❤✐s ✇❛②✱ ✇❡ ♣r❡s❡♥t ❛❝t✐✈✐t✐❡s t❤❛t ❝❛♥ s❡r✈❡ ❛s ❡①❛♠♣❧❡s ❛♥❞ ❡♥❝♦✉✲

r❛❣❡♠❡♥t t♦ ❢✉t✉r❡ ✇♦r❦ ✇✐t❤ ♦t❤❡r ♣❛rts ♦❢ ❣❡♦♠❡tr② ❛♥❞ ♦t❤❡r ❜r❛♥❝❤❡s ♦❢ ♠❛t❤❡♠❛t✐❝s✳

✈✐✐

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✶✳✶ ❇♦❛✈❡♥t✉r❛ ❈❛✈❛❧✐❡r✐ ✲ ❋♦♥t❡✿ ♠❛t❤❡✉s♠❛t❤✐❝❛✳❜❧♦❣s♣t✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺

✶✳✷ ❆♣❧✐❝❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺

✶✳✸ ▲❡♦♥❤❛r❞ P❛✉❧ ❊✉❧❡r ✲ ❋♦♥t❡✿ ✇✇✇✳❡①♣❧✐❝❛t♦r✐✉♠✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

✸✳✶ P✐râ♠✐❞❡s ❞♦ ❊❣✐t♦ ✲ ❋♦♥t❡✿ ✇✇✇✳❛s❝♦♥✈❡rs❛✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✸✳✷ ❈✐❧✐♥❞r♦ ❈✐r❝✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✸✳✸ ❈♦♥❡ ✲ ❋♦♥t❡✿ ❧♦❛❞✐♥❣✐♥❢♦r♠❛t✐♦♥s✳❜❧♦ ❣s♣♦t✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✸✳✹ ❊s❢❡r❛ ❞❡ r❡✈♦❧✉çã♦ ✲ ❋♦♥t❡✿ ✇✇✇✳♠✉♥❞♦❡❞✉❝❛❝❛♦✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✸✳✺ P♦❧✐❡❞r♦s ✲ ❋♦♥t❡✿ ❤tt♣✿✴✴✇✇✇✳❡❞✉❝✳❢❝✳✉❧✳♣t✴✐❝♠✴✐❝♠✾✾✴✐❝♠✶✻✴❝✉r✐♦s✐❞❛❞❡s✳❤t♠ ✶✾

✹✳✶ P❛r❛❧❡❧❡♣í♣❡❞♦ ❝♦♠ ❛r❡st❛s ✐❣✉❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✹✳✷ P❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ❛r❡st❛s 1✱ 1 ❡ a ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✹✳✸ P❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ❛r❡st❛s a✱ 1 ❡ b ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✹✳✹ P❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ❛r❡st❛s a✱ b ❡ c ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✹✳✺ P❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ❛r❡st❛s a✱ b ❡ c ✲ ❋♦♥t❡✿ ❛♥♦ss❛❡s❝♦❧❛✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✹✳✻ ❘♦♠❜♦❡❞r♦s ✲ ❋♦♥t❡✿ ✇✇✇✳♣♦rt❛❧❡s❝♦❧❛r✳♥❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✹✳✼ P❛r❛❧❡❧❡♣í♣❡❞♦ ✲ ❋♦♥t❡✿ ♠❛t❡♠❛t✐❝❛❝♦❡s✳❜❧♦❣s♣♦t✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✺✳✶ ❈á❧❝✉❧♦ ❞❡ ➪r❡❛s ✲ ❋♦♥t❡✿ ♠❛✉r♦✇❡✐❣❡❧✳❜❧♦❣s♣♦t✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✺✳✷ ❋✐❣✉r❛s P❧❛♥❛s ✲ ❋♦♥t❡✿ ▼❛t❡♠át✐❝❛ ▼✉❧t✐♠í❞✐❛✱ ●✉✐❛ ❞♦ Pr♦❢❡ss♦r✱ ●❡♦♠❡✲tr✐❛ ❡ ▼❡❞✐❞❛s✱ ❯♥✐❝❛♠♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✈✐✐✐

▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ✐①

✺✳✸ ❙ó❧✐❞♦s ✲ ❋♦♥t❡✿♦❜❛r✐❝❡♥tr♦❞❛♠❡♥t❡✳❜❧♦❣s♣♦t✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✺✳✹ ▼♦❡❞❛s r❡♣r❡s❡♥t❛♥❞♦ s❡❝çõ❡s✳ ❋♦♥t❡✿ ♦❜❛r✐❝❡♥tr♦❞❛♠❡♥t❡✳❜❧♦❣s♣♦t✳❝♦♠ ✳ ✳ ✷✻

✺✳✺ Pr♦❝❡❞✐♠❡♥t♦ ❞❡ ❈❛✈❛❧✐❡r✐ ■ ✲ ❋♦♥t❡✿ ♦❜❛r✐❝❡♥tr♦❞❛♠❡♥t❡✳❜❧♦❣s♣♦t✳❝♦♠ ✳ ✳ ✳ ✷✼

✺✳✻ Pr♦❝❡❞✐♠❡♥t♦ ❞❡ ❈❛✈❛❧✐❡r✐ ■■ ✲ ❋♦♥t❡✿ ♦❜❛r✐❝❡♥tr♦❞❛♠❡♥t❡✳❜❧♦❣s♣♦t✳❝♦♠ ✳ ✳ ✳ ✷✼

✻✳✶ P✐râ♠✐❞❡ ❲✐♥❣❡♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✻✳✷ ❊❞✐çã♦ ❞❛ P✐râ♠✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✻✳✸ ❘❡❧❛çã♦ ❞❡ ❊✉❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✻✳✹ P♦❧í❣♦♥♦ ♥♦ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

✻✳✺ P❛r❛❧❡❧❡♣í♣❡❞♦ ♥♦ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✻✳✻ ❊s❢❡r❛ ♥♦ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶

✻✳✼ ❉❡❝á❣♦♥♦ ♥♦ ●❡♦❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✻✳✽ Pr✐s♠❛ t❡♥❞♦ ❝♦♠♦ ❜❛s❡ ✉♠ ❞❡❝á❣♦♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✻✳✾ Pr✐s♠❛ P❡♥t❛❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✻✳✶✵ P❧❛♥✐✜❝❛çã♦ ❞♦ Pr✐s♠❛ P❡♥t❛❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✻✳✶✶ Pr✐s♠❛ P❡♥t❛❣♦♥❛❧ ♥♦ P♦❧② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✻✳✶✷ Pr✐s♠❛ P❡♥t❛❣♦♥❛❧ ❘❡❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹

✼✳✶ P✐râ♠✐❞❡ ❞❡ ❇❛s❡ ◗✉❛❞r❛♥❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻

✼✳✷ P✐râ♠✐❞❡ ❞❡ ❇❛s❡ ❍❡①❛❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✼✳✸ ❙ó❧✐❞♦s ❞❡ ◆❛t✉r❡③❛ ❉✐❢❡r❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✽✳✶ ❘❡❣✐ã♦ ❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✽✳✷ ❘❡❣✐ã♦ ◗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

✽✳✸ ➪r❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

✽✳✹ ➪r❡❛ ❝♦♠♣r❡❡♥❞✐❞❛ ❡♥tr❡ ❛ ❝✉r✈❛ y = x2 ❡ ♦ ❡✐①♦ ① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

✽✳✺ P❛r❛❧❡❧❡♣í♣❡❞♦ P ✲ ❋♦♥t❡✿ ■♥❢♦❡s❝♦❧❛✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷

▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ①

✽✳✻ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ tr✐❛♥❣✉❧❛r ✲ ❋♦♥t❡✿ ❊❧♦♥ ▲❛❣❡s ▲✐♠❛ ✲ ▼❡❞✐❞❛ ❡❋♦r♠❛ ❡♠ ●❡♦♠❡tr✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✽✳✼ P❡♥tá❣♦♥♦ ❡♠ n− 2 tr✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹

✽✳✽ P✐râ♠✐❞❡ P❡♥t❛❣♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✽✳✾ ❈❧❡♣s✐❞r❛ ❡ ❆♥t✐❝❧❡♣s✐❞r❛✳ ❋♦♥t❡✿ ♦❜❛r✐❝❡♥tr♦❞❛♠❡♥t❡✳❜❧♦❣s♣♦t✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✹✺

✽✳✶✵ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛✳ ❋♦♥t❡✿ ♦❜❛r✐❝❡♥tr♦❞❛♠❡♥t❡✳❜❧♦❣s♣♦t✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻

✾✳✶ ❊♥❡♠ ✷✵✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

✾✳✷ ❯❋❯ ✲ ▼● ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

✾✳✸ ❯❋❘❙ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵

✾✳✹ ❊◆❊▼✱ ✷✵✶✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

✾✳✺ ❈✐❧✐♥❞r♦ ✐♥s❝r✐t♦✳ ❋♦♥t❡✿ ✇✇✇✳♠❛❡str♦♣❧♦♠❡r♦✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

✾✳✻ ❙❡♠✐❡s❢❡r❛✳ ❋♦♥t❡✿ ✇✇✇✳♠❛❡str♦♣❧♦♠❡r♦✳❝♦♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

✾✳✼ ❊①tr❛t♦ ❞❡ ❚♦♠❛t❡✿ ❘❡❛❧✐❞❛❞❡❀ ❈✐❧✐♥❞r♦✿ ▼♦❞❡❧♦ ▼❛t❡♠át✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

✶✵✳✶ Pr✐s♠❛ ❚r✐❛♥❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻

✶✵✳✷ ◗✉❛❞r❛❞♦ ✐♥s❝r✐t♦ ❡♠ ✉♠ ❝ír❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

✶✵✳✸ P❛r❛❧❡❧❡♣í♣❡❞♦ ❡ ❝✉❜♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

✶✵✳✹ Pr✐s♠❛ ♦❜❧íq✉♦✳ ❋♦♥t❡✿✇✇✇✳♠s♣s✳❡♥❣✳❜r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

✶✵✳✺ ❚r✐â♥❣✉❧♦ ❘❡tâ♥❣✉❧♦✳ ❋♦♥t❡✿ ❜r❛✐♥❧②✳❝♦♠✳❜r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽

✶✵✳✻ Pr✐s♠❛ ❍❡①❛❣♦♥❛❧✳ ❋♦♥t❡✿ ❜r❛✐♥❧②✳❝♦♠✳❜r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

✶✵✳✼ ❚r✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❏❉❍ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵

✶✵✳✽ ❚r✐â♥❣✉❧♦ ❘❡tâ♥❣✉❧♦ ❍❏❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵

✶✵✳✾ P✐râ♠✐❞❡ ❚r✐❛♥❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷

✶✵✳✶✵P✐râ♠✐❞❡ ◗✉❛❞r❛♥❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸

■♥tr♦❞✉çã♦

❆♥❛❧✐s❛♥❞♦ ♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ✭P❈◆✬s✮✱ ✈❡r✐✜❝❛♠♦s q✉❡ ♦s ❛❧✉♥♦s ❞♦❊♥s✐♥♦ ▼é❞✐♦ ❞❡✈❡♠ t❡r ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ♣❧❡♥♦ ❛ r❡s♣❡✐t♦ ❞❛ ●❡♦♠❡tr✐❛ P❛♥❛ ❡ ❊s♣❛❝✐❛❧❡ s❡r❡♠ ❝❛♣❛③❡s ❞❡ r❡❧❛❝✐♦♥❛r ❝♦♥❝❡✐t♦s ❞❡st❛s ❞✉❛s ♣❛rt❡s ❛❧é♠ ❞❡ q✉❡st✐♦♥❛r ❛ r❡❛❧✐❞❛❞❡✱✉t✐❧✐③❛♥❞♦ ♦ r❛❝✐♦❝í♥✐♦ ❧ó❣✐❝♦✱ ❛ ❝r✐❛t✐✈✐❞❛❞❡ ❡ t❛♠❜é♠ ❛ t❡❝♥♦❧♦❣✐❛✳ P♦❞❡♠♦s ♣❡r❝❡❜❡r✱ q✉❡❞❡ ✉♠❛ ♠❛♥❡✐r❛ ❣❡♥❡r❛❧✐③❛❞❛✱ ❛ ▼❛t❡♠át✐❝❛✱ ❛ss✐♠ ❝♦♠♦ ✉♠❛ ❞❡ s✉❛s r❛♠✐✜❝❛çõ❡s q✉❡ é❛ ●❡♦♠❡tr✐❛✱ ❛♣r❡s❡♥t❛ ✉♠❛ ❣r❛♥❞❡ r❡❥❡✐çã♦ ❥✉♥t♦ ❛♦s ❛❧✉♥♦s✳ P♦❞❡ ✲ s❡ ❡①♣❧✐❝❛r ❡ss❡ ❢❛t♦❞❡✈✐❞♦ ❛♦ ❡①❝❡ss✐✈♦ ✉s♦ ❞❡ ❢ór♠✉❧❛s ♣r❡❞❡t❡r♠✐♥❛❞❛s ❡ ❛♦ ❡♥s✐♥♦ ♠❡❝â♥✐❝♦ ❝♦♠ r❡s♦❧✉çã♦❞❡ ❛t✐✈✐❞❛❞❡s s❡♠ ✉♠❛ ❛♣❧✐❝❛çã♦ ♥♦ ❝♦t✐❞✐❛♥♦ ❡ s❡♠ ✉♠ ❛tr❛t✐✈♦ q✉❡ ♠✉✐t❛s ✈❡③❡s ❡stá ♥♦♣ró♣r✐♦ ❞❡s❡♥❤♦ ♦✉ ♥❛ r❡❧❛çã♦ ❝♦♠ ♦ ❞✐❛ ❛ ❞✐❛ ❞♦s ❞✐s❝❡♥t❡s✳ ❯♠ ❞♦s ♣r✐♥❝✐♣❛✐s ❞❡s❛✜♦s❞♦ ♣r♦❢❡ss♦r é ❜✉s❝❛r ✉♠❛ ❢♦r♠❛ ✐♥♦✈❛❞♦r❛ ♣❛r❛ ✉♠ ❡♥s✐♥♦ ♠❛✐s ❛tr❛t✐✈♦ ❡ q✉❡ ❡st✐♠✉❧❡ ♦s❛❧✉♥♦s ❛ ❜✉s❝❛r❡♠ r❡s✉❧t❛❞♦s✱ ✈✐s✉❛❧✐③❛♥❞♦ ♦s ♠❡s♠♦s ❡ t❛♠❜é♠ ❝♦♥str✉✐♥❞♦ s✐t✉❛çõ❡s ♥❛sq✉❛✐s ♦ ❛♣r❡♥❞✐③❛❞♦ s❡ t♦r♥❡ ♠❛✐s s✐❣♥✐✜❝❛t✐✈♦✳ ❆ss✐♠✱ ❛ ❡s❝♦❧❤❛ ❞♦s ♠❡❝❛♥✐s♠♦s ❡ r❡❝✉rs♦s❞✐❞át✐❝♦s ❜❡♠ ❝♦♠♦ ❛ ♠❛♥❡✐r❛ ❞❡ ♠❛♥✐♣✉❧❛r t❛✐s r❡❝✉rs♦s é ❡①tr❡♠❛♠❡♥t❡ ✐♠♣♦rt❛♥t❡✳

P❛r❛ ❡❢❡t✐✈❛♠❡♥t❡ r❡❛❧✐③❛r♠♦s ♦ ❡♥s✐♥♦ ❞❛ ●❡♦♠❡tr✐❛ ❡ ❣❡r❛r♠♦s ✉♠❛ ❛♣r❡♥❞✐③❛❣❡♠s✐❣♥✐✜❝❛t✐✈❛✱ ❞❡✈❡♠♦s ❡st❛r ❛t❡♥t♦s ❛♦s ❥á ❡①✐st❡♥t❡s ♥♦s ❛❥✉❞❛♠ ❛ ❛♣r❡s❡♥t❛r ❡ ❥✉st✐✜❝❛r❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s ♥ã♦ ♠✉✐t♦ ✉t✐❧✐③❛❞❛s ♥♦ ❞✐❛ ❛ ❞✐❛ ❞♦s ♣r♦❢❡ss♦r❡s ❡ ❛❧✉♥♦s✳

❉✉r❛♥t❡ ♦s ♣ró①✐♠♦s ❝❛♣ít✉❧♦s✱ ✐r❡♠♦s ❛♣r❡s❡♥t❛r ❝♦♥str✉çõ❡s ❞❡ ❞❡t❡r♠✐♥❛❞♦s só❧✐❞♦s✱♦s ❝❤❛♠❛❞♦s só❧✐❞♦s ❞❡ P❧❛tã♦ ❡ ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s q✉❡ s❡ ❢❛③❡♠ ✐♠♣♦rt❛♥t❡s✱ ❛❧é♠ ❞❡❛❧❣✉♥s ❡①❡r❝í❝✐♦s ♥♦s q✉❛✐s ❛♣❧✐❝❛♠♦s ♦ q✉❡ ❢♦✐ ❛♣r❡s❡♥t❛❞♦✱ t✉❞♦ ✐st♦ ♠♦str❛♥❞♦ t❛♠❜é♠ ♦❡♠❜❛s❛♠❡♥t♦ ❞♦s P❈◆s ♣❛r❛ ❥✉st✐✜❝❛r t❛❧ tr❛❜❛❧❤♦ ❡ s✉❛ ❞❡✈✐❞❛ ✐♠♣♦rtâ♥❝✐❛✱ s❡♠ ❡sq✉❡❝❡r♦ ❊◆❊▼✱ q✉❡ t❡♠ s✐❞♦ ❛❞♦t❛❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛ ♣❛r❛ ♦ ❛❝❡ss♦ ❞❡ ❛❧✉♥♦s ♥❛s ✐♥st✐t✉✐çõ❡s ❞❡❡♥s✐♥♦ s✉♣❡r✐♦r✳

❆♣r❡s❡♥t❛r❡♠♦s t❛♠❜é♠✱ ♦s ♠❛t❡r✐❛✐s ❝♦♥❝r❡t♦s✱ q✉❡ ♣♦❞❡rã♦ s❡r ✉s❛❞♦s ♣❛r❛ ❛ ✈✐s✉❛❧✐✲③❛çã♦ ❞❡ ❡❧❡♠❡♥t♦s ❡ ♦❜t❡♥çã♦ ❞❡ r❡s✉❧t❛❞♦s ❛ ❝❡r❝❛ ❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ só❧✐❞♦✳❖ ✉s♦ ❞❡ s♦❢t✇❛r❡s t❛♠❜é♠ t❡rá ✉♠ ❝❛♣ít✉❧♦ r❡s❡r✈❛❞♦✱ ♦♥❞❡ ❛♣r❡s❡♥t❛r❡♠♦s ❝♦♥str✉çõ❡s❞❡ ✈ár✐♦s só❧✐❞♦s ❣❡♦♠étr✐❝♦s ❡ s✉❛s ♣r✐♥❝✐♣❛✐s ❝❛r❛❝t❡ríst✐❝❛s ❛❧é♠ ❞❡ ✈✐s✉❛❧✐③❛çõ❡s ❞✐❢❡r❡♥✲❝✐❛❞❛s✳

■♥tr♦❞✉çã♦ ✷

❆♣r❡s❡♥t❛♠♦s t❛♠❜é♠ ♠❛♥❡✐r❛s ❛❧t❡r♥❛t✐✈❛s ❡ ♠❛♥❡✐r❛s ✉s✉❛✐s ♣❛r❛ ♦ ❝á❧❝✉❧♦✱ ❛ ✈✐s✉❛❧✐✲③❛çã♦ ❞♦ ✈♦❧✉♠❡ ❡ ár❡❛s ❞❡ só❧✐❞♦s✱ ✉s❛♥❞♦ ❝♦♠♦ r❡❝✉rs♦ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧❧✐❡r✐✱ ♦ ❚❡♦r❡♠❛❞❡ ❊✉❧❡r ❡ s♦❢t✇❛r❡s ❣rá✜❝♦s ❝♦♠♦ ♦ ●❡♦❣❡❜r❛✱ ❛❧é♠ ❞❡ ♠❛t❡r✐❛✐s ❝♦♥❝r❡t♦s✳

❈♦♥s✐❞❡r❛♥❞♦ t♦❞❛s ❡st❛s ✐♥❢♦r♠❛çõ❡s✱ ♣r♦♣♦r❡♠♦s ❝♦♠ ❡st❡ ♣r♦❥❡t♦ ✉♠❛ ❛♣r❡s❡♥t❛çã♦❛♦s ♣r♦❢❡ss♦r❡s ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛♦s ❛❧✉♥♦s ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ❞✐❢❡r❡♥❝✐❛❞❛ ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❛●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❛ ❛♣❧✐❝❛çã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ♦ ❝á❧❝✉❧♦❞❡ ✈♦❧✉♠❡s ❜❡♠ ❝♦♠♦ ♦ ❛✉①í❧✐♦ ❞❡ s♦❢t✇❛r❡s ❝♦♠♦ ♦ ●❡♦❣❡❜r❛ ♣❛r❛ ❛ ✈✐s✉❛❧✐③❛çã♦ ❡ ♦❝♦♠♣❧❡♠❡♥t♦ ❞❛s ✐♥❢♦r♠❛çõ❡s ❡ ♦ ❛♣r❡♥❞✐③❛❞♦✳

❈❛♣ít✉❧♦ ✶

❈♦♥t❛♥❞♦ ✉♠ ♣♦✉❝♦ ❞❡ ❍✐stór✐❛

❆ ❤✐stór✐❛ ♥♦s ❞✐③ q✉❡ ❞❡s❞❡ ♦s ♣r✐♠ór❞✐♦s ❞❛ ♦r❣❛♥✐③❛çã♦ ❤✉♠❛♥❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♥❛s♦r❣❛♥✐③❛çõ❡s ❛❣rí❝♦❧❛s✱ t❡♠♦s ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❝á❧❝✉❧♦✱ ❛♥á❧✐s❡ ❡ ✉s♦ ❞❡ ✈♦❧✉♠❡s ❞❡ só❧✐❞♦s❣❡♦♠étr✐❝♦s✳ ❊✉❝❧✐❞❡s ❥á ❝✐t❛✈❛ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❡ ✈♦❧✉♠❡s ❡♠ s❡✉s tr❛❜❛❧❤♦s✱ ♥♦ ❧✐✈r♦ ❳■■❞♦s ❊❧❡♠❡♥t♦s ❞❡ ❊✉❝❧✐❞❡s✱ ♦❜s❡r✈❛♠♦s ❛❧❣✉♥s t❡♦r❡♠❛s q✉❡ r❡❧❛❝✐♦♥❛♠ ❛ ✐❞❡✐❛ ❞❡ ✈♦❧✉♠❡s❡♠ ♠❡s♠♦ ❞❡✜♥✐✲❧♦✳ ❖ tr❛❜❛❧❤♦ ❝♦♠ ✈♦❧✉♠❡s ❡ ár❡❛s s❡ ❞❡✉ ❞❡✈✐❞♦ ❛ ✉♠❛ ♥❡❝❡ss✐❞❛❞❡✜❧♦só✜❝❛ ❡ t❡♦❧ó❣✐❝❛✱ ❝♦♠♦ ♥♦s ❛♣❛r❡❝❡ ❡♠ ✈ár✐❛s s✐t✉❛çõ❡s ♥❛s ❝✐✈✐❧✐③❛çõ❡s ❣r❡❣❛s ❡ ❡♠ ❝✐✲✈✐❧✐③❛çõ❡s ♠❛✐s ❛♥t✐❣❛s✱ ♦♥❞❡ ❛s ✜❣✉r❛s ❣❡♦♠étr✐❝❛s ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ s✉❛s ár❡❛s ❡ s❡✉s✈♦❧✉♠❡s ❡r❛♠ ✉s❛❞♦s ♣❛r❛ ❥✉st✐✜❝❛r s✐t✉❛çõ❡s ♥❛s q✉❛✐s ♥ã♦ s❡ ❞❡♣❡♥❞✐❛ ❛♣❡♥❛s ❞❛ ❝r❡♥ç❛t❡♦❧ó❣✐❝❛ ♠❛s s✐♠ ❞❡ s✐t✉❛çõ❡s ❡♠ q✉❡ ♥ú♠❡r♦s ❡ ❢♦r♠❛s r❡♣r❡s❡♥t❛✈❛♠ ❛çõ❡s ❡ ♠❡❞✐❞❛st♦♠❛❞❛s ♣♦r ❞✐✈✐♥❞❛❞❡s✳ ❖s ❝á❧❝✉❧♦s ❞❡ ár❡❛s ❡ ❞♦s ♣ró♣r✐♦s ✈♦❧✉♠❡s ❝✐t❛❞♦s ❛♥t❡r✐♦r♠❡♥t❡❡r❛♠ ❢❡✐t♦s✱ ✐♥✐❝✐❛❧♠❡♥t❡✱ ❞❡ ❢♦r♠❛ ❛♣r♦①✐♠❛❞❛✱ ♣♦ré♠ ♦s ♠ét♦❞♦s✱ ❢♦r❛♠ s❡♥❞♦ ❛♣❡r❢❡✐ç♦✲❛❞♦s ❛té q✉❡ ♥❛ ❛♥t✐❣✉✐❞❛❞❡✱ ❡♠ ❝✐✈✐❧✐③❛çõ❡s ♠❛✐s ❛✈❛♥ç❛❞❛s ❝♦♠♦ ♦s ❡❣í♣❝✐♦s ❡ ♦s ❣r❡❣♦s✱♦❜t❡✈❡✲s❡ ❢ór♠✉❧❛s ❡ ❛❜str❛çõ❡s ♠✉✐t♦ s♦✜st✐❝❛❞❛s✳ ❆❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❢ór♠✉❧❛s ❝♦♠♣❧❡①❛s❡ ❡str✉t✉r❛❞❛s ♣❛r❛ ❛❧❣✉♥s só❧✐❞♦s ❡s♣❡❝í✜❝♦s ❝♦♠♦✿ ♣r✐s♠❛s✱ ♣✐râ♠✐❞❡s ❡ ❝✐❧✐♥❞r♦s✳ P♦❞❡♠♦s❛✐♥❞❛ ♣❡♥s❛r ♣r✐♠✐t✐✈❛♠❡♥t❡ ❛❝❡r❝❛ ❞❡ ❞❡t❡r♠✐♥❛r ♦ ✈♦❧✉♠❡ ✉s❛♥❞♦ ✉♠❛ ✉♥✐❞❛❞❡ ❝♦♠♦ r❡✲❢❡rê♥❝✐❛✱ ❝♦♠♦ ✉♠ ❝✉❜♦ ✉♥✐tár✐♦✱ q✉❡ s❡r✈✐r✐❛ ❞❡ r❡❢❡rê♥❝✐❛ ♣❛r❛ s❛❜❡r♠♦s q✉❛♥t❛s ✈❡③❡s ✉♠só❧✐❞♦ q✉❛❧q✉❡r ❝♦♥té♠ ❡ss❡ t❛❧ ❝✉❜♦ ✉♥✐tár✐♦✳ ❈♦♠♦ ❞✐t♦✱ ❢♦r♠❛ ❡①tr❡♠❛♠❡♥t❡ ♣r✐♠✐t✐✈❛✳

❖ ✐t❛❧✐❛♥♦✱ ♥❛s❝✐❞♦ ❡♠ ▼✐❧ã♦✱ ❋r❛♥❝❡s❝♦ ❇♦❛✈❡♥t✉r❛ ❈❛✈❛❧✐❡r✐✱ ❢♦✐ ✉♠ ❞♦s ♠❛t❡♠át✐❝♦s♠❛✐s ✐♥✢✉❡♥t❡s ❞❡ s✉❛ é♣♦❝❛ ❡ ❢♦✐ ♦ ✐♥st✐❣❛❞♦r ❞❡ ✉♠ ♠ét♦❞♦ ♠❛✐s ❛♣✉r❛❞♦ ♣❛r❛ ♦ ❝á❧❝✉❧♦❞❡ ✈♦❧✉♠❡s ❡ ár❡❛s✱ ❑❡♣❧❡r ❢♦✐ ❞❡s❛✜❛❞♦ ❛ ❡♥❝♦♥tr❛r ár❡❛s ❡ ✈♦❧✉♠❡s ❞❡ ❛❧❣✉♠❛s ✜❣✉r❛s❡ só❧✐❞♦s ❣❡♦♠étr✐❝♦s✱ ✐♥❞✐❝❛♥❞♦ ❛ss✐♠ ✉♠ ❡stí♠✉❧♦ ♣❛r❛ q✉❡ ❈❛✈❛❧✐❡r✐ ❡st✉❞❛ss❡ ✉♠❛ ♠❛✲♥❡✐r❛ ♣❛r❛ ❡❢❡t✉❛r t❛✐s ❝á❧❝✉❧♦s ❝♦♠♦ ❝✐t❛❞♦ ❡♠ ❬✽❪✳ ❈❛✈❛❧✐❡r✐ ❢♦✐ ✉♠ ❞✐s❝í♣✉❧♦ ❞❡ ●❛❧✐❧❡✉●❛❧✐❧❡✐✱ s❡♥❞♦ ❡st❡ q✉❡♠ ♦ ✐♥❞✐❝♦✉ ♣❛r❛ ♦❝✉♣❛r ✉♠❛ ❝át❡❞r❛ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇♦❧♦♥❤❛❞❡ ✶✻✷✾ ❛ ✶✻✹✼✱ ❛♥♦ ❞❡ s✉❛ ♠♦rt❡✳ ❊❧❡ ♦❝✉♣❛✈❛ ♣❛r❛❧❡❧❛♠❡♥t❡ ❛♦ ❝❛r❣♦ ❞❡ ♣r♦❢❡ss♦r ✉♠❛❛t✐✈✐❞❛❞❡ r❡❧✐❣✐♦s❛ ♥♦ ♠♦♥❛stér✐♦ ❞❡ ❙ã♦ ❏❡rô♥✐♠♦✱ ♣♦✐s ❡r❛ ✉♠ ❥❡s✉❛❞♦ ❡ ♥ã♦ ✉♠ ❥❡s✉ít❛

✸

✹

❝♦♠♦ é ❛✜r♠❛❞♦✱ ❡①❡r❝❡♥❞♦ t❛♠❜é♠ ✉♠❛ ❛t✐✈✐❞❛❞❡ ❝♦♠♦ ❛strô♥♦♠♦✳ ❊st❛❜❡❧❡❝❡✉ ❡♥tã♦✉♠❛ ✐♠♣♦rt❛♥t❡ ❝♦♥❝❧✉sã♦ ❛ r❡s♣❡✐t♦ ❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦✱ ❛tr❛✈és ❞❡ ♣❛rt❡s ❞♦ ♠❡s♠♦✱q✉❡ sã♦ s❡❝çõ❡s ♦❜t✐❞❛s ❛tr❛✈és ❞❡ ❝♦rt❡s ❞♦ só❧✐❞♦ ♦ q✉❡ é ❝❤❛♠❛❞♦ ❤♦❥❡ ❞❡ Pr✐♥❝í♣✐♦ ❞❡❈❛✈❛❧✐❡r✐✳ ❖ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ t❡♠ ❝♦♠♦ ❜❛s❡ ♦ ♠ét♦❞♦ ❞♦s ✐♥❞✐✈✐sí✈❡✐s q✉❡ ❝♦♠❡ç♦✉ ❛s❡ ❞❡s❡♥✈♦❧✈❡r ❛ ♣❛rt✐r ❞❡ ✶✽✷✻ ❡ q✉❡ ❢♦✐ ❞✐✈✉❧❣❛❞♦ ❛tr❛✈és ❞♦ ❧✐✈r♦ ●❡♦♠❡tr✐❛ ■♥❞✐✈✐s✐❜✐❧✐❜✉s✭●❡♦♠❡tr✐❛ ❞♦s ■♥❞✐✈✐sí✈❡✐s✮✱ ♣✉❜❧✐❝❛❞♦ ✐♥✐❝✐❛❧♠❡♥t❡ ♥♦ ❛♥♦ ❞❡ ✶✻✸✺✳ ❆ ✐❞❡✐❛ ❛♣r❡s❡♥t❛❞❛ ♣♦r❈❛✈❛❧✐❡r✐ ♥ã♦ ❡r❛ ❞❡ t♦❞♦ ♥♦✈❛✱ ♣♦✐s ♦ ♣ró♣r✐♦ ●❛❧✐❧❡✉ ❥á ❤❛✈✐❛ t✐❞♦ ❛ ♥❡❝❡ss✐❞❛❞❡ ❡ ♦ ❡stí♠✉❧♦♣❛r❛ ♦ ❡♥❝♦♥tr♦ ❞♦s ✈♦❧✉♠❡s ❛❧é♠ ❞♦s ❣r❡❣♦s ❛♥t✐❣♦s q✉❡ t❛♠❜é♠ ✐♥st✐❣❛r❛♠ ❛ ❡♥❝♦♥tr❛r✈♦❧✉♠❡ ❞❡ só❧✐❞♦s ✉s❛♥❞♦ t❛✐s ✐❞❡✐❛s ♣♦ré♠ ❛ ❞❡♠♦♥str❛çã♦ ❞❛s ♠❡s♠❛s ❡r❛♠ ♦❜t✐❞❛s ❞❡♠❛♥❡✐r❛ ❞✐❢❡r❡♥t❡ ♣❡❧♦ ❢❛t♦ ❞♦s ❝♦♥❤❡❝✐♠❡♥t♦s s❡r❡♠ ❜❡♠ ❞✐❢❡r❡♥t❡s✳ ❖ ❜ás✐❝♦ ❞♦ Pr✐♥❝í♣✐♦❞❡ ❈❛✈❛❧✐❡r✐ ❝♦♥s✐st✐❛ ♥♦ s❡❣✉✐♥t❡✿ ❯♠❛ ✜❣✉r❛ ♣❧❛♥❛ s❡r✐❛ ❢♦r♠❛❞❛ ♣♦r ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡❝♦r❞❛s ♣❛r❛❧❡❧❛s ❡♥tr❡ s✐ ❡ ✉♠❛ ✜❣✉r❛ só❧✐❞❛ ✭✉♠ só❧✐❞♦ ❣❡♦♠étr✐❝♦✮ s❡r✐❛ ❢♦r♠❛❞♦ ♣♦r ✉♠❛✐♥✜♥✐❞❛❞❡ ❞❡ s❡❝çõ❡s ♣❛r❛❧❡❧❛s ❡♥tr❡ s✐✱ ♣♦r ✐ss♦ ❡ss❛s ❝♦r❞❛s ❡ ❡ss❛s s❡❝çõ❡s sã♦ ❝❤❛♠❛❞❛s❞❡ ✐♥❞✐✈✐sí✈❡✐s✳ ❊ss❛s ✐❞❡✐❛s ❛♣r❡s❡♥t❛❞❛s ♣♦r ❈❛✈❛❧✐❡r✐ s♦❜r❡ ♦s ✐♥❞✐✈✐sí✈❡✐s✱ ❡♥✈♦❧✈✐❛♠ ✉♠❛❞✐✜❝✉❧❞❛❞❡ ❞❡ ❡♥t❡♥❞✐♠❡♥t♦✱ ♣♦✐s ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❧ó❣✐❝♦ ❝♦♠♦ ♣♦❞❡rí❛♠♦s ✐♠❛❣✐♥❛r ✉♠❛✜❣✉r❛ ✜♥✐t❛✱ ❧✐♠✐t❛❞❛✱ ❢♦r♠❛❞❛ ♣♦r ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡ ❡❧❡♠❡♥t♦s ✐♥❞✐✈✐sí✈❡✐s❄ ❊♠ q✉❛✐s s✐✲t✉❛çõ❡s ♣♦❞❡♠♦s ✉s❛r ❡st❡ ❝♦♥❝❡✐t♦❄ ❖ ♠ét♦❞♦ ❞♦s ✐♥❞✐✈✐sí✈❡✐s ❡ ♦✉tr♦s ❡q✉✐✈❛❧❡♥t❡s ❛ ❡❧❡✱❢♦r❛♠ ❡❢❡t✐✈❛♠❡♥t❡ ✉s❛❞♦s ♣♦r ❚♦rr✐❝❡❧❧✐✱ ❋❡r♠❛t✱ P❛s❝❛❧✱ ❙❛✐♥t✲❱✐♥❝❡♥t✱ ❇❛rr♦✇ ❡ ♦✉tr♦sq✉❡ ♥♦ ❝✉rs♦ ❞❡ s❡✉ tr❛❜❛❧❤♦ ❝❤❡❣❛r❛♠ ❛ r❡s✉❧t❛❞♦s ❡q✉✐✈❛❧❡♥t❡s à ✐♥t❡❣r❛çã♦ ❞❡ ❡①♣r❡ssõ❡s❝♦♠♦ xn✱ senθ✱ sen2θ ❡ θ.senθ✳

❖ tr❛t❛❞♦ ✐♥✐❝✐❛❧ ❞❡ ❈❛✈❛❧✐❡r✐ é ❞❡♠❛s✐❛❞❛♠❡♥t❡ ❝♦♥❢✉s♦✱ s❡♥❞♦ ❞❡ ❞✐❢í❝✐❧ ❡♥t❡♥❞✐♠❡♥t♦✱♣♦ré♠ s❡ ❡st✉❞❛❞♦ ❝♦♥s❡❣✉❡✲s❡ ❛♣r❡s❡♥t❛r ✉♠❛ ❝♦❧♦❝❛çã♦ ❞❡♥tr♦ ❞❛ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛ ❡ ❞❡♥tr♦❞❛ ❣❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧✳ ❖ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣r♦♣r✐❛♠❡♥t❡ ❞✐t♦ é ♠✉✐t♦ ✉s❛❞♦ ❞❡♥tr♦ ❞❛●❡♦♠❡tr✐❛ ❊s♣❛❝✐❛❧ ❢❛❝✐❧✐t❛♥❞♦ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❡ ❛ ❛❝❡✐t❛çã♦ ❞♦ ❝á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡ ❞❡ ✉♠só❧✐❞♦ ❣❡♦♠étr✐❝♦✳

❈♦♥s✐❞❡r❡ ❞♦✐s só❧✐❞♦s X ❡ Y ❝♦♠ ♠❡s♠❛ ❛❧t✉r❛✳ ❖s ✐♥✜♥✐t♦s ♣❧❛♥♦s ♣❛r❛❧❡❧♦s ❡♥tr❡ s✐❛♦ ✐♥t❡r❝❡♣t❛r❡♠ ♦s só❧✐❞♦s X ❡ Y ✈ã♦ ❞❡t❡r♠✐♥❛r ✐♥✜♥✐t❛s s❡❝çõ❡s q✉❡ ❞❛rã♦ ❛ ✐❞❡✐❛ ❞♦s✐♥❞✐✈✐sí✈❡✐s q✉❡ tê♠ ár❡❛s ✐❣✉❛✐s ❡ ♣♦rt❛♥t♦ ♦s só❧✐❞♦s tê♠ ✈♦❧✉♠❡s ✐❣✉❛✐s✳ ❆❧❣✉♥s ♦✉tr♦st❡♦r❡♠❛s ❡st❛❜❡❧❡❝✐❞♦s ♣♦r ❈❛✈❛❧✐❡r✐ q✉❡ r❡❧❛❝✐♦♥❛♠ ✐♥❞✐✈✐sí✈❡✐s ♣♦❞❡♠ s❡r ✈✐st♦s ❞❡♥tr♦ ❞❛❣❡♦♠❡tr✐❛ ♣❧❛♥❛✱ t❛✐s ❝♦♠♦ ♦s q✉❡ tr❛❜❛❧❤❛♠ ❡ss❡s ✐♥❞✐✈✐sí✈❡✐s ❡♠ ♣❛r❛❧❡❧♦❣r❛♠♦s ❡ tr✐â♥❣✉❧♦s❢♦r♠❛❞♦s ♣❡❧❛s ❞✐❛❣♦♥❛✐s ❞♦s ♠❡s♠♦s✳

❖ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ r❡♣r❡s❡♥t❛ ✉♠❛ ❢❡rr❛♠❡♥t❛ s✐❣♥✐✜❝❛t✐✈❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s❡ ❞❡ ✈♦❧✉♠❡s✱ ♣♦❞❡♥❞♦ r❡s♦❧✈❡r ♠✉✐t♦s ♣r♦❜❧❡♠❛s ❞❡ ♠❡♥s✉r❛çã♦ q✉❡ às ✈❡③❡s ♣♦❞❡r✐❛♠r❡q✉❡r❡r té❝♥✐❝❛s ❛✈❛♥ç❛❞❛s ❞❡ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ❡ ✐♥t❡❣r❛❧ q✉❡ ♥ã♦ sã♦ ❛♣r❡s❡♥t❛❞❛s ❡❞❡❞✉③✐❞❛s ❡♠ ♥í✈❡❧ ♠é❞✐♦ ❡ ❢✉♥❞❛♠❡♥t❛❧✳ ❊❧❡ ❡♥tã♦ ❢❛❝✐❧✐t❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❡ ❛ ✐❞❡✐❛ ❜ás✐❝❛♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞♦s ✈♦❧✉♠❡s ❞❡ só❧✐❞♦s ❣❡♦♠étr✐❝♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❛r❛ ❛q✉❡❧❡s q✉❡ ❡stã♦s❡♥❞♦ ❛♣r❡s❡♥t❛❞♦s à ❣❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧ ❡ ♣❛r❛ ♦s q✉❡ tê♠ r❡s✐stê♥❝✐❛ ❡ ❞✐✜❝✉❧❞❛❞❡ ❝♦♠ ❛♠❡s♠❛✳

✺

P♦❞❡♠♦s ❞❡❞✉③✐r ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛✱ ✉s❛♥❞♦ ❛❧❣✉♠❛s ✐❞❡✐❛s ❜ás✐❝❛s ❛ r❡s♣❡✐t♦❞♦s ✐♥❞✐✈✐sí✈❡✐s ❡st❛❜❡❧❡❝✐❞♦s ✐♥✐❝✐❛❧♠❡♥t❡ ♣♦r ❈❛✈❛❧✐❡r✐✱ ❡ tr❛❜❛❧❤❛♥❞♦ ❝♦♠ ❞♦✐s só❧✐❞♦s ❞❡♥❛t✉r❡③❛ ❞✐❢❡r❡♥t❡✱ ♠❛s s❡❝❝✐♦♥❛♥❞♦ ❛♠❜♦s ♣♦r ✉♠ ♠❡s♠♦ ♣❧❛♥♦ ❡ ♦❜t❡♥❞♦ s❡❝çõ❡s ❜❡♠❞✐❢❡r❡♥t❡s✱ ♣♦ré♠ ❝♦♠ ❛ ♠❡s♠❛ ár❡❛ ❡ ❛ss✐♠ ❡♥tã♦ ♦s só❧✐❞♦s ❛♣r❡s❡♥t❛❞♦s ✐♥✐❝✐❛❧♠❡♥t❡ tê♠♦ ♠❡s♠♦ ✈♦❧✉♠❡✳ ❉❡ ✉♠❛ ♠❛♥❡✐r❛ ❣❡r❛❧✱ ❛❝❡✐t❛✲s❡ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❝♦♠♦ ✉♠ ❛①✐♦♠❛❡ ❞❡ ❢♦r♠❛ ✐♥t✉✐t✐✈❛✱ ♥ã♦ ❤❛✈❡♥❞♦ ❛ss✐♠ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡ ❞❡♠♦♥str❛r✳

❋✐❣✉r❛ ✶✳✶✿ ❇♦❛✈❡♥t✉r❛ ❈❛✈❛❧✐❡r✐ ✲ ❋♦♥t❡✿ ♠❛t❤❡✉s♠❛t❤✐❝❛✳❜❧♦❣s♣t✳❝♦♠

❆❧é♠ ❞❛s ✐♥❢♦r♠❛çõ❡s ❛ r❡s♣❡✐t♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✱ ♣♦❞❡♠♦s t❛♠❜é♠ ✉s❛r ♦t❡♦r❡♠❛ ❞❡ ❊✉❧❡r ♣❛r❛ ❡st❛❜❡❧❡❝❡r ❛s r❡❧❛çõ❡s ❡♥tr❡ ❛s ❢❛❝❡s✱ ♦s ❧❛❞♦s ❡ ♦s ✈ért✐❝❡s ❞❡ ✉♠♣♦❧✐❡❞r♦ ♣❛r❛ ♣♦❞❡r♠♦s ❡str✉t✉r❛r ♦ ❝á❧❝✉❧♦ ❞♦ ✈♦❧✉♠❡ ❞♦ ♠❡s♠♦✱ q✉❛♥❞♦ s❡ tr❛t❛ ❞❡ ✉♠♣r✐s♠❛ ✭r❡t♦ ♦✉ ♦❜❧✐q✉♦✮✱ ✈❡r✐✜❝❛♠♦s q✉❡ ❡st❡s sã♦ ♣♦❧✐❡❞r♦s ❝♦♥✈❡①♦s✱ ♣♦rt❛♥t♦ ♣♦❧✐❡❞r♦s❡✉❧❡r✐❛♥♦s✳ ❱❛❧❡ ❧❡♠❜r❛r q✉❡ ♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r r❡❧❛❝✐♦♥❛ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❢❛❝❡s✱ ✈ért✐❝❡s ❡❛r❡st❛s ❞❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦✱ ❡ ❡st❛s ✐♥❢♦r♠❛çõ❡s s❡rã♦ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛♦ ❡♥❝♦♥tr♦ ❞❛s q✉❛♥t✐❞❛❞❡s ❞❡ ❧❛❞♦s ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❞❛s ár❡❛s ❞❡ ♣♦❧í❣♦♥♦s q✉❡ ❢♦r♠❛♠❛s s❡❝çõ❡s ❞❡ ✉♠ só❧✐❞♦ ❣❡♦♠étr✐❝♦✳ ❖❜s❡r✈❡ ♦ ❡①❡♠♣❧♦ ❞❛ ❋✐❣✉r❛ ✶✳✷ q✉❡ t❡♠♦s ❛ s❡❣✉✐r ❡♠q✉❡ ❛♥❛❧✐s❛r❡♠♦s ✉♠ q✉❛❞r✐❧át❡r♦ ❝♦♥✈❡①♦ q✉❡ ♣♦❞❡rá s❡r ❛ s❡❝çã♦ ❞❡ ✉♠ só❧✐❞♦ ❝♦♥✈❡①♦ ❞♦q✉❛❧ ❡st❛r❡♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ❝❛❧❝✉❧❛r ♦ ✈♦❧✉♠❡✳

❋✐❣✉r❛ ✶✳✷✿ ❆♣❧✐❝❛çã♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r

❯s❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❧❛❞♦s✱ ❞❡ ❛r❡st❛s ❡ ❢❛❝❡sq✉❡ ♦ ♣♦❧✐❡❞r♦ ✭só❧✐❞♦✮✱ r❡s✉❧t❛♥t❡ ❞❛ s♦❜r❡♣♦s✐çã♦ ❞❡ ✐♥✜♥✐t❛s s❡❝çõ❡s ❝♦♠♦ ❡st❛✳

✻

❋✐❣✉r❛ ✶✳✸✿ ▲❡♦♥❤❛r❞ P❛✉❧ ❊✉❧❡r ✲ ❋♦♥t❡✿ ✇✇✇✳❡①♣❧✐❝❛t♦r✐✉♠✳❝♦♠

❉❡✈❡♠♦s ❧❡♠❜r❛r ❞❡ q✉❡✱ ❡♠ s❡ tr❛t❛♥❞♦ ❞❡ ✉♠ ♣♦❧✐❡❞r♦✱ ♣♦❞❡♠♦s ✉s❛r ♦ ❚❡♦r❡♠❛ ❞❡❊✉❧❡r ♣❛r❛ ❡♥❝♦♥tr❛r♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ ♥ú♠❡r♦ ❞❡ ❢❛❝❡s ❞♦ ♠❡s♠♦ ❡ ❛ss✐♠ s❡♥❞♦ ♦ s❡✉✈♦❧✉♠❡ ♣♦❞❡rá s❡r ♦❜t✐❞♦ ❛tr❛✈és ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳

➱ ✐♠♣♦rt❛♥t❡ ❛♣r❡s❡♥t❛r ♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r✱ q✉❡ ❢♦✐ ❞❡s❝♦❜❡rt♦ ❡♠ ✶✼✺✽✱ ❡ q✉❡ ♣♦❞❡ s❡r❡♥✉♥❝✐❛❞♦ ❛ss✐♠✿ ❙❡ ✉♠ ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ t❡♠ V ✈ért✐❝❡s✱ A ❛r❡st❛ ❡ F ❢❛❝❡s✱ ✈❛❧❡ ❛ s❡❣✉✐♥t❡r❡❧❛çã♦✿

V + F = A+ 2

❊♠ q✉❡ A é ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s ❞♦ ♣♦❧✐❡❞r♦✱ F ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s ❡ V ♦ ♥ú♠❡r♦ ❞❡✈ért✐❝❡s✳

❆♣❡♥❛s ✉♠❛ ✐♥❢♦r♠❛çã♦ ❤✐stór✐❝❛✱ ♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r t❡♠ ✈ár✐❛s ❞❡♠♦♥str❛çõ❡s ❝♦♠♦❛ q✉❡ ✐r❡♠♦s ♠♦str❛r ❛❣♦r❛✱ ❡ q✉❡ é ❛♣r❡s❡♥t❛❞❛ ❡♠ ❬✻❪

❉❡✜♥✐çã♦ ✶✳✶ ❙✉♣❡r❢í❝✐❡ ♣♦❧✐é❞r✐❝❛ ❧✐♠✐t❛❞❛ ❝♦♥✈❡①❛ ❙✉♣❡r❢í❝✐❡ ♣♦❧✐é❞r✐❝❛ ❧✐♠✐t❛❞❛

❝♦♥✈❡①❛ é ❛ r❡✉♥✐ã♦ ❞❡ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♣♦❧í❣♦♥♦s ♣❧❛♥♦s ❡ ❝♦♥✈❡①♦s ✭♦✉ r❡❣✐õ❡s ♣♦✲

❧✐❣♦♥❛✐s ❝♦♥✈❡①❛s ✮✱ t❛✐s q✉❡✿

❛✮ ❉♦✐s ♣♦❧í❣♦♥♦s ♥ã♦ ❡stã♦ ♥♦ ♠❡s♠♦ ♣❧❛♥♦❀

❜✮ ❈❛❞❛ ❧❛❞♦ ❞❡ ♣♦❧í❣♦♥♦ ♥ã♦ ❡stá ❡♠ ♠❛✐s q✉❡ ❞♦✐s ♣♦❧í❣♦♥♦s❀

❝✮ ❍❛✈❡♥❞♦ ❧❛❞♦s ❞❡ ♣♦❧í❣♦♥♦s q✉❡ ❡stã♦ ❡♠ ✉♠ só ♣♦❧í❣♦♥♦✱ ❡❧❡s ❞❡✈❡♠ ❢♦r♠❛r ✉♠❛

✜❣✉r❛ ♣♦❧✐❣♦♥❛❧ ❢❡❝❤❛❞❛✱ ♣❧❛♥❛ ♦✉ ♥ã♦✱ ❝❤❛♠❛❞❛ ❝♦♥t♦r♥♦❀

❞✮ ❖ ♣❧❛♥♦ ❞❡ ❝❛❞❛ ♣♦❧í❣♦♥♦ ❞❡✐①❛ ♦s ❞❡♠❛✐s ♥✉♠ ♠❡s♠♦ s❡♠✐❡s♣❛ç♦ ✭❝♦♥❞✐çã♦ ❞❡

❝♦♥✈❡①✐❞❛❞❡✮✳

❆s s✉♣❡r❢í❝✐❡s ♣♦❧✐é❞r✐❝❛s ❧✐♠✐t❛❞❛s ❝♦♥✈❡①❛s q✉❡ tê♠ ❝♦♥t♦r♥♦ sã♦ ❝❤❛♠❛❞❛s ❛❜❡rt❛s✳❆s q✉❡ ♥ã♦ tê♠ ❝♦♥t♦r♥♦ sã♦ ❝❤❛♠❛❞❛s ❞❡ ❢❡❝❤❛❞❛s✳

✼

P❛rt❡ ■

P♦r ✐♥❞✉çã♦ ✜♥✐t❛ r❡❢❡r❡♥t❡ ❛♦ ♥ú♠❡r♦ ❞❡ ❢❛❝❡s✱ ✈❛♠♦s ♣r♦✈❛r✱ ♣r❡❧✐♠✐♥❛r♠❡♥t❡✱ q✉❡ ♣❛r❛✉♠❛ s✉♣❡r❢í❝✐❡ ♣♦❧✐é❞r✐❝❛ ❧✐♠✐t❛❞❛ ❝♦♥✈❡①❛ ❛❜❡rt❛✱ ✈❛❧❡ ❛ r❡❧❛çã♦✿

Va + Fa − Aa = 1

❡♠ q✉❡✿Va é ♦ ♥ú♠❡r♦ ❞❡ ✈ért✐❝❡s✱Fa é ♦ ♥ú♠❡r♦ ❞❡ ❢❛❝❡s ❡Aa é ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s ❞❛ s✉♣❡r❢í❝✐❡ ♣♦❧✐é❞r✐❝❛ ❧✐♠✐t❛❞❛ ❡ ❛❜❡rt❛✳

✐✮ P❛r❛ Fa = 1✳ ◆❡st❡ ❝❛s♦ ❛ s✉♣❡r❢í❝✐❡ s❡ r❡❞✉③ ❛ ✉♠ ♣♦❧í❣♦♥♦ ♣❧❛♥♦ ❝♦♥✈❡①♦ ❞❡ n ❧❛❞♦s❡✱ ❡♥tã♦✱ Va ❂ n✱ Aa ❂ n✳ ❚❡♠♦s ❛ss✐♠✿

❱a − Aa + Fa = n− n+ 1 = 1✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ Va − Aa + Fa = 1

▲♦❣♦✱ ❛ r❡❧❛çã♦ ❡stá ✈❡r✐✜❝❛❞❛ ♣❛r❛✿Fa = 1

✳

✐✐✮ ❆❞♠✐t✐♥❞♦ q✉❡ ❛ r❡❧❛çã♦ ✈❛❧❡ ♣❛r❛ ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ F ′ ❢❛❝❡s✭ q✉❡ ♣♦ss✉✐ V ′ ✈ért✐❝❡s ❡ A′

❛r❡st❛s ✮✱ ✈❛♠♦s ♣r♦✈❛r t❛♠❜é♠ q✉❡ ✈❛❧❡ ♣❛r❛ ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ F ′ + 1 ❢❛❝❡s ✭q✉❡ ♣♦ss✉✐F ′ + 1 = F1 ❢❛❝❡s✱ Va ✈ért✐❝❡s ❡ Aa ❛r❡st❛s✮✳

P♦r ❤✐♣ót❡s❡✱ ♣❛r❛ ❛ s✉♣❡r❢í❝✐❡ ❞❡ F ′ ❢❛❝❡s✱ A′ ❛r❡st❛s ❡ V ′ ✈❛❧❡ ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦✿

V ′ − A′ + F ′ = 1

❆❝r❡s❝❡♥t❛♥❞♦ ❛ ❡ss❛ s✉♣❡r❢í❝✐❡ ✭q✉❡ é ❛❜❡rt❛✮ ✉♠❛ ❢❛❝❡ ❞❡ p ❛r❡st❛s ✭❧❛❞♦s✮ ❡ ❝♦♥s✐❞❡r❛♥❞♦q✉❡ q ❞❡ss❛s ❛r❡st❛s ✭❧❛❞♦s✮ ❝♦✐♥❝✐❞❡♠ ❝♦♠ ❛s ❛r❡st❛s ❥á ❡①✐st❡♥t❡s✱ ♦❜t❡♠♦s ✉♠❛ ♥♦✈❛s✉♣❡r❢í❝✐❡ ❝♦♠ Fa ❢❛❝❡s✱ Aa ❛r❡st❛s ❡ Va ✈ért✐❝❡s t❛✐s q✉❡✿

Fa = F ′ + 1

Aa ❂ A′ + p ✲ q✱ ✭q ❛r❡st❛s ❝♦✐♥❝✐❞✐r❛♠✮

Va ❂ V ′ + p ✲ (q + 1)✱ ✭s❡ q ❛r❡st❛s ❝♦✐♥❝✐❞❡♠✱ q + 1 ✈ért✐❝❡s ❝♦✐♥❝✐❞❡♠✮

✽

❋♦r♠❛♥❞♦ ❛ss✐♠ ❛ ❡①♣r❡ssã♦ Va − Aa + Fa ❡ s✉❜st✐t✉✐♥❞♦ ♦s ✈❛❧♦r❡s ❡♥❝♦♥tr❛❞♦s ❛♥t❡r✐♦r✲♠❡♥t❡✱ t❡r❡♠♦s✿

Va − Aa + Fa = V ′ + p− (q + 1)− (A′ + p− q) + (F ′ + 1) ❂

V ′ + p− q − 1− A′ − p+ q + F ′ + 1 = V ′ − A′ + F ′

❊ ❛ss✐♠ ❝♦♠♦ Va − Aa + Fa = V ′ − A′ + F ′ ♣r♦✈❛♠♦s q✉❡ ❡ss❛ ❡①♣r❡ssã♦ ♥ã♦ s❡ ❛❧t❡r❛ s❡❛❝r❡s❝❡♥t❛♠♦s ✭♦✉ r❡t✐r❛♠♦s✮ ✉♠❛ ❢❛❝❡ ❞❛ s✉♣❡r❢í❝✐❡✳ ❈♦♠♦ ♣♦r ❤✐♣ót❡s❡✱ V ′ − A′ + F ′ = 1✱✈❡♠ q✉❡✿

Va − Aa + Fa = 1

❖ q✉❡ ♣r♦✈❛ ❛ r❡❧❛çã♦ ♣r❡❧✐♠✐♥❛r✳

P❛rt❡ ■■

❚♦♠❡♠♦s ❛ s✉♣❡r❢í❝✐❡ ❞❡ q✉❛❧q✉❡r ♣♦❧✐❡❞r♦ ❝♦♥✈❡①♦ ♦✉ q✉❛❧q✉❡r s✉♣❡r❢í❝✐❡ ♣♦❧✐é❞r✐❝❛❝♦♥✈❡①❛ ❧✐♠✐t❛❞❛ ❡ ❢❡❝❤❛❞❛ ✭❝♦♠ V ✈ért✐❝❡s✱ A ❛r❡st❛s ❡ F ❢❛❝❡s ✮ ❡ ❞❡❧❛ r❡t✐r❛♠♦s ✉♠❛ ❢❛❝❡✳

❋✐❝❛♠♦s ❡♥tã♦ ❝♦♠ ✉♠❛ s✉♣❡r❢í❝✐❡ ❛❜❡rt❛ ✭❝♦♠ Va ✈ért✐❝❡s✱ Aa ❛r❡st❛s ❡ Fa ❢❛❝❡s✮ ♣❡❧❛q✉❛❧ ✈❛❧❡ ❛ r❡❧❛çã♦✿

Va − Aa + Fa = 1 ❡ ❝♦♠♦ Va = V ✱ Aa = A ❡ Fa = F − 1✱ ✈❡♠ V − A+ F − 1 = 1✱ ♦✉s❡❥❛✿

V − F = A+ 2✳

❊ ❛ss✐♠ ✜♥❛❧♠❡♥t❡ ♣r♦✈❛♠♦s ♦ ❚❡♦r❡♠❛ ❞❡ ❊✉❧❡r✳

❈❛♣ít✉❧♦ ✷

❖ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✱ ♦s P❛râ♠❡tr♦s

❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ❡ ♦ ❊◆❊▼

✷✳✶ P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s

❖s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ❝♦♥st✐t✉❡♠ ✉♠ r❡❢❡r❡♥❝✐❛❧ ♣❛r❛ ❛ ❡❞✉❝❛çã♦ ♥♦ ❊♥✲s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ ❡ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ❡♠ t♦❞♦ ♦ ♣❛ís✳ ❙✉❛ ❢✉♥çã♦ é ♦r✐❡♥t❛r ❡ ❣❛r❛♥t✐r ❛❝♦❡rê♥❝✐❛ ❞♦s ✐♥✈❡st✐♠❡♥t♦s ♥♦ s✐st❡♠❛ ❡❞✉❝❛❝✐♦♥❛❧✱ s♦❝✐❛❧✐③❛♥❞♦ ❞✐s❝✉ssõ❡s✱ ♣❡sq✉✐s❛s ❡ r❡✲❝♦♠❡♥❞❛çõ❡s✳ ❖ ❝♦♥❥✉♥t♦ ❞❛s ♣r♦♣♦s✐çõ❡s ❡①♣r❡ss❛s ♥♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐sr❡s♣♦♥❞❡ à ♥❡❝❡ss✐❞❛❞❡ ❞❡ r❡❢❡r❡♥❝✐❛✐s ❛ ♣❛rt✐r ❞♦s q✉❛✐s ♦ s✐st❡♠❛ ❡❞✉❝❛❝✐♦♥❛❧ ❞♦ P❛ís s❡♦r❣❛♥✐③❡✱ ❛ ✜♠ ❞❡ ❣❛r❛♥t✐r q✉❡ s❡❥❛♠ r❡s♣❡✐t❛❞❛s ❛s ❞✐✈❡rs✐❞❛❞❡s✳

❆ ▲❡✐ ❞❡ ❉✐r❡tr✐③❡s ❡ ❇❛s❡s ❞❛ ❊❞✉❝❛çã♦ ◆❛❝✐♦♥❛❧ ❡①♣❧✐❝✐t❛ q✉❡ ♦ ❊♥s✐♥♦ ▼é❞✐♦ é ❛✏❡t❛♣❛ ✜♥❛❧ ❞❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛✑ ✭❆rt✳✸✻✮✱ ♦ q✉❡ ❝♦♥❝♦rr❡ ♣❛r❛ ❛ ❝♦♥str✉çã♦ ❞❡ s✉❛ ✐❞❡♥t✐✲❞❛❞❡✳ ❖ ❊♥s✐♥♦ ▼é❞✐♦ ♣❛ss❛ ❛ t❡r ❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❛ t❡r♠✐♥❛❧✐❞❛❞❡✱ ♦ q✉❡ s✐❣♥✐✜❝❛ ❛ss❡❣✉r❛r❛ t♦❞♦s ♦s ❝✐❞❛❞ã♦s ❛ ♦♣♦rt✉♥✐❞❛❞❡ ❞❡ ❝♦♥s♦❧✐❞❛r ❡ ❛♣r♦❢✉♥❞❛r ♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❛❞q✉✐r✐❞♦s♥♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧❀ ❛♣r✐♠♦r❛r ♦ ❡❞✉❝❛♥❞♦ ❝♦♠♦ ♣❡ss♦❛ ❤✉♠❛♥❛❀ ♣♦ss✐❜✐❧✐t❛r ♦ ♣r♦ss❡✲❣✉✐♠❡♥t♦ ❞❡ ❡st✉❞♦s❀ ❣❛r❛♥t✐r ❛ ♣r❡♣❛r❛çã♦ ❜ás✐❝❛ ♣❛r❛ ♦ tr❛❜❛❧❤♦ ❡ ❛ ❝✐❞❛❞❛♥✐❛❀ ❞♦t❛r ♦❡❞✉❝❛♥❞♦ ❞♦s ✐♥str✉♠❡♥t♦s q✉❡ ♦ ♣❡r♠✐t❛♠ ✏❝♦♥t✐♥✉❛r ❛♣r❡♥❞❡♥❞♦✑✳ ❖ ❊♥s✐♥♦ ▼é❞✐♦✱ ♣♦r✲t❛♥t♦✱ é ❛ ❡t❛♣❛ ✜♥❛❧ ❞❡ ✉♠❛ ❡❞✉❝❛çã♦ ❞❡ ❝❛rát❡r ❣❡r❛❧✱ ❛✜♥❛❞❛ ❝♦♠ ❛ ❝♦♥t❡♠♣♦r❛♥❡✐❞❛❞❡✱❝♦♠ ❛ ❝♦♥str✉çã♦ ❞❡ ❝♦♠♣❡tê♥❝✐❛s ❜ás✐❝❛s✱ q✉❡ s✐t✉❡♠ ♦ ❛❧✉♥♦ ❝♦♠♦ s✉❥❡✐t♦ ♣r♦❞✉t♦r ❞❡❝♦♥❤❡❝✐♠❡♥t♦ ❡ ♣❛rt✐❝✐♣❛♥t❡ ❞♦ ♠✉♥❞♦ ❡♠ q✉❡ ❡❧❡ ❡stá ✐♥s❡r✐❞♦✱ ♣❛rt✐❝✐♣❛♥❞♦ ❞♦ tr❛❜❛❧❤♦♥♦ q✉❛❧ ❡❧❡ ❞❡s❡❥❡✱ ❛t✉❛♥❞♦ ❝♦♠♦ ✏s✉❥❡✐t♦ ❡♠ s✐t✉❛çã♦✑ ✲ ❝✐❞❛❞ã♦✳

❙❡❣✉♥❞♦ ♦s P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s✱ ❛ ❛♣r❡♥❞✐③❛❣❡♠ ♥❛ ár❡❛ ❞❡ ❈✐ê♥❝✐❛s ❞❛◆❛t✉r❡③❛✱ ▼❛t❡♠át✐❝❛ ❡ s✉❛s ❚❡❝♥♦❧♦❣✐❛s ✐♥❞✐❝❛ q✉❡ ♦ ❛❧✉♥♦ ❞❡✈❡ t❡r ❛ ❝♦♠♣r❡❡♥sã♦ ❡

✾

✷✳✶✳ P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ✶✵

❛ ✉t✐❧✐③❛çã♦ ❞♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❝✐❡♥tí✜❝♦s✱ ♣❛r❛ ❡①♣❧✐❝❛r ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞♦ ♠✉♥❞♦✱ ❜❡♠❝♦♠♦ ♣❧❛♥❡❥❛r✱ ❡①❡❝✉t❛r ❡ ❛✈❛❧✐❛r ❛s ❛çõ❡s ❞❡ ✐♥t❡r✈❡♥çã♦ ♥♦ s❡✉ ❝♦t✐❞✐❛♥♦✳

◆♦ tó♣✐❝♦ ✺ ❞❛s ❉✐r❡tr✐③❡s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦✭❉❈◆❊▼ ✮✱ ♦❜✲s❡r✈❛♠♦s ❛ s❡❣✉✐♥t❡ ♦r✐❡♥t❛çã♦✿

❖ ❛❧✉♥♦ ❞❡✈❡rá s❡r ❝❛♣❛③ ❞❡✿

• ❛♥❛❧✐s❛r q✉❛❧✐t❛t✐✈❛♠❡♥t❡ ❞❛❞♦s q✉❛♥t✐t❛t✐✈♦s✱ r❡♣r❡s❡♥t❛❞♦s ❣rá✜❝❛ ♦✉ ❛❧❣❡❜r✐❝❛♠❡♥t❡✱r❡❧❛❝✐♦♥❛❞♦s ❛ ❝♦♥t❡①t♦s só❝✐♦ ❡❝♦♥ô♠✐❝♦s✱ ❝✐❡♥tí✜❝♦s ♦✉ ❝♦t✐❞✐❛♥♦s❀

• ✲ ✐❞❡♥t✐✜❝❛r✱ r❡♣r❡s❡♥t❛r ❡ ✉t✐❧✐③❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❣❡♦♠étr✐❝♦ ♣❛r❛ ❛♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❛❧❡✐t✉r❛✱ ❞❛ ❝♦♠♣r❡❡♥sã♦ ❡ ❞❛ ❛çã♦ s♦❜r❡ ❛ r❡❛❧✐❞❛❞❡✳

❆♥❛❧✐s❛♥❞♦ ♦s P❈◆s✱ ✈❡r✐✜❝❛♠♦s ❛ ✐♠❡♥s❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ s❡ tr❛❜❛❧❤❛r ❛ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛❡ ❡s♣❛❝✐❛❧ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♣♦✐s ♦ ❝♦t✐❞✐❛♥♦ ❞♦s ❛❧✉♥♦s é ❜❛s❡❛❞♦ ❡♠ ❝✐♠❛ ❞❡ ✜❣✉r❛s ❡♣r♦❜❧❡♠❛s q✉❡ ❧❤❡s ❞ã♦ ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❡st❛r❡♠ ✐♥s❡r✐❞♦s ♥♦ ♣r♦❝❡ss♦ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠❞❡ ♠❛♥❡✐r❛ ❛ ♠❛♥✐♣✉❧❛r ❢ór♠✉❧❛s ❝♦♠ ❝♦♥❝❡✐t♦s ♣r❡✈✐❛♠❡♥t❡ ❛♣r❡s❡♥t❛❞♦s ❡ ❡st✐♣✉❧❛❞♦s♣❡❧♦ ♣r♦❢❡ss♦r ❡ ♣❡❧♦s ♠❛t❡r✐❛✐s ❞✐❞át✐❝♦s ❛♣r❡s❡♥t❛❞♦s ❛♦s ♠❡s♠♦s✳ ❚❡♠♦s✱ s❡❣✉♥❞♦ ♦P❈◆ ❞❡ ▼❛t❡♠át✐❝❛✱ q✉❡ ❛♣r❡s❡♥t❛r s✐t✉❛çõ❡s q✉❡ r❡♣r❡s❡♥t❡♠ ❞❡s❛✜♦s ♣❛r❛ ♦s ❛❧✉♥♦s✱ q✉❡s✐❣♥✐✜q✉❡♠ ♥ã♦ ❛♣❡♥❛s ❛ ♠❡♠♦r✐③❛çã♦ ❞❡ ❢ór♠✉❧❛s✱ ♠❛s ❛ ❛♣❧✐❝❛çã♦ ❞❛s ♠❡s♠❛s ❥✉♥t♦ ❝♦♠✉♠ r❛❝✐♦❝í♥✐♦ ♠❛✐s ❡❧❛❜♦r❛❞♦ ❡ ❡str✉t✉r❛❞♦ ♣❡❧♦ ♣ró♣r✐♦ ❛❧✉♥♦✱ ♥ã♦ t❡♥❞♦ ❡st❡ ❢❡✐t♦ ✉♠❛s✐♠♣❧❡s r❡♣❡t✐çã♦ ❞❡ ♣r♦❝❡ss♦s ❡ ♥ã♦ ❞❡✐①❛♥❞♦ t♦t❛❧♠❡♥t❡ ❞❡ ❧❛❞♦ ❛❧❣✉♠❛s ❛t✐✈✐❞❛❞❡s ♥❛sq✉❛✐s s❡ ❛♣r❡s❡♥t❡ ✐❞❡✐❛s ❞♦ t✐♣♦ ✏❝❛❧❝✉❧❡✑✱ ✏r❡s♦❧✈❛✑✱ ✏❡♥❝♦♥tr❡✑✱ ♣♦✐s ❡st❛s ✐rã♦ s❡r ♥❡❝❡ssár✐❛s♣❛r❛ q✉❡ ♦ ❛❧✉♥♦ ❛♣r❡♥❞❛ ♦s ♣r♦❝❡❞✐♠❡♥t♦s té❝♥✐❝♦s ❞❛ s♦❧✉çã♦ ❞♦s ♣r♦❜❧❡♠❛s✳

➱ ❜♦♠ ♦❜s❡r✈❛r♠♦s t❛♠❜é♠✱ q✉❡ ❝♦♥❝❡✐t♦s ♠❛t❡♠át✐❝♦s✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ ❝♦♥❝❡✐t♦s❣❡♦♠étr✐❝♦s✱ ❞❡✈❡♠ s❡r ❛❜♦r❞❛❞♦s ❡ ❛♣r❡s❡♥t❛❞♦s ❞❡ ✉♠❛ ♠❛♥❡✐r❛ q✉❡ ♥ã♦ s❡ ♣r❡♥❞❛ s♦♠❡♥t❡♥♦ ❝♦t✐❞✐❛♥♦ ❡ ♥✉♠❛ ❛♣❧✐❝❛çã♦ ♣rát✐❝❛ ❡ ❝♦♥❝r❡t❛✱ ♣♦rq✉❡ ❡♠ ❛❧❣✉♠❛s s✐t✉❛çõ❡s ♥ã♦ t❡r❡♠♦s✉♠❛ ✈✐s✉❛❧✐③❛çã♦ ♥♦ ❡s♣❛ç♦ q✉❡ ♥♦s ❝❡r❝❛ ❞❛q✉❡❧❛s ❞❡✜♥✐çõ❡s ❡ s✐t✉❛çõ❡s ❛♣r❡s❡♥t❛❞❛s✳ ❚❡✲♠♦s q✉❡ ✐♥❝❡♥t✐✈❛r ✉♠ ❡♥✈♦❧✈✐♠❡♥t♦ ♠❛t❡♠át✐❝♦ ❡♠ ♥í✈❡❧ ▼é❞✐♦ ❡ ❋✉♥❞❛♠❡♥t❛❧✱ ❞❡ ❢♦r♠❛q✉❡ ♦s ❞✐s❝❡♥t❡s s❡ ❢❛♠✐❧✐❛r✐③❡♠ ❝♦♠ ❛ q✉❡stã♦ ❞❡ ❡st❛r ✉s❛♥❞♦ ❝♦♥❝❡✐t♦s ❡ r❛❝✐♦❝í♥✐♦s q✉❡♦s ❛❥✉❞❡♠ ❡♠ ♦✉tr❛s s✐t✉❛çõ❡s✱ ❡♠ ♦✉tr♦s ♥í✈❡✐s ♠❛t❡♠át✐❝♦s ❡ t❛♠❜é♠ ❡♠ ♦✉tr❛s ár❡❛s ❞♦❝♦♥❤❡❝✐♠❡♥t♦✱ ♥ã♦ ✜❝❛♥❞♦ ❞❡♣❡♥❞❡♥t❡s ❞❡ t❡r❡♠ s❡♠♣r❡ ✉♠❛ s✐t✉❛çã♦ ♣rát✐❝❛ ♣❛r❛ ♦ ✉s♦♦✉ ✉♠❛ ❥✉st✐✜❝❛t✐✈❛ ❞❡ ❡st✉❞♦✱ ♠❛s q✉❡ ❡st✐♠✉❧❡♠ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❡ ♦❛♣r❡ç♦ ♣❡❧♦ r❛❝✐♦❝í♥✐♦ ❧ó❣✐❝♦ ♠❛t❡♠át✐❝♦✱ ♣❡❧❛ ✐♠♣♦rtâ♥❝✐❛ ❡ ❞❡♣❡♥❞ê♥❝✐❛ ❞❡ ♦✉tr❛s ár❡❛s ❡❞❛ ♣ró♣r✐❛ ♠❛t❡♠át✐❝❛ ❞❛q✉✐❧♦ q✉❡ s❡ ❡stá ❡st✉❞❛♥❞♦✳

❖ ❡♥s✐♥♦ ❞❛ ❣❡♦♠❡tr✐❛ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ❞❡✈❡ s❡r ❡♥❝❛r❛❞♦ ❝♦♠♦ ✉♠❛ ❢♦r♠❛ ❞❡ ❧❡✈❛r ♦❛❧✉♥♦ ❛ q✉❡st✐♦♥❛r ✈ár✐❛s s✐t✉❛çõ❡s ❡♠ q✉❡ ❡❧❡ ❡stá ✐♥s❡r✐❞♦✱ s❡❥❛ ♥❛ ❡s❝♦❧❛ s❡❥❛ ♥♦ s❡✉ ❞✐❛ ❛❞✐❛ ❢♦r❛ ❞♦ ❛♠❜✐❡♥t❡ ❡s❝♦❧❛r ❡♠ q✉❡ ❡❧❡ ❞❡✈❡rá s❡r ❝❛♣❛③ ❞❡ r❡❝♦♥❤❡❝❡r ❡ q✉❡st✐♦♥❛r ❛q✉✐❧♦❞♦ q✉❛❧ ❡❧❡ t❡✈❡ ❝♦♥t❛t♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳

✷✳✶✳ P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s ✶✶

❖ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❡ s✉❛s ❛♣❧✐❝❛çõ❡s sã♦ ❡st✉❞❛❞♦s ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡ ♥❛ s❡❣✉♥❞❛sér✐❡ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❡ ❛♣r❡s❡♥t❛✲s❡✱ ❝♦♠♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ ❢✉♥❞❛♠❡♥t❛❧ ❡ ❜ás✐❝❛ ♣❛r❛ ♦❡♥t❡♥❞✐♠❡♥t♦ ❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s s✐♠♣❧❡s ♦✉ ♠❛✐s ❡❧❛❜♦r❛❞♦s ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ ❛♦❝á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦ tr✐❞✐♠❡♥s✐♦♥❛❧✱ ❜❡♠ ❝♦♠♦ ♦ ❛♣r✐♠♦r❛♠❡♥t♦ ❞♦ ❝á❧❝✉❧♦ ❡❛♣❧✐❝❛çã♦ ❞❡ ár❡❛s ❞❡ ✜❣✉r❛s ♣❧❛♥❛s✳ ❊❧❡ é ❛ ♣r✐♥❝✐♣❛❧ ❢❡rr❛♠❡♥t❛ ❡❧❡♠❡♥t❛r ♣❛r❛ ❛ ♦❜t❡♥çã♦❞❡ ❡①♣r❡ssõ❡s ❞♦ ✈♦❧✉♠❡ ❞❡ só❧✐❞♦s✱ é ✉t✐❧✐③❛❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ ♥❛ ❞❡❞✉çã♦ ❞❛ ❡①♣r❡ssã♦❞♦ ✈♦❧✉♠❡ ❞❡ ❝♦♥❡s ❡ ❞❡ ❡s❢❡r❛s✳ ❆❧é♠ ❞✐ss♦✱ ♣❡r♠✐t❡ ♦ ❝á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡ ❞❡ só❧✐❞♦s ♥ã♦r❡❣✉❧❛r❡s✳ ❆ss✐♠✱ ♣❡❧❛ s✉❛ ✐♠♣♦rtâ♥❝✐❛✱ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❞❡✈❡ s❡r ❡♥s✐♥❛❞♦ ♥♦ ❊♥s✐♥♦▼é❞✐♦✳

P❛r❛ ✉♠ ❜♦♠ ❡♥t❡♥❞✐♠❡♥t♦ ❡ ❝♦♠♣r❡ssã♦ ♣❡r❢❡✐t❛ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✱ ❢❛③✲s❡ ♥❡❝❡s✲sár✐♦ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❜ás✐❝♦ ❞❡ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛ ♦ q✉❡ ❞❡✈❡rá s❡r ❛♣r❡s❡♥t❛❞♦ ❛♦s ❛❧✉♥♦s❝♦♠ ❛♥t❡❝❡❞ê♥❝✐❛ ♣❡❧♦s ♣r♦✜ss✐♦♥❛✐s q✉❡ tr❛❜❛❧❤❛♠ ❡♠ ❛♥♦s ❛♥t❡r✐♦r❡s✱ ♦✉ ❞❡ ✉♠❛ ♠❛♥❡✐r❛♠❛✐s s✉❝✐♥t❛✱ ❞✉r❛♥t❡ ♦ ♣ró♣r✐♦ ❛♥♦ ❧❡t✐✈♦ ❡♠ q✉❡ s❡ ❡stá ❛♣r❡s❡♥t❛♥❞♦ ❛s ♥♦çõ❡s ❜ás✐❝❛s ❞❡só❧✐❞♦s ❣❡♦♠étr✐❝♦s✳ ❖s ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❞♦s q✉❛✐s ♥♦ss♦s ❞✐s❝❡♥t❡s ❥❛♠❛✐s ♣♦❞❡rã♦ s❡r ♣r✐✲✈❛❞♦s ❞❡ s❡✉ ❝♦♥❤❡❝✐♠❡♥t♦✱ t❛✐s ❝♦♠♦ ❛s ♥♦çõ❡s ❜ás✐❝❛s ❞❡ ♣♦❧í❣♦♥♦s ❡ s✉❛s ár❡❛s✱ ❛s ♥♦çõ❡s❣❡r❛✐s ❞❡ ♣❧❛♥♦✱ r❡t❛ ❡ ♣♦♥t♦ sã♦ ♦ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ♣❛r❛ q✉❡ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ s❡❥❛❛♣r❡s❡♥t❛❞♦✱ ❡♥t❡♥❞✐❞♦ ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❢❛③❡r ❝♦♠ q✉❡ ❡st❡s ❡❞✉❝❛♥❞♦s s❡❥❛♠ ❝❛♣❛③❡s ❞❡r❡❝♦♥❤❡❝❡r✱ ❞✐❢❡r❡♥❝✐❛r ❡ ♠❛♥✐♣✉❧❛r ❝♦♠ ❝á❧❝✉❧♦s ❞❡ ár❡❛s✱ ♣❡rí♠❡tr♦s ❡ ✈♦❧✉♠❡s ❞❡ ✜❣✉r❛s❜✐❞✐♠❡♥s✐♦♥❛✐s ❡ tr✐❞✐♠❡♥s✐♦♥❛✐s✳ ◆ã♦ ♣♦❞❡♠♦s ❞❡✐①❛r ❞❡ ❝✐t❛r t❛♠❜é♠ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡♦s ❛❧✉♥♦s t❡r❡♠ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❛ r❡s♣❡✐t♦ ❞❛ ❣❡♦♠❡tr✐❛ ❞❡ ♣♦s✐çã♦ ♣❛r❛ s❛❜❡r ❝♦♠♦ ♦ssó❧✐❞♦s ♣♦❞❡♠ ❡st❛r r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ♦ ♣❧❛♥♦ ❛ r❡t❛ ❡ ♦ ♣♦♥t♦✳

❙❛❜❡♠♦s ❞❛ ✐♠❡♥s❛ ❞✐✜❝✉❧❞❛❞❡ q✉❡ t❡♠♦s ♥♦ ♥♦ss♦ ❞✐❛ ❛ ❞✐❛ ❡ ❞♦s ❞❡s❛✜♦s ❝♦♠ ♦s q✉❛✐s♥♦s ❞❡♣❛r❛♠♦s ❛♦ tr❛❜❛❧❤❛r♠♦s ❝♦♠ ♥♦ss♦s ❛❧✉♥♦s ❛ ❣❡♦♠❡tr✐❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛ ❣❡♦♠❡tr✐❛❡s♣❛❝✐❛❧ ♥♦ q✉❡ ❞✐③ r❡s♣❡✐t♦ ❛♦ ❞❡s❡♥❤♦✱ ♥❛ s✉❛ ❝♦♥str✉çã♦ ♦✉ ❛té ♥❛ ♣ró♣r✐❛ ❛♥á❧✐s❡ ❞❡ ❢♦r♠❛sq✉❡ ❡stã♦ ♣r♦♥t❛s✳ ◆♦ss♦s ❛❧✉♥♦s✱ ♥❛ s✉❛ ❣r❛♥❞❡ ♠❛✐♦r✐❛✱ tê♠ ✉♠❛ ❞✐✜❝✉❧❞❛❞❡ ♠✉✐t♦ ❣r❛♥❞❡❡♠ r❡❛❧✐③❛r ❞❡s❡♥❤♦s ❡ ❛té ❝❡rt♦ ♣♦♥t♦ ❡♠ ❢❛③❡r ✉♠❛ ❛♥á❧✐s❡ ❞♦s ♠❡s♠♦s✳ Pr❡❝✐s❛♠♦s ❡♥tã♦❢❛③❡r ❝♦♠ q✉❡ ❡❧❡s ♣❛ss❡♠ ❛ tr❛❜❛❧❤❛r ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ♠❛✐s ❢❛♠✐❧✐❛r ❝♦♠ ❛s ❢♦r♠❛s✱ ♣❛r❛❡♥tã♦ ❡♥t❡♥❞❡r ♠❡❧❤♦r ♦s ❝♦♥❝❡✐t♦s ❣❡♦♠étr✐❝♦s✱ s✉❛ ✐♠♣♦rtâ♥❝✐❛ ♣❛r❛ ❛❧❣✉♠❛s ❞❡❞✉çõ❡s❡ ❞❡✜♥✐çõ❡s ❛ s❡r❡♠ ❡st✉❞❛❞❛s ❡ ♥ã♦ só ♦s ❡♥❝❛r❡♠ ❝♦♠♦ ✉♠❛ ❞✐✜❝✉❧❞❛❞❡ ❛ ♠❛✐s ♣❛r❛ ♦❛♣r❡♥❞✐③❛❞♦ ❡ ♦ ❡st✉❞♦ ❞❛ ❣❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧ ❡ ♣❧❛♥❛✳ P♦❞❡♠♦s ❛✐♥❞❛ ✉s❛r✱ ❤♦❥❡ ❡♠ ❞✐❛✱♦✉tr♦s r❡❝✉rs♦s q✉❡ ♥ã♦ s❡❥❛♠ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♦s ♠❛✐s tr✐✈✐❛✐s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ r❡❝✉rs♦s❡❧❡trô♥✐❝♦s ❡ s♦❢t✇❛r❡s ♠❛t❡♠át✐❝♦s ♣❛r❛ ♦ ❛✉①í❧✐♦ ♥❛ ❞❡❞✉çã♦ ❡ ♥❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s❝♦♠ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳

❱❛♠♦s ❛♣r❡s❡♥t❛r ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❝♦♠♦ s❡♥❞♦ ✉♠ ❡❧♦ ❡♥tr❡ ❛ t❡♦r✐❛ ❡ ❛ ♣rát✐❝❛♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s s✐♠♣❧❡s ♦✉ ❜❡♠ ♠❛✐s ❝♦♠♣❧❡①♦s q✉❡ ❡♥✈♦❧✈❛♠ t❛♠❜é♠ ♦✉tr❛sár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❝♦♠ r❡❝✉rs♦s ♠❛✐s ♠♦❞❡r♥♦s ♣❛r❛ ❛ s✉❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦✳ ❖♣r✐♥❝í♣✐♦ s❡ t♦r♥❛ ♣rát✐❝♦ ❛ ♣❛rt✐r ❞♦ ♠❡♠❡♥t♦ ❡♠ q✉❡ s❡ t❡♠ ❛ s✉❛ ❛♣r❡s❡♥t❛çã♦ ❛tr❛✈és❞❡ ✉♠❛ s♦❜r❡♣♦s✐çã♦ ❞❡ ✜❣✉r❛s ❝♦♠ ❛ ♠❡s♠❛ ár❡❛✳ ❋✐❝❛ ❝❧❛r♦ ♣❛r❛ t♦❞♦s q✉❡ ♦ ♦❜s❡r✈❛♠

✷✳✷✳ ❊①❛♠❡ ◆❛❝✐♦♥❛❧ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ✲ ❊◆❊▼ ✶✷

♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ q✉❡ ❛♦ s♦❜r❡♣♦r♠♦s ➇✐♥✜♥✐t❛s➈ ✜❣✉r❛s ❞❡ ♠❡s♠❛ ár❡❛✱ ♠❛s ❞❡ ❢♦r♠❛s❣❡♦♠étr✐❝❛s ❞✐❢❡r❡♥t❡s✱ ♣♦❞❡r❡♠♦s t❡r ♦ ♠❡s♠♦ ✈♦❧✉♠❡✳ ❈❛❜❡ ❛q✉✐✱ ❡st❛❜❡❧❡❝❡r♠♦s ✉♠❛❞✐❢❡r❡♥ç❛ ❡♥tr❡ ✈♦❧✉♠❡ ❡ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ✉♠ só❧✐❞♦✳ ❖ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦ é ♦ q✉❛♥t♦ ❡❧❡♦❝✉♣❛ ♥♦ ❡s♣❛ç♦✱ ❧❡♠❜r❛♥❞♦✲s❡ s❡♠♣r❡ ❞❡ q✉❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦✱ t❡♦r✐❝❛♠❡♥t❡✱ é ❞❡✜♥✐❞♦❝♦♠♦ s❡♥❞♦ ♦ ♣r♦❞✉t♦ ❞❛s s✉❛s três ❞✐♠❡♥sõ❡s✱ ❝♦♠♣r✐♠❡♥t♦✱ ❧❛r❣✉r❛ ❡ ❛❧t✉r❛✱ ♣♦❞❡♥❞♦ t❡r✈❛r✐❛çõ❡s ❞❡♣❡♥❞❡♥❞♦ ❞♦ só❧✐❞♦ ❡st✉❞❛❞♦✳ ❆ ❝❛♣❛❝✐❞❛❞❡ é ♦ q✉❛♥t♦ ♦ só❧✐❞♦ ❛r♠❛③❡♥❛ ❡♠s❡✉ ✐♥t❡r✐♦r✳

➱ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ r❡ss❛❧t❛r♠♦s q✉❡ ❛♦ ♥♦s ❜❛s❡❛r♠♦s ♥♦s P❈◆s ❞❡ ♠❛✲t❡♠át✐❝❛✱ ✈❛♠♦s ❛♣r❡s❡♥t❛r ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♠♦♠❡♥t♦ ♦ ✉s♦ ❡ ♦ ❛♣♦✐♦ ❞❡ ♠❛t❡r✐❛✐s❝♦♥❝r❡t♦s ❡ ❞❡ r❡❝✉rs♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣❛r❛ ❛ ❛♣r❡s❡♥t❛çã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❡ s❡✉s❞❡s❞♦❜r❛♠❡♥t♦s ❡ ❛♣❧✐❝❛çõ❡s ♥♦ ❝♦t✐❞✐❛♥♦ ❡ ❝♦♠♦ ❥á ❝✐t❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ♣❛r❛ ❞❡❞✉çõ❡s ❞❡r❛❝✐♦❝í♥✐♦s q✉❡ ♣♦ss❛♠ s❡r ✉s❛❞♦s ❡♠ ♦✉tr❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❡ ❡♠ ♦✉tr❛s ❞✐s❝✐♣❧✐♥❛s✱❛❧é♠ ❞❡ s❡r❡♠ ❛♣r♦✈❡✐t❛❞♦s ❞❡♥tr♦ ❞❛ ♣ró♣r✐❛ ♠❛t❡♠át✐❝❛✳

✷✳✷ ❊①❛♠❡ ◆❛❝✐♦♥❛❧ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ✲ ❊◆❊▼

❊♠ s✉❛ ▼❛tr✐③ ❞❡ ❘❡❢❡rê♥❝✐❛✱ ♦ ❊①❛♠❡ ◆❛❝✐♦♥❛❧ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♣r❡❝♦♥✐③❛ ♦s s❡❣✉✐♥t❡stó♣✐❝♦s✿

✷✳✷✳✶ ❊■❳❖❙ ❈❖●◆■❚■❱❖❙ ✭❝♦♠✉♥s ❛ t♦❞❛s ❛s ár❡❛s ❞❡ ❝♦♥❤❡❝✐✲

♠❡♥t♦✮

■✳ ❉♦♠✐♥❛r ❧✐♥❣✉❛❣❡♥s ✭❉▲✮✿ ❞♦♠✐♥❛r ❛ ♥♦r♠❛ ❝✉❧t❛ ❞❛ ▲í♥❣✉❛ P♦rt✉❣✉❡s❛ ❡ ❢❛③❡r✉s♦ ❞❛s ❧✐♥❣✉❛❣❡♥s ♠❛t❡♠át✐❝❛✱ ❛rtíst✐❝❛ ❡ ❝✐❡♥tí✜❝❛ ❡ ❞❛s ❧í♥❣✉❛s ❡s♣❛♥❤♦❧❛ ❡ ✐♥❣❧❡s❛✳

■■✳ ❈♦♠♣r❡❡♥❞❡r ❢❡♥ô♠❡♥♦s ✭❈❋✮✿ ❝♦♥str✉✐r ❡ ❛♣❧✐❝❛r ❝♦♥❝❡✐t♦s ❞❛s ✈ár✐❛s ár❡❛s ❞♦ ❝♦✲♥❤❡❝✐♠❡♥t♦ ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ ❢❡♥ô♠❡♥♦s ♥❛t✉r❛✐s✱ ❞❡ ♣r♦❝❡ss♦s ❤✐stór✐❝♦✲❣❡♦❣rá✜❝♦s✱❞❛ ♣r♦❞✉çã♦ t❡❝♥♦❧ó❣✐❝❛ ❡ ❞❛s ♠❛♥✐❢❡st❛çõ❡s ❛rtíst✐❝❛s✳

■■■✳ ❊♥❢r❡♥t❛r s✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛ ✭❙P✮✿ s❡❧❡❝✐♦♥❛r✱ ♦r❣❛♥✐③❛r✱ r❡❧❛❝✐♦♥❛r✱ ✐♥t❡r♣r❡t❛r❞❛❞♦s ❡ ✐♥❢♦r♠❛çõ❡s r❡♣r❡s❡♥t❛❞♦s ❞❡ ❞✐❢❡r❡♥t❡s ❢♦r♠❛s✱ ♣❛r❛ t♦♠❛r ❞❡❝✐sõ❡s ❡ ❡♥❢r❡♥t❛rs✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛✳

■❱✳ ❈♦♥str✉✐r ❛r❣✉♠❡♥t❛çã♦ ✭❈❆✮✿ r❡❧❛❝✐♦♥❛r ✐♥❢♦r♠❛çõ❡s✱ r❡♣r❡s❡♥t❛❞❛s ❡♠ ❞✐❢❡r❡♥✲t❡s ❢♦r♠❛s✱ ❡ ❝♦♥❤❡❝✐♠❡♥t♦s ❞✐s♣♦♥í✈❡✐s ❡♠ s✐t✉❛çõ❡s ❝♦♥❝r❡t❛s✱ ♣❛r❛ ❝♦♥str✉✐r ❛r❣✉✲♠❡♥t❛çã♦ ❝♦♥s✐st❡♥t❡✳

✷✳✷✳ ❊①❛♠❡ ◆❛❝✐♦♥❛❧ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ✲ ❊◆❊▼ ✶✸

❱✳ ❊❧❛❜♦r❛r ♣r♦♣♦st❛s ✭❊P✮✿ r❡❝♦rr❡r ❛♦s ❝♦♥❤❡❝✐♠❡♥t♦s ❞❡s❡♥✈♦❧✈✐❞♦s ♥❛ ❡s❝♦❧❛ ♣❛r❛❡❧❛❜♦r❛çã♦ ❞❡ ♣r♦♣♦st❛s ❞❡ ✐♥t❡r✈❡♥çã♦ s♦❧✐❞ár✐❛ ♥❛ r❡❛❧✐❞❛❞❡✱ r❡s♣❡✐t❛♥❞♦ ♦s ✈❛❧♦r❡s❤✉♠❛♥♦s ❡ ❝♦♥s✐❞❡r❛♥❞♦ ❛ ❞✐✈❡rs✐❞❛❞❡ ❝✉❧t✉r❛❧✳

❖ ❊①❛♠❡ ◆❛❝✐♦♥❛❧ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❛tr❛✈és ❞❛s ❍❛❜✐❧✐❞❛❞❡s ❡ ❈♦♠♣❡tê♥❝✐❛s ❞❡ ❝❛❞❛ár❡❛ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♣r❡❝♦♥✐③❛♠ q✉❡ ♦ ❛❧✉♥♦ ❛♦ ✜♥❛❧ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❞❡✈❡ s❡r ❝❛♣❛③ ❞❡✿

➪r❡❛ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦✿ ▼❛t❡♠át✐❝❛ ❡ s✉❛s t❡❝♥♦❧♦❣✐❛s

✷✳✷✳✷ ❈♦♠♣❡tê♥❝✐❛ ❞❡ ár❡❛ ✷ ✲ ❯t✐❧✐③❛r ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❣❡♦♠étr✐❝♦

♣❛r❛ r❡❛❧✐③❛r ❛ ❧❡✐t✉r❛ ❡ ❛ r❡♣r❡s❡♥t❛çã♦ ❞❛ r❡❛❧✐❞❛❞❡ ❡ ❛❣✐r

s♦❜r❡ ❡❧❛✳

❍✻ ✲ ■♥t❡r♣r❡t❛r ❛ ❧♦❝❛❧✐③❛çã♦ ❡ ❛ ♠♦✈✐♠❡♥t❛çã♦ ❞❡ ♣❡ss♦❛s✴♦❜❥❡t♦s ♥♦ ❡s♣❛ç♦ tr✐❞✐♠❡♥✲s✐♦♥❛❧ ❡ s✉❛ r❡♣r❡s❡♥t❛çã♦ ♥♦ ❡s♣❛ç♦ ❜✐❞✐♠❡♥s✐♦♥❛❧✳

❍✼ ✲ ■❞❡♥t✐✜❝❛r ❝❛r❛❝t❡ríst✐❝❛s ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ♦✉ ❡s♣❛❝✐❛✐s✳

❍✽ ✲ ❘❡s♦❧✈❡r s✐t✉❛çã♦✲♣r♦❜❧❡♠❛ q✉❡ ❡♥✈♦❧✈❛ ❝♦♥❤❡❝✐♠❡♥t♦s ❣❡♦♠étr✐❝♦s ❞❡ ❡s♣❛ç♦ ❡❢♦r♠❛✳

❍✾ ✲ ❯t✐❧✐③❛r ❝♦♥❤❡❝✐♠❡♥t♦s ❣❡♦♠étr✐❝♦s ❞❡ ❡s♣❛ç♦ ❡ ❢♦r♠❛ ♥❛ s❡❧❡çã♦ ❞❡ ❛r❣✉♠❡♥t♦s♣r♦♣♦st♦s ❝♦♠♦ s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s ❞♦ ❝♦t✐❞✐❛♥♦✳

✷✳✷✳✸ ❈♦♠♣❡tê♥❝✐❛ ❞❡ ár❡❛ ✸ ✲ ❈♦♥str✉✐r ♥♦çõ❡s ❞❡ ❣r❛♥❞❡③❛s ❡

♠❡❞✐❞❛s ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❞❛ r❡❛❧✐❞❛❞❡ ❡ ❛ s♦❧✉çã♦ ❞❡ ♣r♦✲

❜❧❡♠❛s ❞♦ ❝♦t✐❞✐❛♥♦✳

❍✶✵ ✲ ■❞❡♥t✐✜❝❛r r❡❧❛çõ❡s ❡♥tr❡ ❣r❛♥❞❡③❛s ❡ ✉♥✐❞❛❞❡s ❞❡ ♠❡❞✐❞❛✳

❍✶✶ ✲ ❯t✐❧✐③❛r ❛ ♥♦çã♦ ❞❡ ❡s❝❛❧❛s ♥❛ ❧❡✐t✉r❛ ❞❡ r❡♣r❡s❡♥t❛çã♦ ❞❡ s✐t✉❛çã♦ ❞♦ ❝♦t✐❞✐❛♥♦✳

❈❛♣ít✉❧♦ ✸

◆♦çõ❡s ■♥✐❝✐❛✐s ❛ r❡s♣❡✐t♦ ❞♦s ♣♦❧✐❡❞r♦s

❞❡ P❧❛tã♦ ❡ ❛❧❣✉♥s só❧✐❞♦s

❆♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s q✉❡ ❡stã♦ r❡❧❛❝✐♦♥❛❞♦s ❝♦♠ ❛ ❣❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧✳ ❚r❛✲❜❛❧❤❛r❡♠♦s ❝♦♠ só❧✐❞♦s ❡ ♦❜s❡r✈❛r❡♠♦s q✉❡ ❝❛❞❛ ✉♠ ❞❡st❡s só❧✐❞♦s é ❝♦♠♣♦st♦ ♣♦r s✉❛s✉♣❡r❢í❝✐❡ ❡ ♣♦r s❡✉ ✐♥t❡r✐♦r✳ ■♥✐❝✐❛❧♠❡♥t❡ s❡rã♦ ❛♣r❡s❡♥t❛❞❛s ❛❧❣✉♠❛s ♥♦çõ❡s ❜ás✐❝❛s ❛❝❡r❝❛❞♦ ❡s♣❛ç♦ tr✐❞✐♠❡♥s✐♦♥❛❧ ❡ ❞❛ ♠❛♥❡✐r❛ ❝♦♠♦ sã♦ ❢♦r♠❛❞♦s ♦s ♣♦❧✐❡❞r♦s ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ❣❡✲r❛❧✳ ❊♠ s❡❣✉✐❞❛ s❡rá ❢❡✐t♦ ✉♠ ❜r❡✈❡ ❝♦♠❡♥tár✐♦ s♦❜r❡ ✜❣✉r❛s ❣❡♦♠étr✐❝❛s ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡✐♥tr♦❞✉③✐r ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣♦❧✐❡❞r♦s ♣❛r❛ q✉❡ ♣♦ss❛♠♦s ❛♣r❡s❡♥t❛r ♣♦st❡r✐♦r♠❡♥t❡ ♦s ♣r✐s♠❛s❡ ❛s ♣✐râ♠✐❞❡s✱ ❜❡♠ ❝♦♠♦ ❛❧❣✉♥s ❡①❡♠♣❧♦s ♣❛r❛ s❡ t❡r ✉♠❛ ✈✐s✉❛❧✐③❛çã♦ ❞♦s ♠❡s♠♦s✳

◆♦ ❡♥❝❡rr❛♠❡♥t♦ ❞❡ss❡ tr❛❜❛❧❤♦ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❡①❡r❝í❝✐♦s ♣❛r❛ ✉♠❛ ❛♣❧✐❝❛çã♦ ❞❡❛❧❣✉♥s ❞♦s ❝♦♥❝❡✐t♦s ❛♣r❡s❡♥t❛❞♦s✳

❯♠❛ ♣r✐♠❡✐r❛ ✐❞❡✐❛ ♣❛r❛ ❞❡✜♥✐r ♦s ♣♦❧✐❡❞r♦s é ❛ s❡❣✉✐♥t❡✿ ✏P♦❧✐❡❞r♦ é ✉♠❛ r❡✉♥✐ã♦ ❞❡ ✉♠♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♣♦❧í❣♦♥♦s ♣❧❛♥♦s✱ ♦♥❞❡ ❝❛❞❛ ❧❛❞♦ ❞❡ ✉♠ ❞❡ss❡s ♣♦❧í❣♦♥♦s é t❛♠❜é♠ ❧❛❞♦❞❡ ✉♠✱ ❡ ❛♣❡♥❛s ✉♠✱ ♦✉tr♦ ♣♦❧í❣♦♥♦✑✳ ❈❛❞❛ ✉♠ ❞❡ss❡s ♣♦❧í❣♦♥♦s ❝❤❛♠❛✲s❡ ❢❛❝❡ ❞♦ ♣♦❧✐❡❞r♦✱❝❛❞❛ ❧❛❞♦ ❝♦♠✉♠ ❛ ❞✉❛s ❢❛❝❡s ❝❤❛♠❛✲s❡ ✉♠❛ ❛r❡st❛ ❞♦ ♣♦❧✐❡❞r♦ ❡ ❝❛❞❛ ✈ért✐❝❡ ❞❡ ✉♠❛ ❢❛❝❡é t❛♠❜é♠ ❝❤❛♠❛❞♦ ❞❡ ✈ért✐❝❡ ❞♦ ♣♦❧✐❡❞r♦✳ ❚♦❞♦ ♣♦❧✐❡❞r♦✱ ❧✐♠✐t❛ ✉♠❛ r❡❣✐ã♦ ❞♦ ❡s♣❛ç♦✱❝❤❛♠❛❞❛ ❞❡ ✐♥t❡r✐♦r ❞❡ss❡ ♣♦❧✐❡❞r♦✳ ❆ ár❡❛ ♦❝✉♣❛❞❛ ♣❡❧❛ s✉♣❡r❢í❝✐❡ ♣♦❞❡ s❡r ♦❜t✐❞❛ ♣♦r♠❡✐♦ ❞❡ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❞❡ ✜❣✉r❛s ♣❧❛♥❛s✱ ❡♥q✉❛♥t♦ q✉❡ s❡✉ ✈♦❧✉♠❡ ♥❡♠ s❡♠♣r❡ s❡rá ✉♠❛t❛r❡❢❛ ♠✉✐t♦ ❢á❝✐❧ ❡ ❛ss✐♠✱ ✉t✐❧✐③❛r❡♠♦s ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ♦❜t❡r♠♦s ♦s ✈♦❧✉♠❡s❞❡ ❞❡t❡r♠✐♥❛❞♦s ♣♦❧✐❡❞r♦s q✉❡ sã♦ ❞❡ ✐♥t❡r❡ss❡✳

✶✹

✸✳✶✳ P♦❧✐❡❞r♦s ❊s♣❡❝✐❛✐s ✶✺

✸✳✶ P♦❧✐❡❞r♦s ❊s♣❡❝✐❛✐s

❉❛r❡♠♦s t♦t❛❧ ❛t❡♥çã♦ ❛ ♣❛rt✐r ❞❡ss❡ ♠♦♠❡♥t♦ ❛ ❛❧❣✉♥s ♣♦❧✐❡❞r♦s ❡s♣❡❝✐❛✐s✱ ❡♠ ♣❛rt✐❝✉❧❛r♦s ♣r✐s♠❛s ❡ ❛s ♣✐râ♠✐❞❡s✳ ❱❛♠♦s ✈❡r✐✜❝❛r q✉❡ ❡ss❡s ♣♦❧✐❡❞r♦s tê♠ ❛❧❣✉♠❛s ♣❛rt✐❝✉❧❛r✐❞❛❞❡s❡ ♣♦r ✐ss♦ ♠❡s♠♦ ♣♦❞❡♠♦s ✉sá✲❧❛s ♥❛ r❡s♦❧✉çã♦ ❞❡ ✈ár✐♦s ♣r♦❜❧❡♠❛s✳ ❋♦❝❛r❡♠♦s ❛ ♥♦ss❛❛t❡♥çã♦ ♥♦s ♣r♦❜❧❡♠❛s q✉❡ ❡♥✈♦❧✈❡♠ ár❡❛s ❡ ✈♦❧✉♠❡s ❞❡ t❛✐s ♣♦❧✐❡❞r♦s✱ ár❡❛s ❞❡ s✉❛s ❢❛❝❡s✱r❡❧❛çõ❡s ❡♥tr❡ ♦s s❡✉s ❧❛❞♦s✱ s✉❛s ❛r❡st❛s ❡ s❡✉s ✈ért✐❝❡s ❛❧é♠ ❞❛s s✐t✉❛çõ❡s ❡♠ q✉❡ t❡♥❤❛♠♦sq✉❡ ✉s❛r ♣r♦♣r✐❡❞❛❞❡s ❡ ❝♦♥❝❡✐t♦s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ❛ ❝♦♠♣r❡❡♥sã♦ ❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✳

✸✳✶✳✶ Pr✐s♠❛s

❚❡①t♦s ❤✐stór✐❝♦s ♠♦str❛♠ q✉❡ ♦ ♣r✐s♠❛ é ✉♠❛ ✜❣✉r❛ ❣❡♦♠étr✐❝❛ ❝♦♥❤❡❝✐❞❛ ❞❡s❞❡ ❛♥t❡s❞❡ ✷✵✵✵ ❛✳❈✳✱ ♣♦✐s✱ s❡❣✉♥❞♦ ❬✽❪✱ ♦s ❡st✉❞✐♦s♦s ❞❛ é♣♦❝❛ ❥á ♠♦str❛♠✲s❡ ❢❛♠✐❧✐❛r✐③❛❞♦s ❝♦♠♦ ✈♦❧✉♠❡ ❞♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ r❡t♦ r❡tâ♥❣✉❧♦ ❡✱ ♠❛✐s ❣❡r❛❧♠❡♥t❡✱ ❞♦ ✈♦❧✉♠❡ ❞♦ ♣r✐s♠❛ r❡t♦❞❡ ❜❛s❡ tr❛♣❡③♦✐❞❛❧✳ ❊st✉❞♦s ♣r♦❞✉③✐❞♦s ❤✐st♦r✐❝❛♠❡♥t❡ ♠♦str❛♠ q✉❡ ❞✐✈❡rs♦s ❡st✉❞✐♦s♦s❞❡❞✐❝❛r❛♠✲s❡ ❛♦ ❡st✉❞♦ ❞♦ ♣r✐s♠❛✳ ❉❡♥tr❡ ❡st❡s ❡st✉❞✐♦s♦s ♣♦❞❡♠♦s ❞❡st❛❝❛r P❧❛tã♦✱ ❉❡♠ó✲❝r✐t♦ ❡ ❆rq✉✐♠❡❞❡s✳ ❇❛s❡❛❞♦ ♥❡st❡s ❡ ♦✉tr♦s ❞❛❞♦s✱ ❞✐✈❡rs♦s ♠❛t❡♠át✐❝♦s ❞❡❞✐❝❛r❛♠✲s❡ ❛♦❡st✉❞♦ ❞♦ ♣r✐s♠❛ ❝♦♠ ♦❜❥❡t✐✈♦s ❞✐✈❡rs♦s✱ ♥❡st❡ tr❛❜❛❧❤♦ s❡rá ❛❜♦r❞❛❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣r✐s♠❛s❡❣✉♥❞♦ ❛❧❣✉♥s ❛✉t♦r❡s ❝♦♥t❡♠♣♦râ♥❡♦s✳ ❆ ❞❡✜♥✐çã♦ ❢♦r♠❛❧ ❞❡ ✉♠ ♣r✐s♠❛✱ s❡❣✉♥❞♦ ❬✻❪✱é ❛ s❡❣✉✐♥t❡✿ ❈♦♥s✐❞❡r❡♠♦s ✉♠❛ r❡❣✐ã♦ ♣♦❧✐❣♦♥❛❧ ❝♦♥✈❡①❛ ♣❧❛♥❛ ✭♣♦❧í❣♦♥♦ ♣❧❛♥♦ ❝♦♥✈❡①♦✮A1A2...An ❞❡ n ❧❛❞♦s ❡ ✉♠❛ r❡t❛ r ♥ã♦ ♣❛r❛❧❡❧❛ ♥❡♠ ❝♦♥t✐❞❛ ♥♦ ♣❧❛♥♦ ❞❛ r❡❣✐ã♦ ✭♣♦❧í❣♦♥♦✮✳❈❤❛♠❛ ✲ s❡ ♣r✐s♠❛ ✐❧✐♠✐t❛❞♦ ❝♦♥✈❡①♦ ♦✉ ♣r✐s♠❛ ❝♦♥✈❡①♦ ✐♥❞❡✜♥✐❞♦ à r❡✉♥✐ã♦ ❞❛s r❡t❛s ♣❛✲r❛❧❡❧❛s ❛ r ❡ q✉❡ ♣❛ss❛♠ ♣❡❧♦s ♣♦♥t♦s ❞❛ r❡❣✐ã♦ ♣♦❧✐❣♦♥❛❧ ❞❛❞❛✳ ❈♦♠♦ ❝✐t❛❞♦ ❡♠ ❬✼❪✱ ♦♣r✐s♠❛ é ✉♠ ❝✐❧✐♥❞r♦ ❝✉❥❛s ❜❛s❡s sã♦ ♣♦❧í❣♦♥♦s✳ ❆❧❣✉♥s ❡①❡♠♣❧♦s ✐♠♣♦rt❛♥t❡s ❞❡ ♣r✐♠❛s q✉❡♣♦❞❡♠♦s ❝✐t❛r sã♦ ♦s ❝✉❜♦s✱ ♦s ♣❛r❛❧❡❧❡♣í♣❡❞♦s✱ q✉❡ sã♦ ♣r✐♠❛s ❞❡ ❜❛s❡s q✉❛❞r❛♥❣✉❧❛r❡s✱♣r✐s♠❛ ❤❡①❛❣♦♥❛❧ r❡❣✉❧❛r ❞❡♥tr❡ ♦✉tr♦s✳ ■♠♣♦rt❛♥t❡ ❞✐③❡r♠♦s q✉❡ ♦ ♥♦♠❡ ❞❡ ✉♠ ♣r✐s♠❛ é❞❛❞♦ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❢♦r♠❛ ❞❡ s✉❛ ❜❛s❡ ❡ q✉❡ ✉♠ ♣r✐s♠❛ r❡❣✉❧❛r é ❛q✉❡❧❡ ❡♠ q✉❡ ❡st❛ ❜❛s❡t❡♠ t♦❞♦s ♦s s❡✉s ❧❛❞♦s ❝♦♠ ❛ ♠❡s♠❛ ♠❡❞✐❞❛✳

✸✳✶✳✷ P✐râ♠✐❞❡s

❖❜s❡r✈❛♥❞♦ ❛s ♣✐râ♠✐❞❡s ❞♦ ❊❣✐t♦✱ ❋✐❣✉r❛ ✸✳✶ ✈❡r✐✜❝❛♠♦s q✉❡ ♦ ❡st✉❞♦ ❞❡ss❡ t✐♣♦ ❞❡♣♦❧✐❡❞r♦ t❡♠ ❞❡s♣❡rt❛❞♦ ✐♥t❡r❡ss❡ ❤á ♠✐❧❤❛r❡s ❞❡ ❛♥♦s✳ P♦❞❡♠♦s ♦❜s❡r✈❛r ✐ss♦ ❛tr❛✈és ❞❛❣r❛♥❞❡ ♣✐râ♠✐❞❡ ❞❡ ●✐③é ❝♦♥str✉í❞❛ ♣♦r ✈♦❧t❛ ❞❡ ✷✻✵✵ ❛✳❈✳

❆ ❞❡✜♥✐çã♦ ❢♦r♠❛❧ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ é ❞❛❞❛ ❡♠ ❬✻❪ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

❈♦♥s✐❞❡r❡♠♦s ✉♠❛ r❡❣✐ã♦ ♣♦❧✐❣♦♥❛❧ ♣❧❛♥♦ ❝♦♥✈❡①❛ ✭♣♦❧í❣♦♥♦ ♣❧❛♥♦ ✲ ❝♦✈❡①♦✮ A1A2...An

❞❡ n ❧❛❞♦s ❡ ✉♠ ♣♦♥t♦ V ❢♦r❛ ❞❡ s❡✉ ♣❧❛♥♦✳ ❈❤❛♠❛ ✲ s❡ ♣✐râ♠✐❞❡ ✐❧✐♠✐t❛❞❛ ❝♦♥✈❡①❛ ♦✉

✸✳✷✳ ❙ó❧✐❞♦s ❢♦r♠❛❞♦s ♣♦r s✉♣❡r❢í❝✐❡s ♥ã♦ ♣❧❛♥❛s ✶✻

♣✐râ♠✐❞❡ ❝♦♥✈❡①❛ ✐♥❞❡✜♥✐❞❛ à r❡✉♥✐ã♦ ❞❛s s❡♠✐rr❡t❛s ❞❡ ♦r✐❣❡♠ ❡♠ V ❡ q✉❡ ♣❛ss❛♠ ♣❡❧♦s♣♦♥t♦s ❞❛ r❡❣✐ã♦ ♣♦❧✐❣♦♥❛❧ ✭♣♦❧í❣♦♥♦✮ ❞❛❞❛✳ ❯♠❛ ♣✐râ♠✐❞❡✱ ❝♦♠♦ ❝✐t❛❞♦ ❡♠ ❬✼❪✱ s❡rá ✉♠ ❝♦♥❡❝♦♠ ❜❛s❡ ♣♦❧✐❣♦♥❛❧ ❡ ❛ss✐♠ ❝♦♠♦ ♥♦s ♣r✐s♠❛s ♣♦❞❡♠♦s ❞✐③❡r q✉❡ ✉♠❛ ♣✐râ♠✐❞❡ é r❡❣✉❧❛rq✉❛♥❞♦ ❛ s✉❛ ❜❛s❡ é ✉♠ ♣♦❧í❣♦♥♦ r❡❣✉❧❛r ✭t♦❞♦s ♦s ❧❛❞♦s ❝♦♠ ❛ ♠❡s♠❛ ♠❡❞✐❞❛✮✳

❋✐❣✉r❛ ✸✳✶✿ P✐râ♠✐❞❡s ❞♦ ❊❣✐t♦ ✲ ❋♦♥t❡✿ ✇✇✇✳❛s❝♦♥✈❡rs❛✳❝♦♠

✸✳✷ ❙ó❧✐❞♦s ❢♦r♠❛❞♦s ♣♦r s✉♣❡r❢í❝✐❡s ♥ã♦ ♣❧❛♥❛s

❱❡❥❛♠♦s ✉♠❛ ✐♥tr♦❞✉çã♦ ❜r❡✈❡ ❛ r❡s♣❡✐t♦ ❞❡ s✉♣❡r❢í❝✐❡s r❡❣r❛❞❛s ❞❡s❡♥✈♦❧✈í✈❡✐s ❝✐❧í♥❞r✐❝❛s❝♦♠♦ ❛♣❛r❡❝❡♠ ❡♠ ❬✻❪✳

❙✉♣❡r❢í❝✐❡s r❡❣r❛❞❛s ❡ ❝✐❧í♥❞r✐❝❛s sã♦ s✉♣❡r❢í❝✐❡s ❣❡r❛❞❛s ♣♦r ✉♠❛ r❡t❛ r ❣❡r❛tr✐③✱ q✉❡ s❡♠❛♥té♠ ♣❛r❛❧❡❧❛ ❛ ✉♠❛ r❡t❛ ❞❛❞❛ s✱ q✉❡ é ❛ ❞✐r❡çã♦✱ ❡ ♣❡r❝♦rr❡ ❡ ♦s ♣♦♥t♦s ❞❡ ✉♠❛ ❧✐♥❤❛❞❛❞❛ t q✉❡ é ❛ ❞✐r❡tr✐③✳ ❊ss❛s s✉♣❡r❢í❝✐❡s sã♦ ❞❡♥♦♠✐♥❛❞❛s ❞❡ r❡❣r❛❞❛s ♣♦r s❡r❡♠ ❣❡r❛❞❛s♣♦r r❡t❛s ❡ ❞❡s❡♥✈♦❧✈í✈❡✐s ♣♦r ♣♦❞❡r❡♠ s❡r ❛♣❧✐❝❛❞❛s✱ ❡st❡♥❞✐❞❛s ♦✉ ❛té ❞❡s❡♥✈♦❧✈✐❞❛s ♥✉♠♣❧❛♥♦ ✭♣❧❛♥✐✜❝❛❞❛s✮ s❡♠ ❞♦❜r❛s ♦✉ r✉♣t✉r❛s ❡♠ s✉❛ ❡str✉t✉r❛✳

❆♣r❡s❡♥t❛♠♦s ❛❧❣✉♠❛s s✉♣❡r❢í❝✐❡s r❡❣r❛❞❛s ❞❡s❡♥✈♦❧✈í✈❡✐s ❝✐❧í♥❞r✐❝❛s✿

✲ s❡ ❛ ❞✐r❡tr✐③ é ✉♠❛ r❡t❛ ♥ã♦ ♣❛r❛❧❡❧❛ à r❡t❛ s✱ ❛ s✉♣❡r❢í❝✐❡ ❝✐❧í♥❞r✐❝❛ ❣❡r❛❞❛ s❡rá ✉♠♣❧❛♥♦✳

✲ s❡ ❛ ❞✐r❡tr✐③ é ✉♠ ♣♦❧í❣♦♥♦ ✭❧✐♥❤❛ ♣♦❧✐❣♦♥❛❧ ❢❡❝❤❛❞❛✮✱ ❝✉❥♦ ♣❧❛♥♦ ❝♦♥❝♦rr❡ ❝♦♠ ❛ r❡t❛s✱ ❛ s✉♣❡r❢í❝✐❡ ❝✐❧í♥❞r✐❝❛ ❣❡r❛❞❛ é ✉♠❛ s✉♣❡r❢í❝✐❡ ♣r✐s♠át✐❝❛ ✐❧✐♠✐t❛❞❛✳

✲ s❡ ❛ ❞✐r❡tr✐③ é ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝✉❥♦ ♣❧❛♥♦ ❝♦♥❝♦rr❡ ❝♦♠ ❛ r❡t❛ s✱ ❛ s✉♣❡r❢í❝✐❡ ❝✐❧í♥❞r✐❝❛é ✉♠ ❝✐❧✐♥❞r♦ ❝✐r❝✉❧❛r ✐❧✐♠✐t❛❞❛✳

✸✳✷✳ ❙ó❧✐❞♦s ❢♦r♠❛❞♦s ♣♦r s✉♣❡r❢í❝✐❡s ♥ã♦ ♣❧❛♥❛s ✶✼

❆♣r❡s❡♥t❡♠♦s ❛s s✉♣❡r❢í❝✐❡s r❡❣r❛❞❛s ❞❡s❡♥✈♦❧✈í✈❡✐s ❝ô♥✐❝❛s q✉❡ sã♦ s✉♣❡r❢í❝✐❡s ❣❡r❛❞❛s♣♦r ✉♠❛ r❡t❛ r✱ ❞❡♥♦♠✐♥❛❞❛ ❞❡ ❣❡r❛tr✐③✱ q✉❡ ♣❛ss❛ ♣♦r ✉♠ ♣♦♥t♦ V ✭✈ért✐❝❡✮ ❡ ♣❡r❝♦rr❡ ♦s♣♦♥t♦s ❞❡ ✉♠❛ ❧✐♥❤❛ ❞❛❞❛✱ ♥❛ q✉❛❧ ❞❡♥♦♠✐♥❛r❡♠♦s ❞❡ s ✭❞✐r❡tr✐③✮✱ ❝♦♠ V ❢♦r❛ ❞❡ s✳

❆♣r❡s❡♥t❛♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ s✉♣❡r❢í❝✐❡s r❡❣r❛❞❛s ❞❡s❡♥✈♦❧✈í✈❡✐s ❝ô♥✐❝❛s✿

✲ s❡ ❛ ❞✐r❡tr✐③ é ✉♠ r❡t❛✱ ❛ s✉♣❡r❢í❝✐❡ ❣❡r❛❞❛ é ✉♠ ♣❧❛♥♦✱ ♠❡♥♦s ❛ r❡t❛ ♣❛r❛❧❡❧❛ à ❞✐r❡tr✐③✳

✲ s❡ ❛ ❞✐r❡tr✐③ é ✉♠ r❡t❛✱ ❛ s✉♣❡r❢í❝✐❡ ❣❡r❛❞❛ é ✉♠ ♣❧❛♥♦✱ ♠❡♥♦s ❛ r❡t❛ ♣❛r❛❧❡❧❛ à ❞✐r❡tr✐③✳

✲ s❡ ❛ ❞✐r❡tr✐③ é ✉♠ ♣♦❧í❣♦♥♦ ✭❧✐♥❤❛ ♣♦❧✐❣♦♥❛❧ ❢❡❝❤❛❞❛✮ ❝✉❥♦ ♣❧❛♥♦ ♥ã♦ ❝♦♥té♠ ♦ ✈ért✐❝❡V ✱ ❛ s✉♣❡r❢í❝✐❡ ❝ô♥✐❝❛ ❣❡r❛❞❛ é ❛ r❡✉♥✐ã♦ ❞❛s ❞✉❛s s✉♣❡r❢í❝✐❡s ❞❡ â♥❣✉❧♦s ♣♦❧✐é❞r✐❝♦s♦♣♦st❛s ♣❡❧♦ ✈ért✐❝❡ ✭s✉♣❡r❢í❝✐❡s ❞❡ ♣✐râ♠✐❞❡s ✐❧✐♠✐t❛❞❛s✮✳

✲ s❡ ❛ ❞✐r❡tr✐③ é ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❝✉❥♦ ♣❧❛♥♦ ♥ã♦ ❝♦♥té♠ ♦ ✈ért✐❝❡✱ ❛ s✉♣❡r❢í❝✐❡ ❝ô♥✐❝❛❣❡r❛❞❛ é ✉♠❛ s✉♣❡r❢í❝✐❡ ❝ô♥✐❝❛ ❝✐r❝✉❧❛r ✭❞❡ ❞✉❛s ❢♦❧❤❛s✮✳

✸✳✷✳✶ ❈✐❧✐♥❞r♦

❈♦♥s✐❞❡r❡♠♦s ✉♠ ❝ír❝✉❧♦ ✭r❡❣✐ã♦ ❝✐r❝✉❧❛r✮ ❞❡ ❝❡♥tr♦ ♥♦ ♣♦♥t♦ A ❡ r❛✐♦ r✱ q✉❡ ❡st❡❥❛s✐t✉❛❞♦ ❡♠ ✉♠ ♣❧❛♥♦ α✱ ❡ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ BC✱ ♥ã♦ ♥✉❧♦✱ ♥ã♦ ♣❛r❛❧❡❧♦ ❡ ♥ã♦ ❝♦♥t✐❞♦ ♥♦♣❧❛♥♦ α✳ ❉❡♥♦♠✐♥❛♠♦s ❞❡ ❝✐❧✐♥❞r♦ ❝✐r❝✉❧❛r à r❡✉♥✐ã♦ ❞♦s s❡❣♠❡♥t♦s ❝♦♥❣r✉❡♥t❡s ❡ t❛♠❜é♠♣❛r❛❧❡❧♦s ❛♦ s❡❣♠❡♥t♦ BC✱ ❝♦♠ ✉♠❛ ❞❡ s✉❛s ❡①tr❡♠✐❞❛❞❡s ♥♦s ♣♦♥t♦s ❞♦ ❝ír❝✉❧♦ ❡ s✐t✉❛❞♦s❡♠ ✉♠ ♠❡s♠♦ s❡♠✐❡s♣❛ç♦ ❞♦s ❞❡t❡r♠✐♥❛❞♦s ♣♦r α✳

❱❡❥❛ ♥❛ ❋✐❣✉r❛ ✸✳✷✿

❋✐❣✉r❛ ✸✳✷✿ ❈✐❧✐♥❞r♦ ❈✐r❝✉❧❛r

✸✳✸✳ ➪r❡❛s ❡ ✈♦❧✉♠❡s ✶✽

✸✳✷✳✷ ❈♦♥❡s

❈♦♥s✐❞❡r❡♠♦s ✉♠ ❝ír❝✉❧♦ ❞❡ ❝❡♥tr♦ O ❡ r❛✐♦ r s✐t✉❛❞♦ ❡♠ ✉♠ ♣❧❛♥♦ α ❡ ✉♠ ♣♦♥t♦ V

❢♦r❛ ❞❡st❡ ♣❧❛♥♦✳ ❉❡♥♦♠✐♥❛♠♦s ❞❡ ❝♦♥❡ ❝✐r❝✉❧❛r à r❡✉♥✐ã♦ ❞♦s s❡❣♠❡♥t♦s ❞❡ r❡t❛ ❝♦♠ ✉♠❛❡①tr❡♠✐❞❛❞❡ ❡♠ V ❡ ❛ ♦✉tr❛ ♥♦s ♣♦♥t♦s ❞♦ ❝ír❝✉❧♦✳

❱❡❥❛ ❛ ❋✐❣✉r❛ ✸✳✸✿

❋✐❣✉r❛ ✸✳✸✿ ❈♦♥❡ ✲ ❋♦♥t❡✿ ❧♦❛❞✐♥❣✐♥❢♦r♠❛t✐♦♥s✳❜❧♦ ❣s♣♦t✳❝♦♠

✸✳✷✳✸ ❊s❢❡r❛s

❈♦♥s✐❞❡r❡♠♦s ✉♠ ♣♦♥t♦ O ❡ ✉♠ s❡❣♠❡♥t♦ ❞❡ ♠❡❞✐❞❛ r✳ ❈❤❛♠❛✲s❡ ❡s❢❡r❛ ❞❡ ❝❡♥tr♦ O ❡r❛✐♦ r ❛♦ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s P ❞♦ ❡s♣❛ç♦ t❛✐s q✉❡ ❛ ❞✐stâ♥❝✐❛ OP s❡❥❛ ♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛ r✳

P♦❞❡♠♦s ❝♦♥s❡❣✉✐r ✉♠❛ ❡s❢❡r❛✱ r♦t❛❝✐♦♥❛♥❞♦ ✉♠ s❡♠✐❝ír❝✉❧♦ ❡♠ t♦r♥♦ ❞♦ ❡✐①♦ q✉❡ ❝♦♥té♠♦ s❡✉ ❞✐â♠❡tr♦✳

❖❜s❡r✈❡ ❛ ❋✐❣✉r❛ ✸✳✹ ♦♥❞❡ ❛♣r❡s❡♥t❛♠♦s ✉♠❛ ❡s❢❡r❛ ❞❡ r❡✈♦❧✉çã♦✳

❋✐❣✉r❛ ✸✳✹✿ ❊s❢❡r❛ ❞❡ r❡✈♦❧✉çã♦ ✲ ❋♦♥t❡✿ ✇✇✇✳♠✉♥❞♦❡❞✉❝❛❝❛♦✳❝♦♠

✸✳✸ ➪r❡❛s ❡ ✈♦❧✉♠❡s

■♥t✉✐t✐✈❛♠❡♥t❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦ é ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❡s♣❛ç♦ ♣♦r ❡❧❡ ♦❝✉♣❛❞♦✱ ♦✉ t❛♠✲❜é♠ ♦ q✉❛♥t♦ ✏❝❛❜❡✑ ❞❡♥tr♦ ❞❛q✉❡❧❡ só❧✐❞♦✳ P♦❞❡✲s❡ ♦❜s❡r✈❛r ❡st❡ ❢❛t♦ ✐♥t✉✐t✐✈♦ q✉❛♥❞♦ s❡

✸✳✹✳ ❊①❡♠♣❧♦s ❞❡ P♦❧✐❡❞r♦s ✶✾

♦❜s❡r✈❛ ❛❧❣✉♠❛s ❡♠❜❛❧❛❣❡♥s ❞❡ ♣r♦❞✉t♦s ♦❜s❡r✈❛❞♦s ♥♦ ♠❡r❝❛❞♦✳ P❛r❛ ❡①♣r✐♠✐r ❡ss❛ ✏q✉❛♥✲t✐❞❛❞❡ ❞❡ ❡s♣❛ç♦ ♦❝✉♣❛❞♦✑ ❛tr❛✈és ❞❡ ✉♠ ♥ú♠❡r♦✱ ❞❡✈❡♠♦s ❝♦♠♣❛rá✲❧❛ ❝♦♠ ✉♠❛ ✉♥✐❞❛❞❡❀♦ r❡s✉❧t❛❞♦ ❞❡ss❛ ❝♦♠♣❛r❛çã♦ s❡rá ♦ q✉❡ ❡❢❡t✐✈❛♠❡♥t❡ ❝❤❛♠❛r❡♠♦s ❞❡ ✈♦❧✉♠❡✳

❆ ✉♥✐❞❛❞❡ ❞❡ ✈♦❧✉♠❡ ❛❞♦t❛❞❛ s❡rá ♦ ❝✉❜♦ ❞❡ ❛r❡st❛ 1✳ P❛r❛ ❝❛❞❛ t✐♣♦ ❞❡ ✉♥✐❞❛❞❡ ❞❡❝♦♠♣r✐♠❡♥t♦✱ t❡r❡♠♦s ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ✈♦❧✉♠❡ ❝♦rr❡s♣♦♥❞❡♥t❡✳ ❙❡✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ✉♥✐❞❛❞❡❞❡ ❝♦♠♣r✐♠❡♥t♦ ❢♦r ❝❡♥tí♠❡tr♦ ✭❝♠ ✮✱ ❡♥tã♦ ❛ ✉♥✐❞❛❞❡ ❝♦rr❡s♣♦♥❞❡♥t❡ ❞❡ ✈♦❧✉♠❡ s❡rá ♦❝❡♥tí♠❡tr♦ ❝ú❜✐❝♦ (cm3)✳ ❆ss✐♠ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦ S ❞❡✈❡ s❡r ♦ ♥ú♠❡r♦ ❞❡ ✈❡③❡s q✉❡❡ss❡ só❧✐❞♦ ❝♦♥té♠ ♦ ❝✉❜♦ ✉♥✐tár✐♦✳ ▼❛s✱ ❝♦♠♦ ❡ss❡ só❧✐❞♦ ♣♦❞❡ t❡r ✉♠❛ ❢♦r♠❛ ❜❛st❛♥t❡✐rr❡❣✉❧❛r✱ ♥ã♦ ✜❝❛ ❝❧❛r♦ ♦ ♥ú♠❡r♦ ❞❡ ✈❡③❡s q✉❡ ♦ só❧✐❞♦ ❝♦♥té♠ ♦ ❝✉❜♦✳

❆ q✉❡stã♦ ❞❛s ár❡❛s ❛ s❡r❡♠ ❝❛❧❝✉❧❛❞❛s✱ ✈❛♠♦s ♥♦s ❜❛s❡❛r ♥❛ ❣❡♦♠❡tr✐❛ ❞♦ ♣❧❛♥♦✱ ♣♦✐s♣❛r❛ ❡♥❝♦♥tr❛r♠♦s ❛s ár❡❛s ❞❡ s✉♣❡r❢í❝✐❡ ❞♦s ♣♦❧✐❡❞r♦s✱ ✉s❛r❡♠♦s ♦s ❝á❧❝✉❧♦s ❡❢❡t✉❛❞♦s ♣❛r❛♦❜t❡♥çã♦ ❞❛ ár❡❛ ❞❡ ✉♠ ♣♦❧í❣♦♥♦ ♣r❡✈✐❛♠❡♥t❡ ❡st✉❞❛❞♦✱ t❛✐s ❝♦♠♦ tr✐â♥❣✉❧♦s✱ q✉❛❞r✐❧át❡r♦s✱❤❡①á❣♦♥♦s✱ ♦❝tó❣♦♥♦s ❡t❝✳

✸✳✹ ❊①❡♠♣❧♦s ❞❡ P♦❧✐❡❞r♦s

❋✐❣✉r❛ ✸✳✺✿ P♦❧✐❡❞r♦s ✲ ❋♦♥t❡✿ ❤tt♣✿✴✴✇✇✇✳❡❞✉❝✳❢❝✳✉❧✳♣t✴✐❝♠✴✐❝♠✾✾✴✐❝♠✶✻✴❝✉r✐♦s✐❞❛❞❡s✳❤t♠

❉❡♥tr❡ ♦s ❡①❡♠♣❧♦s ❛♣r❡s❡♥t❛❞♦s ❛❝✐♠❛✱ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ♣r✐s♠❛✭❝✉❜♦✮ ❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ ✭t❡tr❛❡❞r♦✮✳ ❆♠❜♦s ♦s ♣♦❧✐❡❞r♦s ❝✐t❛❞♦s ❛♥t❡r✐♦r♠❡♥t❡ sã♦ ❝♦♥s✐✲❞❡r❛❞♦s r❡❣✉❧❛r❡s✳

❈❛♣ít✉❧♦ ✹

❆ ♥♦çã♦ ✐♥t✉✐t✐✈❛ ❛ r❡s♣❡✐t♦ ❞❡ ✈♦❧✉♠❡

❞❡ ✉♠ só❧✐❞♦ ❡ ✈♦❧✉♠❡ ❞❡ ✉♠

♣❛r❛❧❡❧❡♣í♣❡❞♦

❖ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦ é ♦ q✉❡ ❡❧❡ ♦❝✉♣❛ ♥♦ ❡s♣❛ç♦ ❡♠ q✉❡ ❡stá ✐♥s❡r✐❞♦✱ ♦✉ s❡❥❛✱ é♦ q✉❛♥t♦ ❡❧❡ ❡stá ♣r❡❡♥❝❤❡♥❞♦ ❞♦ ❡s♣❛ç♦ ❛♥❛❧✐s❛❞♦✳ ❏á ❢♦✐ ❞✐t♦ ❛♥t❡r✐♦r♠❡♥t❡ ❛ ❞✐❢❡r❡♥ç❛❡♥tr❡ ♦ ✈♦❧✉♠❡ ❡ ❝❛♣❛❝✐❞❛❞❡✱ r❡ss❛❧t❛♥❞♦ q✉❡ ✈❛♠♦s ♥♦s ❝♦♥❝❡♥tr❛r ❝♦♠ ❛✜♥❝♦ ♥♦ ❝á❧❝✉❧♦❞❡ ✈♦❧✉♠❡✳ ❈♦♥s✐❞❡r❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ q✉❡ t❡♥❤❛ ❛s s✉❛s três ❛r❡st❛s ✐❣✉❛✐s ❡ ♠❡❞✐♥❞♦❡①❛t❛♠❡♥t❡ ✶ ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦✱ ❝♦♠♦ ❛♣r❡s❡♥t❛❞♦ ♥❛ ❋✐❣✉r❛ ✹✳✶✿

❋✐❣✉r❛ ✹✳✶✿ P❛r❛❧❡❧❡♣í♣❡❞♦ ❝♦♠ ❛r❡st❛s ✐❣✉❛✐s

❖ q✉❡ ♥❛ ✈❡r❞❛❞❡ s❡rá ✉♠ só❧✐❞♦ ♦ q✉❛❧ ❞❡♥♦♠✐♥❛♠♦s ❞❡ ❝✉❜♦✳

P♦❞❡♠♦s ❡♥tã♦ ❛✜r♠❛r q✉❡ ❡st❡ ❝✉❜♦ s❡rá ❛ ✉♥✐❞❛❞❡ ❞❡ ✈♦❧✉♠❡✱ ♦✉ s❡❥❛✱ s❡ t✐✈❡r♠♦s ✉♠só❧✐❞♦ q✉❡ é ❢♦r♠❛❞♦ ♣♦r n ✏❝✉❜✐♥❤♦s✑ ✐❣✉❛✐s ❛ ❡st❡✱ ❞✐r❡♠♦s q✉❡ ❡st❡ só❧✐❞♦ t❡rá ✉♠ ✈♦❧✉♠❡n✳ ❙❡ ❛✉♠❡♥t❛r♠♦s ✉♠❛ ❞❛s ❛r❡st❛s ❞❡st❡ ❝✉❜♦ ✉♥✐tár✐♦✱ ❝❤❛♠❛♥❞♦ ❞❡ a✱ ❡ ✐s♦❧❛♥❞♦ ❛s♦✉tr❛s ❞✉❛s ❛r❡st❛s✱ t❡r❡♠♦s ❡♥tã♦ q✉❡ ♦ ♥♦✈♦ ✈♦❧✉♠❡ s❡rá a✱ ❛ss✐♠ r❡♣r❡s❡♥t❛❞♦ ♥❛ ❋✐❣✉r❛✹✳✷✿

✷✵

✷✶

❋✐❣✉r❛ ✹✳✷✿ P❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ❛r❡st❛s 1✱ 1 ❡ a

❖❜s❡r✈❛♥❞♦ ❡♥tã♦ q✉❡ ♦ ♥♦✈♦ ✈♦❧✉♠❡ s❡rá 1 · a

❆❣♦r❛✱ s❡ ❡st❡ ♣❛r❛❧❡❧❡♣í♣❡❞♦ t✐✈❡r ✉♠❛ ❞❡ s✉❛s ❛r❡st❛s ✉♥✐tár✐❛s ❛✉♠❡♥t❛❞❛ ♣❛r❛ b✱ ❡ ❛s♦✉tr❛s ❞✉❛s ❛r❡st❛s ✜①❛❞❛s✱ t❡r❡♠♦s ❡♥tã♦ ♦ s❡❣✉✐♥t❡ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❛♣r❡s❡♥t❛❞♦ ♥❛ ❋✐❣✉r❛✹✳✸✿

❋✐❣✉r❛ ✹✳✸✿ P❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ❛r❡st❛s a✱ 1 ❡ b

◆♦✈❛♠❡♥t❡✱ ❢❛③❡♥❞♦ ✉♠❛ ♦❜s❡r✈❛çã♦ ❛ r❡s♣❡✐t♦ ❞❡st❡ ♥♦✈♦ só❧✐❞♦✱ ♦❜s❡r✈❛♠♦s q✉❡ ♦ s❡✉✈♦❧✉♠❡ s❡rá 1 · a · b

❚♦♠❛♥❞♦ ❛ ❛r❡st❛ ✉♥✐tár✐❛ r❡st❛♥t❡ ❡ ♠✉❧t✐♣❧✐❝❛♥❞♦ ✲ ❛ ♣♦r C ❡ ✐s♦❧❛♥❞♦ ❛s ♦✉tr❛s ❞✉❛s❛r❡st❛s✱ ✈❛♠♦s ✜♥❛❧♠❡♥t❡ ♦❜t❡r ♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❛♣r❡s❡♥t❛❞♦ ♥❛ ❋✐❣✉r❛ ✹✳✹✿

❋✐❣✉r❛ ✹✳✹✿ P❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ❛r❡st❛s a✱ b ❡ c

❆ss✐♠ ❡♥tã♦ ♦❜s❡r✈❛♥❞♦ ♦ ✈♦❧✉♠❡ ❞❡st❡ ♣❛r❛❧❡❧❡♣í♣❡❞♦✱ ✈❛♠♦s ♦❜t❡r a·b·c ❡ ❝♦♠♣r♦✈❛♥❞♦q✉❡ ❡♠ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ❞✐♠❡♥sõ❡s a✱ b ❡ c✱ ♦ s❡✉ ✈♦❧✉♠❡ s❡rá ♦❜t✐❞♦ ❛tr❛✈és ❞♦ ♣r♦❞✉t♦❞❡st❛s três ❞✐♠❡♥sõ❡s✳

❆ s✐t✉❛çã♦ ❞♦ ✈♦❧✉♠❡ ❛♣r❡s❡♥t❛❞♦ ❛❝✐♠❛✱ ♣♦❞❡rá s❡r r❡s✉♠✐❞❛ ❛tr❛✈és ❞❛ s✐t✉❛çã♦ ❛♣r❡✲s❡♥t❛❞❛ ♥❛ ❋✐❣✉r❛ ✹✳✺✿

✷✷

❋✐❣✉r❛ ✹✳✺✿ P❛r❛❧❡❧❡♣í♣❡❞♦ ❞❡ ❛r❡st❛s a✱ b ❡ c ✲ ❋♦♥t❡✿ ❛♥♦ss❛❡s❝♦❧❛✳❝♦♠

❈❛❞❛ ✉♠ ❞♦s ♣❡q✉❡♥♦s ♣❛r❛❧❡❧❡♣í♣❡❞♦s q✉❡ ❢♦r♠❛ ❛ ✜❣✉r❛ ♠❛✐♦r✱ é ✉♠ ❞♦s q✉❡ ❞❡♠♦♥s✲tr❛♠♦s ❛♥t❡r✐♦r♠❡♥t❡✳

❈♦♥s✐❞❡r❛♥❞♦ ❡♥tã♦ q✉❡ ❡♠ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ r❡tâ♥❣✉❧♦✱ t❡r❡♠♦s ❛ ❜❛s❡ s❡♥❞♦ ✉♠ r❡tâ♥✲❣✉❧♦ ❞❡ ❞✐♠❡♥sõ❡s ✐❣✉❛✐s a✱ b ❡ c✱ ♣♦❞❡rí❛♠♦s ❡♥tã♦ ❞✐③❡r q✉❡ ♦ ✈♦❧✉♠❡ ❞❡st❡ ♣❛r❛❧❡❧❡♣í♣❡❞♦s❡rá ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞❛ ár❡❛ ❞❛ ❜❛s❡ ♣❡❧❛ ❛r❡st❛ b✱ q✉❡ s❡r✐❛ ❡♥tã♦ ❛ ❛❧t✉r❛ ❞♦ só❧✐❞♦ ❡❛ss✐♠✿

VPARALELEPIPEDO = AREADABASE · ALTURA

❯♠❛ ♦❜s❡r✈❛çã♦ ❛ s❡ ❝✐t❛r✱ ❝♦♠♦ ❛♣r❡s❡♥t❛❞♦ ❡♠ ❬✻❪✱ é ❛ ❞❡ q✉❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ é❝❤❛♠❛❞♦ ❞❡ r♦♠❜♦❡❞r♦ q✉❛♥❞♦ ♣♦ss✉✐ ❛s ❞♦③❡ ❛r❡st❛s ✐❣✉❛✐s ❡♥tr❡ s✐ ❡ ❛ss✐♠ s❡♥❞♦✱ ♣♦❞❡♠♦s❝❤❛♠❛r ✉♠ ❝✉❜♦ ❞❡ r♦♠❜♦❡❞r♦✳ ❱❛❧❡ s❛❧✐❡♥t❛r q✉❡ ♥❡♠ t♦❞♦ r♦♠❜♦❡❞r♦ é ♥❡❝❡ss❛r✐❛♠❡♥t❡✉♠ ❝✉❜♦✳ ❆♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ✐♥❞✐❝❛❞♦s ♥❛ ❋✐❣✉r❛ ✹✳✻✿

❋✐❣✉r❛ ✹✳✻✿ ❘♦♠❜♦❡❞r♦s ✲ ❋♦♥t❡✿ ✇✇✇✳♣♦rt❛❧❡s❝♦❧❛r✳♥❡t

❖✉tr❛ ✐♠♣♦rt❛♥t❡ ✐♥❢♦r♠❛çã♦ ❛ s❡r ❛❜♦r❞❛❞❛ t❛♠❜é♠✱ é ♦ ❢❛t♦ ❞❡ q✉❡ ♦ ❝✉❜♦ é ✉♠ ❝❛s♦❡s♣❛❝✐❛❧ ❞❡ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❡ ♣♦rt❛♥t♦ ♣♦❞❡rá t❛♠❜é♠ s❡r ❝❛❧❝✉❧❛❞♦ ♦ s❡✉ ✈♦❧✉♠❡ ✉s❛♥❞♦ ♦♣r♦❞✉t♦ ❞❡ s✉❛s três ❛r❡st❛s✱ ♣♦ré♠ ✈❛❧❡ s❛❧✐❡♥t❛r q✉❡ ❡♠ ❝✉❜♦ ✭r♦♠❜♦❡❞r♦ r❡t♦r❡tâ♥❣✉❧♦✮✱❛s ❛r❡st❛s s❡rã♦ t♦❞❛s ❝♦♥❣r✉❡♥t❡s ❡ ❛ss✐♠ s❡♥❞♦ ♦ s❡✉ ✈♦❧✉♠❡ s❡rá ❝❛❧❝✉❧❛❞♦ ❞❛ s❡❣✉✐♥t❡♠❛♥❡✐r❛✿

VCUBO = a3 ❡♠ q✉❡ a é ❛r❡st❛ ❞♦ ❝✉❜♦✳

❈♦♥s✐❞❡r❛r❡♠♦s ❛❣♦r❛ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ♦❜❧íq✉♦ ❡ ❛ss✐♠ ✈❡r✐✜❝❛r❡♠♦s q✉❡ ♦ s❡✉ ✈♦❧✉♠❡

✷✸

t❛♠❜é♠ ♣♦❞❡rá s❡r ❝❛❧❝✉❧❛❞♦ ♣❡❧❛ ❢ór♠✉❧❛ ❝✐t❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡✳ ❖❜s❡r✈❡ ❡♥tã♦ ♦ ♣❛r❛❧❡❧❡✲♣í♣❡❞♦ ❞❛ ❋✐❣✉r❛ ✹✳✼✿

❋✐❣✉r❛ ✹✳✼✿ P❛r❛❧❡❧❡♣í♣❡❞♦ ✲ ❋♦♥t❡✿ ♠❛t❡♠❛t✐❝❛❝♦❡s✳❜❧♦❣s♣♦t✳❝♦♠

❱❡r✐✜q✉❡♠♦s q✉❡ ♥♦ ♣r✐s♠❛ ✭♣❛r❛❧❡❧❡♣í♣❡❞♦✮ ❞❛ ❋✐❣✉r❛ ✹✳✼✱ t❡♠♦s ✉♠❛ ❜❛s❡ q✉❡ é ✉♠q✉❛❞r✐❧át❡r♦ ❡ ❛s ❛r❡st❛s ❧❛t❡r❛✐s q✉❡ ♥ã♦ ❝♦✐♥❝✐❞❡♠ ❝♦♠ ❛ ❛❧t✉r❛ ❞♦ só❧✐❞♦✱ ♣♦ré♠ ♥❡st❛s✐t✉❛çã♦ t❡♠♦s q✉❡ ❡st❛s ❛r❡st❛s ❢♦r♠❛♠ ✉♠ â♥❣✉❧♦ α ❝♦♠ ♦ ♣❧❛♥♦ ❞❛ ❜❛s❡✱ s❡♥❞♦ ♣♦rt❛♥t♦♠❛✐♦r q✉❡ ❛ ❛❧t✉r❛✳

❈❛♣ít✉❧♦ ✺

❖ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐

❖ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ é ❛ss✉♠✐❞♦ ❝♦♠♦ ✉♠ ❛①✐♦♠❛ ❡ ♣♦r ✐ss♦ ♥ã♦ s❡ ✈ê ❛ ♥❡❝❡ss✐✲❞❛❞❡ ❞❡ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❢♦r♠❛❧ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❡❧♦ ❢❛t♦ ❞❡ s❡ tr❛t❛r ❞❡ ✉♠ ❛ss✉♥t♦ ❛s❡r ❛❜♦r❞❛❞♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✳ ❊♠ ♥í✈❡❧ ❞❡ ❊♥s✐♥♦ ▼é❞✐♦ s❡♠♣r❡ ✉s❛♠♦s ♦ ♣r✐♥❝í♣✐♦ r❡✲❢❡r✐❞♦ ❛♥t❡r✐♦r♠❡♥t❡ ❝♦♠♦ ✉♠❛ ✏❛✜r♠❛çã♦✑ q✉❡ ♥ã♦ ♥❡❝❡ss✐t❛ ❞❡ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❡ ♣♦r✐ss♦ ♠❡s♠♦ ❝♦♥s✐❞❡r❛♠♦s ♦ ♠❡s♠♦ ❝♦♠♦ ✉♠ ❛rt✐❢í❝✐♦ ♣rát✐❝♦ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ✈ár✐♦s♣r♦❜❧❡♠❛s q✉❡ ❡♥✈♦❧✈❛♠ ✈♦❧✉♠❡ ❞❡ só❧✐❞♦s✱ ✈✐st♦ q✉❡ ❛ ❣r❛♥❞❡ ♠❛✐♦r✐❛ ❞♦s ❛❧✉♥♦s ♥ã♦ t❡♠✉♠❛ ❢❛♠✐❧✐❛r✐❞❛❞❡ ❝♦♠ ❞❡♠♦♥str❛çõ❡s✳ ❱❛❧❡ ❧❡♠❜r❛r q✉❡ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡❈❛✈❛❧✐❡r✐ r❡q✉❡r ❝♦♥❝❡✐t♦s ❛✈❛♥ç❛❞♦s ❞❡ ❚❡♦r✐❛ ❞❛ ▼❡❞✐❞❛ ❡ ❞❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥❝✐❛❧ ❡ ■♥t❡❣r❛❧✱✐♥❢♦r♠❛çõ❡s ❡ ❝♦♥❝❡✐t♦s ♥ã♦ ❛❜♦r❞❛❞♦s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❛ss✐♠✱ ♠❛✐s ✉♠❛ ✈❡③✱ ❥✉st✐✜❝❛✲s❡ ❛s✉❛ ❛♣r❡s❡♥t❛çã♦ ❝♦♠♦ ✉♠ ❛①✐♦♠❛✳

❋❛③ ✲ s❡ ♥❡❝❡ssár✐♦ ❡♥tã♦ ❝❤❛♠❛r ❛ ❛t❡♥çã♦ ❞♦s ❛❧✉♥♦s ♣❛r❛ q✉❡ ✜q✉❡♠ ❛t❡♥t♦s q✉❛♥❞♦❞❛ ✉t✐❧✐③❛çã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✱ ❛♣r❡s❡♥t❛❞♦ ❛q✉✐ ❝♦♠♦ ✉♠ ❛①✐♦♠❛✳ ❖s ❡①❡♠♣❧♦s❛♣r❡s❡♥t❛❞♦s ♥❛ s❡çã♦ ✺✳✶ ♥ã♦ ❝♦♥st✐t✉✐rã♦ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞♦ ♣r✐♥❝í♣✐♦✱ ♠❛s s✐♠ ✉♠❛❜♦❛ ✐♥❞✐❝❛çã♦ ❞❡ q✉❡ ♦ ♠❡s♠♦ é ✈❡r❞❛❞❡✐r♦✳

✺✳✶ Pr♦❝❡❞✐♠❡♥t♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐

Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ✜❣✉r❛s ♣❧❛♥❛s ✲ ❡♥❝♦♥tr♦ ❞❡ ár❡❛s

◆♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♠✉✐t❛s ❞❛s ✈❡③❡s é ❛♣r❡s❡♥t❛❞♦ ❛♣❡♥❛s ♣❛r❛♦ ❝á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡s ❞❡ só❧✐❞♦s ❡ ❛ss✐♠ ❡♥tã♦ s❡ t❡♠ ❛ ♥♦çã♦ ❞❡ ✉s♦ ❛♣❡♥❛s ❡♠ ❢♦r♠❛str✐❞✐♠❡♥s✐♦♥❛✐s✱ ♣♦ré♠✱ ♣♦❞❡♠♦s ❛♣❧✐❝❛r ♦ r❡❢❡r✐❞♦ ♣r✐♥❝í♣✐♦ ♥♦ ♣❧❛♥♦✱ ♦✉ s❡❥❛✱ ♣❛r❛ ♦ tr❛✲❜❛❧❤♦ ❝♦♠♦ ❢♦r♠❛s ❜✐❞✐♠❡♥s✐♦♥❛✐s✳ ❱❛♠♦s ❡♥tã♦ ❛❞♠✐t✐r ❛❣♦r❛ q✉❡ t❡♥❤❛♠♦s ✉♠❛ r❡❣✐ã♦❧✐♠✐t❛❞❛ ❞♦ ♣❧❛♥♦ q✉❡ ♣♦ss❛ s❡r ❢❛t✐❛❞❛ ❞❡ t❛❧ ♠❛♥❡✐r❛ q✉❡ ♦❜s❡r✈❡♠♦s ✉♠❛ ✐♥✜♥✐❞❛❞❡ ❞❡

✷✹

✺✳✶✳ Pr♦❝❡❞✐♠❡♥t♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ✷✺

s❡❣♠❡♥t♦s ✏❝♦rt❛♥❞♦✑ ❡st❛ r❡❣✐ã♦ ❧✐♠✐t❛❞❛ ❞♦ ♣❧❛♥♦✳ ❆ r❡✉♥✐ã♦ ❞❡st❡s ✐♥✜♥✐t♦s s❡❣♠❡♥t♦s✭❞❡♣❡♥❞❡♥❞♦ ❞❛ ❢♦r♠❛ ❞❛ r❡❣✐ã♦✮ s❡rá ❝❛r❛❝t❡r✐③❛❞❛ ❝♦♠♦ ❛ ár❡❛ ❞❡st❛ r❡❣✐ã♦✳ ❖❜✈✐❛♠❡♥t❡q✉❛♥❞♦ ❝❛r❛❝t❡r✐③❛♠♦s ❞✉❛s r❡❣✐õ❡s ❞✐st✐♥t❛s ❞♦ ♣❧❛♥♦✱ q✉❡ t❡♥❤❛♠ ♠❡s♠❛s ár❡❛s✱ ❛ r❛③ã♦❡♥tr❡ s❡❣♠❡♥t♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❞❡✈❡rá s❡r 1 ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦✳ ❱❡❥❛♠♦s ❛ s❡❣✉✐r✉♠❛ ✐❧✉str❛çã♦ ♥❛ ❋✐❣✉r❛ ✺✳✶✿

❋✐❣✉r❛ ✺✳✶✿ ❈á❧❝✉❧♦ ❞❡ ➪r❡❛s ✲ ❋♦♥t❡✿ ♠❛✉r♦✇❡✐❣❡❧✳❜❧♦❣s♣♦t✳❝♦♠

❆s ár❡❛s s❡rã♦ ✐❣✉❛✐s✱ ♣♦✐s ❛s r❡t❛s s ❡ t ❡stã♦ ❝♦♠ s❡❣♠❡♥t♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❝♦♠ ♠❡s♠♦❝♦♠♣r✐♠❡♥t♦✱ ♦✉ s❡❥❛✱ ❛s r❛③õ❡s ❡♥tr❡ ♦s s❡❣♠❡♥t♦s s❡rã♦ ✐❣✉❛✐s ❛ 1✳

❱❡❥❛♠♦s ❛❣♦r❛ ❛ ❋✐❣✉r❛ ✺✳✷ ♦♥❞❡ ♣♦❞❡♠♦s ❛♣r❡s❡♥t❛r ♦ ♣r✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ❛♥á❧✐s❡❡ ❝♦♠♣❛r❛çã♦ ❞❡ ár❡❛s ❞❡ ✜❣✉r❛s ♣❧❛♥❛s✿

❋✐❣✉r❛ ✺✳✷✿ ❋✐❣✉r❛s P❧❛♥❛s ✲ ❋♦♥t❡✿ ▼❛t❡♠át✐❝❛ ▼✉❧t✐♠í❞✐❛✱ ●✉✐❛ ❞♦ Pr♦❢❡ss♦r✱ ●❡♦♠❡tr✐❛❡ ▼❡❞✐❞❛s✱ ❯♥✐❝❛♠♣

❊♥tã♦ ♠❛✐s ✉♠❛ ✈❡③✱ ✈❡r✐✜❝❛♠♦s q✉❡ ❞✉❛s r❡❣✐õ❡s ♣❧❛♥❛s t❡rã♦ ♠❡s♠❛ ár❡❛ s❡ ♦s s❡❣✲♠❡♥t♦s ♦❜t✐❞♦s ❡♠ ❝❛❞❛ ✉♠❛ ❞❛s s❡❝çõ❡s tr❛♥s✈❡rs❛✐s t✐✈❡r❡♠ ♦ ♠❡s♠♦ ❝♦♠♣r✐♠❡♥t♦✳

❖ ✉s♦ ❞♦ ♣r✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ✜❣✉r❛s ❜✐❞✐♠❡♥s✐♦♥❛✐s é ❜❡♠ ♠❡♥♦s ✉s❛❞♦ ♣❛r❛ ♦s❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❡ ❛ s✉❛ ❛♣r❡s❡♥t❛çã♦ ♥❡♠ s❡♠♣r❡ é ❢❡✐t❛ ❞✉r❛♥t❡ ❡st❡ ♥í✈❡❧ ❞❡ ❡♥s✐♥♦✳❖ ✐♠♣♦rt❛♥t❡ é s❛❧✐❡♥t❛r q✉❡ ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s t❡♠♦s ♦✉tr♦s r❡❝✉rs♦s ♠❛✐s s✐♠♣❧❡s ❡q✉❡ ♣♦❞❡♠ s❡r ❛♣r❡s❡♥t❛❞♦s s❡♠ ❝❛✉s❛r ♥❡♥❤✉♠ tr❛♥st♦r♥♦✳

Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ♦ ❈á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡s✿

✺✳✶✳ Pr♦❝❡❞✐♠❡♥t♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ✷✻

❱❛♠♦s ❛❞♠✐t✐r ❡♥tã♦ q✉❡ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ s✐♠♣❧❡s✱ t❡♥❤❛♠♦s ✉♠ ❜❧♦❝♦ ❞❡ ❢♦r♠❛ r❡t❛♥✲❣✉❧❛r✱ ♠❛✐s ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❡ q✉❡ ❡st❡ ♣♦ss❛ s❡r ❞✐✈✐❞✐❞♦ ❡♠ ✏❢♦❧❤❛s✑ ❝❛❞❛✈❡③ ♠❛✐s ✜♥❛s✱ q✉❛s❡ tr❛♥s♣❛r❡♥t❡s✱ ❡ q✉❡ ❡st❛s ❢♦❧❤❛s ♣♦ss❛♠ s❡r s♦❜r❡♣♦st❛s✱ ❢♦r♠❛♥❞♦ ♦♣ró♣r✐♦ só❧✐❞♦✳ ❊st❛s ❢♦❧❤❛s✱ ❣❡♦♠❡tr✐❝❛♠❡♥t❡ ❢❛❧❛♥❞♦✱ s❡r✐❛♠ s❡❝çõ❡s ♣❧❛♥❛s q✉❡ sã♦ ❞✐❢❡r❡♥✲❝✐❛❞❛s ♣♦r ✉♠ ✈❛❧♦r ❜❡♠ ♣❡q✉❡♥♦✱ ✉♠❛ ♠❡❞✐❞❛ ❜❡♠ ♣❡q✉❡♥❛ ❡ ❛ss✐♠ s❡♥❞♦✱ ♦ ✈♦❧✉♠❡ ❞❡st❡só❧✐❞♦ s❡rá ❛ r❡✉♥✐ã♦ ❞❡ t♦❞❛s ❛s ✏❢♦❧❤❛s✑ ♦✉ s❡❝çõ❡s ❝✐t❛❞❛s ❛♥t❡r✐♦r♠❡♥t❡ ❡ ✐♥❞❡♣❡♥❞❡♥t❡✲♠❡♥t❡ ❞❛ ♣♦s✐çã♦ ❞❡st❛s ❢♦❧❤❛s ♦ ✈♦❧✉♠❡ s❡rá ♦ ♠❡s♠♦ ❡ ❛ss✐♠ ♣♦❞❡rí❛♠♦s ♣❡♥s❛r ❡♠ t❡r✈ár✐❛s ❢♦r♠❛s ♦✉ ✈ár✐♦s ✏❡st✐❧♦s✑ ♣❛r❛ ❡st❡ ❜❧♦❝♦✱ ♥ã♦ ❛❧t❡r❛♥❞♦ ♦ s❡✉ ✈♦❧✉♠❡✳ ❘❛❝✐♦❝✐♥❛♥❞♦❞❡st❛ ♠❛♥❡✐r❛✱ ✉♠❛ ❝♦♥❝❧✉sã♦ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ é ❛ ❞❡ q✉❡ ♣♦❞❡♠♦s t❡r ❞♦✐s só❧✐❞♦s ❝♦♠♠❡s♠❛ ❛❧t✉r❛ ❡ q✉❡ ♣♦ss❛♠♦s ❣❡r❛r s❡❝çõ❡s ❝♦♠ ❢♦r♠❛s ❞✐❢❡r❡♥t❡s✱ ♠❛s ❝♦♠ ♠❡s♠♦ ✈❛❧♦r♥✉♠ér✐❝♦ ❞❡ ár❡❛ ❞❡ ♠♦❞♦ q✉❡ ♦s s❡✉s ✈♦❧✉♠❡s s❡rã♦ ✐❣✉❛✐s✱ ✐st♦ q✉❡r ❞✐③❡r q✉❡ ❝♦♠♣❛r❛♥❞♦❞♦✐s só❧✐❞♦s ❝♦♠ ♠❡s♠♦ ✈♦❧✉♠❡✱ ❡❧❡s t❡rã♦ ♠❡s♠❛ ❛❧t✉r❛ ❡ ár❡❛s ❞❡ s❡❝çõ❡s ❝♦rr❡s♣♦♥❞❡♥t❡ss❡♥❞♦ ❛s ♠❡s♠❛s✳ ❱❡❥❛ ❛ ❋✐❣✉r❛ ✺✳✸✱ ♦♥❞❡ ♣♦❞❡♠♦s ❝♦♠♣❛r❛r ❞♦✐s só❧✐❞♦s ✲ ✉♠ ❝✐❧✐♥❞r♦ ❡✉♠ ♣r✐s♠❛ ❝♦♠ ❜❛s❡ s❡♥❞♦ ✉♠ q✉❛❞r✐❧át❡r♦✿

❋✐❣✉r❛ ✺✳✸✿ ❙ó❧✐❞♦s ✲ ❋♦♥t❡✿♦❜❛r✐❝❡♥tr♦❞❛♠❡♥t❡✳❜❧♦❣s♣♦t✳❝♦♠

❯♠❛ ♦✉tr❛ s✐t✉❛çã♦ ❜❡♠ ♣rát✐❝❛ ♦♥❞❡ ♣♦❞❡rí❛♠♦s ✈❡r✐✜❝❛r ❛♣❧✐❝❛çã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡❈❛✈❛❧✐❡r✐✱ s❡r✐❛ ❛q✉❡❧❛ ♥❛ q✉❛❧ ❛❣r✉♣❛♠♦s ✈ár✐❛s ♠♦❡❞❛s ❋✐❣✉r❛ ✺✳✹✱ ❡ ❢♦r♠❛♠♦s ✉♠ só❧✐❞♦r❡❞♦♥❞♦ ✭✉♠ ❝✐❧✐♥❞r♦ ❝✐r❝✉❧❛r✮✳ ❆♦ ♠♦✈❡r♠♦s ❡st❛s ♠♦❡❞❛s✱ s❡♠ ❞✐♠✐♥✉✐r ❛ q✉❛♥t✐❞❛❞❡ ❡s❡♠ ❛❧t❡r❛r♠♦s ❛ s✉❛ ❛♥❣✉❧❛çã♦✱ t❡r❡♠♦s ❛ ♦❜s❡r✈❛çã♦ ❞❡ ✈ár✐♦s ♦✉tr♦s só❧✐❞♦s ❞✐❢❡r❡♥t❡s ❞❡✉♠ ❝✐❧✐♥❞r♦ ✱ ♠❛s q✉❡ ❛♦ ✜♥❛❧✱ t❡rã♦ ♦ ♠❡s♠♦ ✈♦❧✉♠❡✱ ♣♦✐s ❡st❛♠♦s ✏tr❛❜❛❧❤❛♥❞♦✑ s❡♠♣r❡❝♦♠ ❛ ♠❡s♠❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♠♦❡❞❛s✳

❋✐❣✉r❛ ✺✳✹✿ ▼♦❡❞❛s r❡♣r❡s❡♥t❛♥❞♦ s❡❝çõ❡s✳ ❋♦♥t❡✿ ♦❜❛r✐❝❡♥tr♦❞❛♠❡♥t❡✳❜❧♦❣s♣♦t✳❝♦♠

❆s ♠♦❡❞❛s ❡stã♦ ❢✉♥❝✐♦♥❛♥❞♦ ❝♦♠♦ ❛s ❢♦❧❤❛s ❛♣r❡s❡♥t❛❞❛s ❛♥t❡r✐♦r♠❡♥t❡ ❡ ♦ só❧✐❞♦✱ ❛r❡✉♥✐ã♦ ❞❡st❛s ♠♦❡❞❛s✳

❆ ❛♣r❡s❡♥t❛çã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✱ ❢♦✐ ❢❡✐t❛ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞❡ ✈♦❧✉♠❡s ❞❡ só❧✐❞♦s❣❡♦♠étr✐❝♦s ❡ ❡st❛ s❡♥❞♦ r❡❧❛❝✐♦♥❛❞❛ ❛tr❛✈és ❞❡ só❧✐❞♦s ❝♦♠ ❝❛r❛❝t❡ríst✐❝❛s ❞❡ ♣r✐s♠❛s✱ ♣♦ré♠

✺✳✶✳ Pr♦❝❡❞✐♠❡♥t♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ✷✼

♣♦❞❡♠♦s ❡st❡♥❞❡r ♦ t❛❧ ♣r✐♥❝í♣✐♦ ♣❛r❛ s❡r ❛♥❛❧✐s❛❞♦ ❡ ❡①❡♠♣❧✐✜❝❛❞♦ ❝♦♠ ♣✐râ♠✐❞❡s ❡ t❛♠❜é♠só❧✐❞♦s ❞❡ s✉♣❡r❢í❝✐❡s ♥ã♦ ♣❧❛♥❛s✱ t❛✐s ❝♦♠♦✱ ♦s ❝✐❧✐♥❞r♦s✱ ❝♦♥❡s ❡ ❡s❢❡r❛s✱ q✉❡ sã♦ ❡st✉❞❛❞♦s♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ✐♥❝❧✉s✐✈❡ ❛♣r❡s❡♥t❛r❡♠♦s ❛ ❞❡❞✉çã♦ ❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛✱ ✉s❛♥❞♦ ♦Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳ ❊st❛ ❛♣r❡s❡♥t❛çã♦ é ❢❡✐t❛ ❞❡ ♠❛♥❡✐r❛ q✉❡ ♣♦ss❛♠♦s ❛♥❛❧✐s❛r s❡çõ❡s♣❧❛♥❛s ❝♦♠ ♠❡s♠❛s ár❡❛s✳ ❱❡❥❛♠♦s ❛ s❡❣✉✐r✱ ❋✐❣✉r❛s ✺✳✺ ❡ ✺✳✻✱ ❛❧❣✉♥s ❡①❡♠♣❧♦s q✉❡ ❝✐t❛♠♦s❛♥t❡r✐♦r♠❡♥t❡✳

❋✐❣✉r❛ ✺✳✺✿ Pr♦❝❡❞✐♠❡♥t♦ ❞❡ ❈❛✈❛❧✐❡r✐ ■ ✲ ❋♦♥t❡✿ ♦❜❛r✐❝❡♥tr♦❞❛♠❡♥t❡✳❜❧♦❣s♣♦t✳❝♦♠

❋✐❣✉r❛ ✺✳✻✿ Pr♦❝❡❞✐♠❡♥t♦ ❞❡ ❈❛✈❛❧✐❡r✐ ■■ ✲ ❋♦♥t❡✿ ♦❜❛r✐❝❡♥tr♦❞❛♠❡♥t❡✳❜❧♦❣s♣♦t✳❝♦♠

❈❛♣ít✉❧♦ ✻

❯s♦ ❞❡ r❡❝✉rs♦s ❡❧❡trô♥✐❝♦s ❡ s♦❢t✇❛r❡s

♥♦ ❡♠♣r❡❣♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐

❉❡✈✐❞♦ ❛♦ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ❞♦s ❥♦✈❡♥s ♣❡❧❛ ✐♥❢♦r♠át✐❝❛✱ ♣♦❞❡♠♦s ❢❛③❡r ✉s♦ ❞❡st❡ ✐♥t❡✲r❡ss❡ ❛ ♥♦ss♦ ❢❛✈♦r✱ ✉s❛♥❞♦ ♦s ♠❡s♠♦s ❝♦♠♦ ❛❧✐❛❞♦s ♣❛r❛ ♦ ❡♥s✐♥♦ ❞❡ ❝♦♥t❡ú❞♦s ❞❡ ♠♦❞♦q✉❡ s♦❢t✇❛r❡s sã♦ ✉s❛❞♦s ❝♦♠♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ ❡ss❡♥❝✐❛❧ ♣❛r❛ ❞❡s♣❡rt❛r ♦ ✐♥t❡r❡ss❡ ❡ ❛ ❝r✐✲❛t✐✈✐❞❛❞❡ ❞♦s ❛❧✉♥♦s ♣❛r❛ ❛ ❛♣❧✐❝❛çã♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❞♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❞❡ ✜❣✉r❛s ♣❧❛♥❛s✱❞❛ ♦❜s❡r✈❛çã♦ ❡ ❝♦♥str✉çã♦ ❞❡ ♣♦❧í❣♦♥♦s r❡❣✉❧❛r❡s ❡ ❝♦♥✈❡①♦s✱ ❛❧é♠ ❞❛ ❝♦♥str✉çã♦ ❡ ❛♥á❧✐s❡s❞❡ ♣❛r❛❧❡❧✐s♠♦s ❡ ♣❡r♣❡♥❞✐❝✉❧❛r✐s♠♦s✳ ❚❡♠♦s q✉❡ s❛❧✐❡♥t❛r q✉❡ ✉♠❛ ❞❛s r❡❝♦♠❡♥❞❛çõ❡s ❞♦P❈◆ é ❛ ✉t✐❧✐③❛çã♦ ❞❡ s♦❢t✇❛r❡s ♣❛r❛ ❛ ♠❡❧❤♦r ❛♥á❧✐s❡ ❡ ❝♦♠♣r❡❡♥sã♦ ❞❡ ❛❧❣✉♥s ❝♦♥t❡ú❞♦s❞❡♥tr♦ ❞❛ ▼❛t❡♠át✐❝❛✱ t♦♠❛♥❞♦ ♦ ❝✉✐❞❛❞♦ ❞❡ ♥ã♦ tr❛♥s❢♦r♠á✲❧♦ ❡♠ ✉♠ r❡❝✉rs♦ q✉❡ ♣♦ss❛♣❛r❡❝❡r ✉♠ ♣r♦❢❡ss♦r ✈✐rt✉❛❧✱ ♠❛s s✐♠ ✉♠ ✏❛❥✉❞❛♥t❡✑ ❞❡st❡ ♣r♦❢❡ss♦r ❡ q✉❡ ♣♦ss❛ s❡r ❡♥tã♦✉♠❛ ❢❡rr❛♠❡♥t❛ ✐♥str✉❝✐♦♥✐st❛✱ q✉❡ s✐r✈❛ ❞❡ ❛❥✉❞❛ ♥❛ tr❛♥s♠✐ssã♦ ❞❡ ❝♦♥❝❡✐t♦s ❡ ❝♦♥❤❡❝✐✲♠❡♥t♦s✱ ♦✉ s❡❥❛✱ ❝♦♥str✉❝✐♦♥✐st❛✱ ♦♥❞❡ ♦ ❛❧✉♥♦ t❡rá ✉♠ ❝♦♥t❛t♦ ❞✐r❡t♦ ❝♦♠ ❛ ♠áq✉✐♥❛ ❡❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❢❛③❡♥❞♦ ❝♦♥str✉çõ❡s ❡ ✐♥t❡r✈❡♥çõ❡s ❝♦♠ ♦ s♦❢t✇❛r❡✳ P♦❞❡♠♦s s❡r ✏❛❥✉❞❛✲❞♦s✑ ♣♦r ❛❧❣✉♥s s♦❢t✇❛r❡s ❞❡ ❢á❝✐❧ ♠❛♥✉s❡✐♦✱ ❝♦♠♦ ♦ ●❡♦❣❡❜r❛✱ q✉❡ s❡rá út✐❧ ♥❛ ❝♦♥str✉çã♦❞❡ ♣♦❧í❣♦♥♦s✱ ♥♦s ❝á❧❝✉❧♦s ❞❡ ár❡❛s ❡ ♠♦♥t❛❣❡♥s ❞❡ só❧✐❞♦s ❣❡♦♠étr✐❝♦s✱ ❜❡♠ ❝♦♠♦ ♦ s❡✉✈♦❧✉♠❡✳ ❈♦♠ ❡st❡ r❡❝✉rs♦✱ ❛❧é♠ ❞❡ t❡r♠♦s ✉♠❛ ❜❛s❡ ♣❛r❛ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✱ ♣♦❞❡✲♠♦s ❛♣r❡s❡♥t❛r ❛♦s ♥♦ss♦s ❡❞✉❝❛♥❞♦s✱ ❝♦♥❝❡✐t♦s ❡ ♠❛♥❡✐r❛s ✐♥t❡r❡ss❛♥t❡s ❞❡ ❡❢❡t✉❛r ❛❧❣✉♠❛s❝♦♥str✉çõ❡s ❡ ❛ ✈✐s✉❛❧✐③❛çã♦ ❞❡ ✜❣✉r❛s ❣❡♦♠étr✐❝❛s✳ ❚❡r❡♠♦s t❛♠❜é♠ ❛ ♦♣çã♦ ❞♦ s♦❢t✇❛r❡❈❛✈♦✱ ♣❛r❛ ❡❢❡t✉❛r ❝á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡ ❞❡ só❧✐❞♦s ❣❡♦♠étr✐❝♦s✱ s❡r✈✐♥❞♦ ❛ss✐♠ ❞❡ ✉♠ ❛❧✐❛❞♦♣❛r❛ ❛❧❣✉♠❛s ❝♦♥❢❡rê♥❝✐❛s ❞❡ r❡s✉❧t❛❞♦s✳ ❖ ✉s♦ ❞♦ s♦❢t✇❛r❡ ✇✐♥❣❡♦♠ t❛♠❜é♠ é ♠✉✐t♦ ✐♥t❡✲r❡ss❛♥t❡✱ ♣♦✐s ❛♣r❡s❡♥t❛ ❛♦ ✉s✉ár✐♦ ❛s ✐♥❢♦r♠❛çõ❡s ♣❡r❢❡✐t❛s ❛ r❡s♣❡✐t♦ ❞❛ ❝♦♥str✉çã♦ ❞❡ ✉♠só❧✐❞♦ ❣❡♦♠étr✐❝♦✱ ❜❡♠ ❝♦♠♦ ❛ ✈✐s✉❛❧✐③❛çã♦ ❞❡ r❡❧❛çõ❡s ✐♠♣♦rt❛♥t❡s✱ ❢❛❝✐❧✐t❛♥❞♦ ❛ss✐♠ ✉♠❛❝♦♠♣❛r❛çã♦ ❡♥tr❡ ár❡❛s ❡ ✈♦❧✉♠❡s✳ ❖s r❡❝✉rs♦s ❞♦ s♦❢t✇❛r❡ P♦❧②✱ ❢❛❝✐❧✐t❛♠ ♥❛ ✈✐s✉❛❧✐③❛çã♦❡♠ ✈ár✐♦s â♥❣✉❧♦s ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ só❧✐❞♦✱ ❛❧é♠ ❞❡ ♣♦❞❡r♠♦s ❝❧❛ss✐✜❝❛r ♦s ♠❡s♠♦s✳ ❈♦♠♦

✷✽

✷✾

♣♦❞❡ s❡r ✈✐st♦ ♥❛s ❋✐❣✉r❛s ✻✳✶✱ ✻✳✷✱ ✻✳✸✳

❋✐❣✉r❛ ✻✳✶✿ P✐râ♠✐❞❡ ❲✐♥❣❡♦♠

❋✐❣✉r❛ ✻✳✷✿ ❊❞✐çã♦ ❞❛ P✐râ♠✐❞❡

▼❛s ❛♦ ♠♦str❛r♠♦s ❡ss❛s ✜❣✉r❛s✱ ♣♦❞❡♠♦s ✐♥s❡r✐r ♥♦s ❛❧✉♥♦s ✉♠❛ ❝✉r✐♦s✐❞❛❞❡ ❡①tr❡♠❛✲♠❡♥t❡ ✐♠♣♦rt❛♥t❡ ♣❛r❛ ❛ ❢✉t✉r❛ ❝♦♥str✉çã♦ ❞❡ ♦✉tr♦s só❧✐❞♦s✳ ❆ ✐❞❡✐❛ ♥ã♦ é ❛♣r❡s❡♥t❛r♠♦ss♦❢t✇❛r❡s q✉❡ ❡❢❡t✉❡♠ ♦ ❝á❧❝✉❧♦✱ ♠❛s q✉❡ ❝♦♥str✉❛♠ ✜❣✉r❛s✱ ♠♦str❡♠ ❡❧❡♠❡♥t♦s ❡ ❛ss✐♠♣♦ss❛♠♦s ❛ ♣❛rt✐r ❞❡st❡ ♠♦♠❡♥t♦ ❡❢❡t✉❛r ♦s ❝á❧❝✉❧♦s ♠❛♥✉❛❧♠❡♥t❡✱ ❧❡♠❜r❛♥❞♦ q✉❡ ❡st❡s♠❡s♠♦s s♦❢t✇❛r❡s tê♠ ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❝❛❧❝✉❧❛r✱ ♥ã♦ s❡♥❞♦ ❡st❡ ♥♦ss♦ ♦❜❥❡t✐✈♦ ✜♥❛❧✳

❖ q✉❡ é ✐♠♣♦rt❛♥t❡ ♠❡♥❝✐♦♥❛r♠♦s é ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ✉♠ ❝♦♥❤❡❝✐♠❡♥t♦ ❜ás✐❝♦ ❛ r❡s♣❡✐t♦❞♦s s♦❢t✇❛r❡s q✉❡ s❡rã♦ ✉s❛❞♦s ♣❛r❛ ✉♠❛ ♠❡❧❤♦r ❛♣r❡♥❞✐③❛❣❡♠ ❡ ✉♠❛ ♠❡❧❤♦r ❞✐♥â♠✐❝❛ ❡♠s❛❧❛ ❞❡ ❛✉❧❛✳

✸✵

❋✐❣✉r❛ ✻✳✸✿ ❘❡❧❛çã♦ ❞❡ ❊✉❧❡r

❯s❛♥❞♦ ♦s r❡❝✉rs♦s ❝✐t❛❞♦s ❛♥t❡r✐♦r♠❡♥t❡ ♣♦❞❡♠♦s ❢❛③❡r ❝♦♠ q✉❡ ♦ ❛❧✉♥♦ s❛✐❜❛✱ ♣♦r❡①❡♠♣❧♦✱ ❝♦♠♣❛r❛r✱ ❛♥❛❧✐s❛r ❡ ✉s❛r ❡♠ ❛❧❣✉♠❛s ♦✉tr❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♦ ❝á❧❝✉❧♦❞❡ ✈♦❧✉♠❡ ✉s❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳ ❖ ❡♠♣r❡❣♦ ❝♦rr❡t♦ ❞❡ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡♣r♦♣r✐❡❞❛❞❡s s❡ ❢❛③ ♥❡❝❡ssár✐♦✱ ♣♦✐s ❝❛s♦ ❝♦♥trár✐♦✱ ♦ ♠❛♥✉s❡✐♦ ❞❡ t❛✐s r❡❝✉rs♦s ♣♦❞❡r✐❛ s❡t♦r♥❛r ✐♥✈á❧✐❞♦ ♦✉ ♣✐♦r✱ s❡r✈✐♥❞♦ ♣❛r❛ ✉♠❛ ✜①❛çã♦ ❡rrô♥❡❛ ❞❡ ❝❡rt♦s ❝♦♥t❡ú❞♦s ♦✉ ♦❜t❡♥çã♦❞❡ ✈❛❧♦r❡s ❡rr❛❞♦s✳

❈♦♠♦ ❞✐t♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ♦s s♦❢t✇❛r❡s ♥ã♦ ✐rã♦ s✐♠♣❧❡s♠❡♥t❡ ❝❛❧❝✉❧❛r ♦s ✈♦❧✉♠❡s ❞❡❛❧❣✉♥s só❧✐❞♦s ♠❛✐s ❝♦♥❤❡❝✐❞♦s✱ ♠❛s ❛♣r❡s❡♥t❛r ♦✉tr❛s ✐♥❢♦r♠❛çõ❡s ❛❧é♠ ❞♦ ❝á❧❝✉❧♦ ♣r♦♣r✐❛✲♠❡♥t❡ ❞✐t♦✳ ❖ q✉❡ ♣♦❞❡r❡♠♦s ✐♥s❡r✐r ❡♠ ♥♦ss♦s ❛❧✉♥♦s s❡rá ♦ ❤á❜✐t♦ ❞❡ ✉s❛r ✉♠❛ ❢❡rr❛♠❡♥t❛♣❛r❛ ❝♦♠♣❛r❛çã♦ ❞❡ r❡s✉❧t❛❞♦s ❡ ♥ã♦ s✐♠♣❧❡s♠❡♥t❡ ✉♠ ✐♥str✉♠❡♥t♦ q✉❡ s❡rá ✉t✐❧✐③❛❞♦ ♣❛r❛✉♠❛ ❝ó♣✐❛ ❞❡ r❡s✉❧t❛❞♦s✱ ♥ã♦ ❛♥❛❧✐s❛♥❞♦ ❛s ♠❡s♠❛s ❡ ♥❡♠ ❝♦♠♣r❡❡♥❞❡♥❞♦ ♦ s❡✉ s✐❣♥✐✜❝❛❞♦❞❡♥tr♦ ❞♦ ❝♦♥t❡①t♦ ❛♣r❡s❡♥t❛❞♦ ♦✉ ❡♠ q✉❡ ❡❧❛ ♣♦❞❡rá s❡r út✐❧ ❡♠ ✉♠❛ s✐t✉❛çã♦ ❢✉t✉r❛✳ ❙❡♠❡st❛s ♥♦çõ❡s ♦ ♥♦ss♦ ❛❧✉♥♦ ♣❛ss❛ ❛ s❡r ✉♠ ♠❡r♦ ❝♦♣✐❛❞♦r ❞❡ r❡s♣♦st❛s ❡ ❞❡ ❢ór♠✉❧❛s✳

❈♦♠♦ ❡①❡♠♣❧♦✱ ♣♦❞❡♠♦s t♦♠❛r ✉♠ ♣r✐s♠❛ ❤❡①❛❣♦♥❛❧ r❡❣✉❧❛r ❋✐❣✉r❛ ✻✳✹✱ ❡ ❛♦ ❝❛❧❝✉❧❛r♠♦s♦ s❡✉ ✈♦❧✉♠❡✱ t❡r❡♠♦s q✉❡ ♦❜r✐❣❛t♦r✐❛♠❡♥t❡ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❛ s✉❛ ❜❛s❡ q✉❡ s❡rá ✉♠ ❤❡①á❣♦♥♦r❡❣✉❧❛r✱ ♣♦❞❡♠♦s ❡♥tã♦ ✉s❛r ♦ ●❡♦❣❡❜r❛ ♣❛r❛ ❡❢❡t✉❛r ❡st❡ ❝á❧❝✉❧♦✱ ✈❡❥❛✿

❋✐❣✉r❛ ✻✳✹✿ P♦❧í❣♦♥♦ ♥♦ ●❡♦❣❡❜r❛

✸✶

❚❡♠♦s ✉♠ ❤❡①á❣♦♥♦ r❡❣✉❧❛r ❞❡ ❧❛❞♦√2 ❞♦ q✉❛❧ ♦ s♦❢t✇❛r❡ ❥á ❝❛❧❝✉❧♦✉ ❛ ár❡❛ ❡✱ ♣♦rt❛♥t♦✱

♦ ❝á❧❝✉❧♦ ❞♦ ✈♦❧✉♠❡ ✜❝❛ ❞❡ ❢á❝✐❧ ❝♦♥❝❧✉sã♦✱ ♣♦✐s ❞❡♣❡♥❞❡♥❞♦ ❞❛ ❛❧t✉r❛ ❞♦ ♣r✐s♠❛✱ ♣♦❞❡♠♦s❛♣❡♥❛s ❛♣❧✐❝❛r ❛ ❢ór♠✉❧❛ ❞❡ ❝á❧❝✉❧♦ ❡ ❛ss✐♠ ♦❜t❡r ♦ ♠❡s♠♦✳

❈♦♠ ❛❧❣✉♥s r❡❝✉rs♦s ❞♦ ●❡♦❣❡❜r❛✱ é ♣♦ssí✈❡❧ t❛♠❜é♠ ❛ ✈✐s✉❛❧✐③❛çã♦ ❞❡ só❧✐❞♦s ❣❡♦♠étr✐✲❝♦s✱ s❡♥❞♦ ❛ss✐♠ ✉♠ r❡❝✉rs♦ ✐♥t❡r❡ss❛♥t❡ ♣❛r❛ q✉❡ ♦s ❛❧✉♥♦s ❝♦♥s✐❣❛♠ ❝♦♥str✉✐r✱ ✈✐s✉❛❧✐③❛r❡ ❝❛❧❝✉❧❛r ✈♦❧✉♠❡s ❡ ár❡❛s ❞❡ só❧✐❞♦s ❣❡♦♠étr✐❝♦s✱ ❝♦♠♦ ♥❛ ❋✐❣✉r❛ ✻✳✺✿✿

❋✐❣✉r❛ ✻✳✺✿ P❛r❛❧❡❧❡♣í♣❡❞♦ ♥♦ ●❡♦❣❡❜r❛

◆❛ ✜❣✉r❛ ❛❝✐♠❛✱ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ ♣r✐s♠❛ q✉❛❞r❛♥❣✉❧❛r✱ ♦♥❞❡ sã♦❛♣r❡s❡♥t❛❞❛s ❛s ❛r❡st❛s✱ ❛s ❜❛s❡s ❡ ❛s ❢❛❝❡s ❧❛t❡r❛✐s✳

❯♠❛ ✐♥❢♦r♠❛çã♦ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡✱ ♣♦✐s t❡♠ ✉♠ ♣❛♣❡❧ ❢✉♥❞❛♠❡♥t❛❧ ♥❛ r❡s♦❧✉çã♦ ❡ ♥❛♦❜s❡r✈❛çã♦ ❞❡ ❡①❡r❝í❝✐♦s ♣♦r ♣❛rt❡ ❞♦s ❛❧✉♥♦s✳ ❱✐s✉❛❧✐③❛♥❞♦ ❛ ❋✐❣✉r❛ ✻✳✺✱ ♦ ❛❧✉♥♦ ❝♦♥s❡❣✉✐rá♣❡r❝❡❜❡r ❡❧❡♠❡♥t♦s ❡ ✐♥❢♦r♠❛çõ❡s ♣r❡❝✐♦s❛s q✉❡ s❡rã♦ út❡✐s ♣❛r❛ ❛ ❝♦♠♣♦s✐çã♦ ❞♦ r❛❝✐♦❝í♥✐♦❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✳

❖ q✉❡ é r❡❧❡✈❛♥t❡ ❝✐t❛r s❡rá ♦ ❢❛t♦ ❞❡ q✉❡ ♦ ❞✐s❝❡♥t❡ t❡rá ✉♠❛ ❛♣r❡s❡♥t❛çã♦ ♣ré✈✐❛ ❛r❡s♣❡✐t♦ ❞♦ s♦❢t✇❛r❡✱ s❛❜❡♥❞♦ ❛ss✐♠ ♠❛♥✐♣✉❧á✲❧♦ ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ❜ás✐❝❛✱ ♠❛s q✉❡ ❧❤❡ ❛❥✉❞❡✳

❱❡❥❛♠♦s ❛❣♦r❛✱ ♥❛ ❋✐❣✉r❛ ✻✳✻✱ ✉♠❛ ❡s❢❡r❛ ❡ ❛ ♣❛rt✐r ❞❛ ♠❡s♠❛✱ ♣♦❞❡r❡♠♦s ❡♥tã♦ ❝❛❧❝✉❧❛r♦ s❡✉ ✈♦❧✉♠❡ ❡ ár❡❛ ❞❡ s✉♣❡r❢í❝✐❡✿

❋✐❣✉r❛ ✻✳✻✿ ❊s❢❡r❛ ♥♦ ●❡♦❣❡❜r❛

✸✷

❖❜s❡r✈❛çã♦✿ ❋✐❣✉r❛ ❝♦♥str✉í❞❛ ♥♦ ●❡♦❣❡❜r❛✱ ✉s❛♥❞♦ r❡❝✉rs♦s ❞❡ ❥❛♥❡❧❛ ✸❉✳ ◆❡st❛ ✜❣✉r❛❝♦♥s❡❣✉✐♠♦s ♦❜s❡r✈❛r ❞❡t❛❧❤❡s ✐♠♣♦rt❛♥t❡s✱ t❛✐s ❝♦♠♦ ❛ ❡q✉❛çã♦ ❞❛ ❡s❢❡r❛ ❡ ❛❧❣✉♥s ♣♦♥t♦s❞❛ ♠❡s♠❛✳

❆s ❋✐❣✉r❛ ✻✳✼ ❡ ✻✳✽ ❛ s❡❣✉✐r ♥♦s ♠♦str❛ ❛ ❝♦♥str✉çã♦ ❞❡ ✉♠ ♣r✐s♠❛ r❡❣✉❧❛r r❡t♦ t❡♥❞♦❝♦♠♦ ❜❛s❡ ✶✵ ❧❛❞♦s ✭❞❡❝á❣♦♥♦✮✱ t♦❞♦s ❡❧❡s ♠❡❞✐♥❞♦ ✷ ✉♥✐❞❛❞❡s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❡ ❛❧t✉r❛✐❣✉❛❧ ❛ ✹ ✉♥✐❞❛❞❡s ❞❡ ❝♦♠♣r✐♠❡♥t♦✳

❋✐❣✉r❛ ✻✳✼✿ ❉❡❝á❣♦♥♦ ♥♦ ●❡♦❣❡❜r❛

❋✐❣✉r❛ ✻✳✽✿ Pr✐s♠❛ t❡♥❞♦ ❝♦♠♦ ❜❛s❡ ✉♠ ❞❡❝á❣♦♥♦

◆❛ ❋✐❣✉r❛ ✻✳✾✱ t❡♠♦s ❛ ❛♣r❡s❡♥t❛çã♦ ❞❡ ✉♠ ♣r✐s♠❛ ♣❡♥t❛❣♦♥❛❧ r❡❣✉❧❛r✱ ❝♦♥str✉í❞♦ ❝♦♠♦ ❛✉①í❧✐♦ ❞♦ ●❡♦❣❡❜r❛ ❡♠ q✉❡ é ❢á❝✐❧ ✈❡r✐✜❝❛r ♦ ❝á❧❝✉❧♦ ❞♦ ✈♦❧✉♠❡ ❞❡st❡ só❧✐❞♦✳

✸✸

❋✐❣✉r❛ ✻✳✾✿ Pr✐s♠❛ P❡♥t❛❣♦♥❛❧

❉♦ ♠❡s♠♦ ♣r✐s♠❛ q✉❡ ❛♣r❡s❡♥t❛♠♦s ❛❝✐♠❛✱ ✈❡r✐✜❝❛r❡♠♦s ♥❛ ❋✐❣✉r❛ ✻✳✶✵✱ ❛ s✉❛ ♣❧❛♥✐✜✲❝❛çã♦ t❛♠❜é♠ ❢❡✐t❛ ❛tr❛✈és ❞♦ s♦❢t✇❛r❡ ●❡♦❣❡❜r❛✳

❋✐❣✉r❛ ✻✳✶✵✿ P❧❛♥✐✜❝❛çã♦ ❞♦ Pr✐s♠❛ P❡♥t❛❣♦♥❛❧

◆♦ s♦❢t✇❛r❡ P♦❧②✱ ♣♦❞❡♠♦s ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ❜❡♠ s✐♠♣❧❡s✱ ❛♣r❡s❡♥t❛r✱ ♣❧❛♥✐✜❝❛r ❡ ♠✉❞❛r❛ ♣♦s✐çã♦ ❞❡ ✈ár✐♦s só❧✐❞♦s✳ ❱❡❥❛ ❛ ❋✐❣✉r❛ ✻✳✶✶✿

❋✐❣✉r❛ ✻✳✶✶✿ Pr✐s♠❛ P❡♥t❛❣♦♥❛❧ ♥♦ P♦❧②

✸✹

❋✐❣✉r❛ ✻✳✶✷✿ Pr✐s♠❛ P❡♥t❛❣♦♥❛❧ ❘❡❣✉❧❛r

❉♦ ♠❡s♠♦ ♣r✐s♠❛ q✉❡ ❛♣r❡s❡♥t❛♠♦s ❛❝✐♠❛✱ ✈❡r✐✜❝❛r❡♠♦s ❛ s✉❛ ♣❧❛♥✐✜❝❛çã♦ t❛♠❜é♠ ❢❡✐t❛❛tr❛✈és ❞♦ s♦❢t✇❛r❡ ●❡♦❣❡❜r❛✳

❈❛♣ít✉❧♦ ✼

❯s♦ ❞❡ ♠❛t❡r✐❛❧ ✏❝♦♥❝r❡t♦✑ ♥♦ ❡st✉❞♦ ❞♦

Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐

❆❧é♠ ❞♦s r❡❝✉rs♦s ❡❧❡trô♥✐❝♦s t❛✐s ❝♦♠♦ s♦❢t✇❛r❡s ❡ ❛♣❧✐❝❛t✐✈♦s✱ ♣♦❞❡♠♦s t❛♠❜é♠ ✉s❛r❝♦♠♦ ♠❛t❡r✐❛❧ ❞❡ ❛♣♦✐♦ à ❛♣r❡♥❞✐③❛❣❡♠✱ ♠❛t❡r✐❛✐s ❝♦♥❝r❡t♦s q✉❡ ♣♦❞❡♠ s❡r ❛❞q✉✐r✐❞♦s ❞❛♠❛♥❡✐r❛ ♠❛✐s s✐♠♣❧❡s ❡ ❛❝❡ssí✈❡❧✳ ❊ss❡s ♠❛t❡r✐❛✐s ♣♦❞❡♠ ❛✉①✐❧✐❛r ❛ ✈✐s✉❛❧✐③❛çã♦ ♣♦r ♣❛rt❡ ❞♦❛❧✉♥♦✱ ❞❡ ❝♦♥❝❡✐t♦s ❡ ✐♥❢♦r♠❛çõ❡s t❡ór✐❝❛s ❛♣r❡s❡♥t❛❞❛s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✱ ❛❧é♠ ❞❡ ♣♦❞❡r❡♠s❡r ♠❛♥✐♣✉❧❛❞♦s ❡ ❝♦♥❢❡❝❝✐♦♥❛❞♦s ♣❡❧♦s ♣ró♣r✐♦s ❞♦❝❡♥t❡s✳ P♦❞❡ ✲ s❡✱ ♣♦r ❡①❡♠♣❧♦✱ ❝♦♥str✉✐r✉♠ ♣r✐s♠❛ ❝♦♠ ✉♠❛ ❝❛✐①❛ ❞❡ s❛♣❛t♦s ♦✉ q✉❛❧q✉❡r ♦✉tr❛ ❡♠❜❛❧❛❣❡♠ q✉❡ ❢❛ç❛ ❝♦♠ q✉❡ s❡❛ss♦❝✐❡ à ✐❞❡✐❛ ❞❡ ✉♠ ♣r✐s♠❛ ✭❜❛s❡s ♣❧❛♥❛s ❡ ♣❛r❛❧❡❧❛s✮✱ ❛❧é♠ ❞✐ss♦✱ ♣♦❞❡♠♦s ❛♣r❡s❡♥t❛r❝♦♥❝❡✐t♦s ❛ r❡s♣❡✐t♦ ❞❡ só❧✐❞♦s r❡❞♦♥❞♦s ❞❡ ✉♠❛ ♠❛♥❡✐r❛ ♠✉✐t♦ ♠❛✐s ❢á❝✐❧ ❞❡ ❡♥t❡♥❞✐♠❡♥t♦❡ ❞❡ ✈✐s✉❛❧✐③❛çã♦ ❝♦♠♦ ❛s ❣❡r❛tr✐③❡s ❞❡ ✉♠ ❝✐❧✐♥❞r♦ ❡ ❛ ❛❧t✉r❛ ❞♦ ♠❡s♠♦ t❛♠❜é♠ ♣♦❞❡♠ s❡r❝♦♥s✐❞❡r❛❞❛s ❝♦♠♦ ❡st❛ ❣❡r❛tr✐③✳

❯♠❛ ❝❛r❛❝t❡ríst✐❝❛ ♠✉✐t♦ ✐♠♣♦rt❛♥t❡ ❞♦ tr❛❜❛❧❤♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ❝♦♠ ♠❛t❡r✐❛❧ ❝♦♥❝r❡t♦é ❛ ❞❡ ♣♦❞❡r♠♦s ♦❜s❡r✈❛r ❡ ❛♣r❡s❡♥t❛r ♣❧❛♥✐✜❝❛çõ❡s ❞❡ só❧✐❞♦s ❡ ❞❡ ❡str✉t✉r❛s ♠❛✐s ❞✐❢í❝❡✐s❞❡ s❡r❡♠ ✈✐s✉❛❧✐③❛❞❛s s♦♠❡♥t❡ ❡♠ três ❞✐♠❡♥sõ❡s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ q✉❛♥❞♦ só t❡♠♦s ❝♦♠♦r❡❝✉rs♦ ❞✐❞át✐❝♦✱ ♦ ❧✐✈r♦ ♦✉ ❛♣♦st✐❧❛✱ ❛❧é♠ ❞♦ ❝❛❞❡r♥♦ ❞♦ ❛❧✉♥♦✳ ❱❛❧❡ ❧❡♠❜r❛r q✉❡ ✉♠❛❞❛s ♠❛✐♦r❡s ❞✐✜❝✉❧❞❛❞❡s ❡♥❝♦♥tr❛❞❛s ♣♦r ♣❛rt❡ ❞♦s ❛❧✉♥♦s é ❛ ❞❡ ♥ã♦ s❛❜❡r❡♠ ❞❡s❡♥❤❛r✱♦✉ ❛♥❛❧✐s❛r ✉♠ ❞❡s❡♥❤♦ ❥á ♣r♦♥t♦✱ ♣♦✐s ❡st❛♠♦s ♥❛ ✈❡r❞❛❞❡ q✉❡r❡♥❞♦ q✉❡ ♦ ❝ér❡❜r♦ ❛ss♦❝✐❡❢♦r♠❛s tr✐❞✐♠❡♥s✐♦♥❛✐s ❛♣r❡s❡♥t❛❞❛s ❡♠ ✉♠ ♣❧❛♥♦ q✉❡ é ❜✐❞✐♠❡♥s✐♦♥❛❧✳ P♦❞❡♠♦s ❡❢❡t✉❛r❝♦rt❡s ♣❛r❛ ♦❜s❡r✈❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ ❡❧❡♠❡♥t♦s ✏♦❜s❝✉r♦s✑✱ tr❛❜❛❧❤❛r ❛ ✐♥s❡rçã♦ ❞❡ ✉♠ só❧✐❞♦❡♠ ♦✉tr♦✱ ♦❜s❡r✈❛♥❞♦ ❛ss✐♠ ❛ ✐♥s❝r✐çã♦ ❡ ❝✐r❝✉♥s❝r✐çã♦ ❞♦s ♠❡s♠♦s✱ ❛❧é♠ ❞❡ ♣♦❞❡r♠♦s❝♦♠♣❛r❛r ár❡❛s ❡ ✈♦❧✉♠❡s✳ ❉❡♣❡♥❞❡♥❞♦ ❞♦ só❧✐❞♦ ❡st✉❞❛❞♦ ♣♦❞❡♠♦s ❛té ❝♦♠♣❛r❛r ✈♦❧✉♠❡❡ ❝❛♣❛❝✐❞❛❞❡✳ ❯♠❛ s✐t✉❛çã♦ ♠✉✐t♦ ❝♦♠✉♠ ❡ ❢á❝✐❧ ❞❡ ❛♣r❡s❡♥t❛r♠♦s ❛♦s ❛❧✉♥♦s ♦ Pr✐♥❝í♣✐♦❞❡ ❈❛✈❛❧✐❡r✐ é ❝♦♠ ✉♠❛ r❡s♠❛ ❞❡ ❢♦❧❤❛s ❞❡ ♣❛♣❡❧ ♦❢í❝✐♦✱ ♦♥❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ❢♦❧❤❛s✱ q✉❛♥❞♦❡♠♣✐❧❤❛❞❛s✱ ❡st❛r✐❛ r❡♣r❡s❡♥t❛♥❞♦ ✉♠❛ s❡❝çã♦ ♣❧❛♥❛ ❡ ❝♦♠♦ t♦❞❛s ❛s ❢♦❧❤❛s tê♠ ❛ ♠❡s♠❛

✸✺

✸✻

❡s♣❡ss✉r❛ ✭♠❡s♠❛ ❛❧t✉r❛ ♥♦ ❝❛s♦ ❞❡ ❝♦♥s✐❞❡r❛r♠♦s s❡❝çõ❡s✮ ❡ sã♦ ✐❞ê♥t✐❝❛s✱ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r♦ ✈♦❧✉♠❡ ❞❡st❛ r❡s♠❛ ✉s❛♥❞♦ ❝♦♥❝❡✐t♦s ❜ás✐❝♦s ❛ r❡s♣❡✐t♦ ❞❡ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❡❛ss✐♠✱ ❡♥❝♦♥tr❛r t❛♠❜é♠ ♦ ✈♦❧✉♠❡ ❞❡ q✉❛❧q✉❡r só❧✐❞♦ ❣❡♦♠étr✐❝♦ ❢♦r♠❛❞♦ ❝♦♠ t♦❞❛s ❡st❛s❢♦❧❤❛s ❡♠♣✐❧❤❛❞❛s✱ ✈✐st♦ q✉❡ sã♦ ❢á❝❡✐s ❞❡ ♠❛♥✐♣✉❧❛r ❡ ✏❞❡❢♦r♠❛r✑ ♣❛r❛ ❛ ✉t✐❧✐③❛çã♦ ❡♠ ♦✉tr♦ssó❧✐❞♦s✳

❆♣r❡s❡♥t❛r❡♠♦s ❛❣♦r❛ ❛❧❣✉♥s ❡①❡♠♣❧♦s ✐♠♣♦rt❛♥t❡s ❛tr❛✈és ❞♦s q✉❛✐s ♣♦❞❡r❡♠♦s ♠♦str❛r❛♦s ❛❧✉♥♦s✱ ✐♥❢♦r♠❛çõ❡s ❝♦♥❝r❡t❛s ❛ r❡s♣❡✐t♦ ❞♦s só❧✐❞♦s ❣❡♦♠étr✐❝♦s ❡ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡ ♦Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳ P♦❞❡♠♦s ✉s❛r ♣♦rt❛✲❝♦♣♦s q✉❡ sã♦ ❞✐str✐❜✉í❞♦s ❡♠ ❜❛r❡s✱ r❡st❛✉r❛♥t❡s❡ ❧❛♥❝❤♦♥❡t❡s✱ ♣❛r❛ ♣♦❞❡r♠♦s ❝♦♥str✉✐r✱ ♣♦r ❡①❡♠♣❧♦✱ ✉♠ ❝✐❧✐♥❞r♦ ❝✐r❝✉❧❛r r❡t♦✱ ♦♥❞❡ ✈❡r✐✜✲❝❛r❡♠♦s q✉❡ ❛ ❥✉st❛♣♦s✐çã♦ ❞❡st❡s ♣♦rt❛✲❝♦♣♦s r❡s✉❧t❛rá ♥♦ ❝✐❧✐♥❞r♦ ❝✐t❛❞♦✱ ♣♦❞❡♠♦s ♦❜t❡r♦✉ ❛té ❝♦♥str✉✐r s✉♣❡r❢í❝✐❡s ❞❡ ♠❡s♠❛ ár❡❛✱ ♠❛s ❝♦♠ ❢♦r♠❛ ❞✐❢❡r❡♥t❡ ❡ ❛ss✐♠ ♦❜s❡r✈❛r❡♠♦sq✉❡ ❛ ❥✉st❛♣♦s✐çã♦ ❞❛s ♠❡s♠❛s ❢♦r♠❛rá só❧✐❞♦s ❣❡♦♠étr✐❝♦s ❞❡ ♠❡s♠♦ ✈♦❧✉♠❡✱ ❥✉st✐✜❝❛♥❞♦❞❡st❛ ♠❛♥❡✐r❛ ♦ ❡♥✉♥❝✐❛❞♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❡ ❢❛③❡♥❞♦ ❝♦♠ ❡❧❡ ✜q✉❡ ✏✈✐s✉❛❧✐③❛❞♦✑✳

❆s ♣✐râ♠✐❞❡s sã♦ ✉♠ ♦✉tr♦ ❡①❡♠♣❧♦ ♣❛r❛ ❛s q✉❛✐s ♦s ♠❛t❡r✐❛✐s ❝♦♥❝r❡t♦s tê♠ ✉♠❛ ❣r❛♥❞❡✐♠♣♦rtâ♥❝✐❛ ❡♠ s✉❛ ❛♣r❡s❡♥t❛çã♦ ❡ ❡♥t❡♥❞✐♠❡♥t♦ ♣♦r ♣❛rt❡ ❞♦s ❛❧✉♥♦s✱ ♣♦✐s ❞❡♣❡♥❞❡♥❞♦ ❞❛♥❛t✉r❡③❛ ❞❛ ♣✐râ♠✐❞❡ ❛ ❝♦♥str✉çã♦ ♥♦ ♣❧❛♥♦ ♦✉ ❛té ❛ ❛♥á❧✐s❡ ❞❡ ✉♠ só❧✐❞♦ ♣r♦♥t♦ ❡♠ ✉♠❧✐✈r♦ ♦✉ ❛♣♦st✐❧❛ s❡ t♦r♥❛ ♠✉✐t♦ ❞✐❢í❝✐❧ ❡ ❡♠ ✉♠ ♠❛t❡r✐❛❧ ❡♠ q✉❡ s❡ ♣♦ss❛ ♠❛♥✐♣✉❧❛r ❛✈✐s✉❛❧✐③❛çã♦✱ ♦ ❝á❧❝✉❧♦ ❡ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ♣♦❞❡ s❡ t♦r♥❛r ♠✉✐t♦ ♠❛✐s ♣r❛③❡r♦s♦ ❡ tr❛♥q✉✐❧♦✳❊♠ ✉♠❛ ♣✐râ♠✐❞❡ ❞❡ ❜❛s❡ ❤❡①❛❣♦♥❛❧ r❡❣✉❧❛r✱ ♣♦❞❡♠♦s✱ ♣♦r ❡①❡♠♣❧♦✱ ❡♥①❡r❣❛r ♦s tr✐â♥❣✉❧♦s❡q✉✐❧át❡r♦s q✉❡ ❢♦r♠❛♠ ♦ ❤❡①á❣♦♥♦ r❡❣✉❧❛r ❜❛s❡ ❡ ❛ss✐♠ ❝♦♥str✉✐r ❢❛❝❡s ✐♥t❡r♥❛s ❡ ♦✉tr♦s❡❧❡♠❡♥t♦s ❞❡ t❛❧ só❧✐❞♦✱ ✐ss♦ s❡♠ ❢❛❧❛r ♥❛ q✉❡stã♦ ❞♦ ✈♦❧✉♠❡ ❞❛ ♠❡s♠❛ q✉❡ é ♦ ♥♦ss♦ ❛❧✈♦♣r✐♥❝✐♣❛❧✳ ❆ s❡❣✉✐r ❛s ❋✐❣✉r❛s ✼✳✶✱ ✼✳✷ ❡ ✼✳✸ sã♦ ❛♣r❡s❡♥t❛❞❛s✳

❋✐❣✉r❛ ✼✳✶✿ P✐râ♠✐❞❡ ❞❡ ❇❛s❡ ◗✉❛❞r❛♥❣✉❧❛r

✸✼

❋✐❣✉r❛ ✼✳✷✿ P✐râ♠✐❞❡ ❞❡ ❇❛s❡ ❍❡①❛❣♦♥❛❧

❋✐❣✉r❛ ✼✳✸✿ ❙ó❧✐❞♦s ❞❡ ◆❛t✉r❡③❛ ❉✐❢❡r❡♥t❡

◆❛s ❋✐❣✉r❛s ✼✳✶✱ ✼✳✷ ❡ ✼✳✸✱ t❡♠♦s só❧✐❞♦s ❞❡ ♥❛t✉r❡③❛ ❞✐❢❡r❡♥t❡ ✭❜❛s❡s ❞✐❢❡r❡♥t❡s✱ só❧✐❞♦sr❡❞♦♥❞♦s ♦✉ ♥ã♦✮✱ ♠❛s q✉❡ tê♠ ♦ ♠❡s♠♦ ✈♦❧✉♠❡✳ ❊st❛ ❛♣r❡s❡♥t❛çã♦ s❡ t♦r♥❛ ♠✉✐t♦ ✐♠♣♦r✲t❛♥t❡ ❛ ♣❛rt✐r ❞♦ ♠♦♠❡♥t♦ ❡♠ q✉❡ ❝♦♥s❡❣✉✐♠♦s ✈❡r ❝♦♥❝r❡t❛♠❡♥t❡ ❛ ✐❞❡✐❛ ❞❡ t❡r♠♦s ❢♦r♠❛s❞✐❢❡r❡♥t❡s ❝♦♠ ♠❡s♠♦ ✈♦❧✉♠❡✳

P♦❞❡♠♦s ♥❛ ✈❡r❞❛❞❡✱ ❝♦♠ ♦ ♠❛t❡r✐❛❧ ❝♦♥❝r❡t♦ ♦❜s❡r✈❛r ❡ ♠❛♥✐♣✉❧❛r ✈ár✐♦s tó♣✐❝♦s ❡ ✈ár✐❛ss✐t✉❛çõ❡s✳

❈❛♣ít✉❧♦ ✽

❉❡♠♦♥str❛çõ❡s ❡ ❡①❡♠♣❧✐✜❝❛çõ❡s

✐♠♣♦rt❛♥t❡s

❱❛♠♦s✱ ❛ ♣❛rt✐r ❞❡ ❛❣♦r❛ ❛♣r❡s❡♥t❛r ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s ❡ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❡ ❛♣❧✐❝❛✲çõ❡s ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ q✉❡ sã♦ ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦ ♠❡s♠♦✳

✽✳✶ ❉❡♠♦♥str❛çã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ♦ ❝á❧✲❝✉❧♦ ❞❡ ✈♦❧✉♠❡s✱ ✉s❛♥❞♦ r❡❝✉rs♦s ❞❡ ❈á❧❝✉❧♦ ❉✐❢❡r❡♥✲❝✐❛❧ ❡ ■♥t❡❣r❛❧ ❝♦♠♦ ❥✉st✐✜❝❛t✐✈❛ ♣❛r❛ ❞♦❝❡♥t❡s✳

❈✐t❛♥❞♦ ❛ ❞❡♠♦♥str❛çã♦ ❢❡✐t❛ ❡♠ ❬✶✽❪✱ t❡♠♦s✿

❘❡ss❛❧t❛♠♦s ❛ ♥♦t❛çã♦ ❛♣r❡s❡♥t❛❞❛ q✉❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦ P s❡rá ✐♥❞✐❝❛❞♦ ♣♦r v(P )

❡ a(R) ✐♥❞✐❝❛ ❛ ár❡❛ ❞❡ ✉♠❛ r❡❣✐ã♦ R ❞♦ ♣❧❛♥♦✳

❈♦♥s✐❞❡r❡♠♦s ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❝❛rt❡s✐❛♥❛s Oxyz✱ ❡ s❡❥❛ P ✉♠ só❧✐❞♦ ✜♥✐t♦❞❡❧✐♠✐t❛❞♦ ♣♦r z = 0✱ z = c > 0 ❡ ♣♦r ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ❣rá✜❝♦s ❞❡ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s❞♦ t✐♣♦ y = f(x, z) ❡ x = g(y, z)✳ P❛r❛ ❝❛❞❛ t t❛❧ q✉❡ 0 ≤ t ≤ c✱ s❡❥❛ Pt ❛ ✐♥t❡rs❡çã♦ ❞❡ P

❝♦♠ ♦ ♣❧❛♥♦ z = t✳ ❙❡❥❛ Q ♦✉tr♦ só❧✐❞♦ ✜♥✐t♦ ❞❡❧✐♠✐t❛❞♦ ♣♦r z = 0✱ z = c > 0 ❡ ♣♦r ✉♠❛q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ ❣rá✜❝♦s ❞❡ ❢✉♥çõ❡s ❝♦♥tí♥✉❛s ❞♦ t✐♣♦ y = f(x, z) ❡ x = g(y, z)✳ P❛r❛❝❛❞❛ t t❛❧ q✉❡ 0 ≤ t ≤ c✱ s❡❥❛ Qt ❛ ✐♥t❡rs❡çã♦ ❞❡ Q ❝♦♠ ♦ ♣❧❛♥♦ z = t✳ ❙✉♣♦♥❤❛♠♦s q✉❡❡①✐st❛ k > 0 t❛❧ q✉❡ a(Pt) = k.a(Qt) ♣❛r❛ t♦❞♦ t✳ ❊♥tã♦ v(P ) = k.v(Q)✳

✸✽

✽✳✷✳ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ✜❣✉r❛s ♣❧❛♥❛s ✲ ❉❡♠♦♥str❛çã♦ ✸✾

❉❡♠♦♥str❛çã♦✿

❉❛ t❡♦r✐❛ ❞❡ ✐♥t❡❣r❛çã♦ ❞❡ ❢✉♥çõ❡s r❡❛✐s t❡♠♦s✿

v(P ) =

∫∫∫

P

dx dy dz =

∫ C

0

∫∫

Pz

dx dy

dz =

∫ C

0

aPz dz =

∫ C

0

KaQz dz = K·∫ C

0

aQz dz

= · · · = Kv(Q)

✽✳✷ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ✜❣✉r❛s ♣❧❛♥❛s ✲ ❉❡♠♦♥s✲tr❛çã♦

❈♦♥s✐❞❡r❡ ❞✉❛s r❡❣✐õ❡s ❞♦ ♣❧❛♥♦✱ ❞❡♥♦♠✐♥❛❞❛s ❞❡ R ❡ Q✱ ✈❛♠♦s ✏✐♥s❡r✐r✑ ❡st❛s ❞✉❛sr❡❣✐õ❡s ❡♠ ♣❧❛♥♦s ❝❛rt❡s✐❛♥♦s ❞✐st✐♥t♦s✱ ❝♦♠♦ ♥❛s ❋✐❣✉r❛s ✽✳✶ ❡ ✽✳✷✿

❋✐❣✉r❛ ✽✳✶✿ ❘❡❣✐ã♦ ❘

❋✐❣✉r❛ ✽✳✷✿ ❘❡❣✐ã♦ ◗

◆❛s ❋✐❣✉r❛s ✽✳✶ ❡ ✽✳✷✱ ❛♥t❡r✐♦r❡s✱ t❡♠♦s ❞✉❛s ❢✉♥çõ❡s ❡♠ ❝❛❞❛ ✉♠❛ ❞❡❧❛s ❡ ❛ r❡❣✐ã♦❧✐♠✐t❛❞❛ ♣❡❧♦s ❣rá✜❝♦s é q✉❡ ❝❤❛♠❛♠♦s ❞❡ r❡❣✐ã♦ R ❡ Q✳

✽✳✸✳ ➪r❡❛ ❞❡ ✉♠❛ r❡❣✐ã♦ ❞❡t❡r♠✐♥❛❞❛ ♣♦r ✉♠❛ ❝✉r✈❛ y = x2 ❡ ♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s✳ ✹✵

❱❛♠♦s ❡♥tã♦ t♦♠❛r ♥❡st❡s ♣❧❛♥♦s✱ r❡t❛s ♣❛r❛❧❡❧❛s ❛♦ ❡✐①♦ ❞❛s ♦r❞❡♥❛❞❛s ❡ q✉❡ ✐♥t❡rs❡❝t❛♠❛s ♥♦ss❛s r❡❣✐õ❡s R ❡ Q✱ ❢♦r♠❛♥❞♦ ♥❡st❛s r❡❣✐õ❡s s❡❣♠❡♥t♦s ❞❡ r❡t❛✳ ❙❡ ♣❛r❛ ❝❛❞❛ ✈❛❧♦r ❞❡x t♦♠❛r♠♦s r❡t❛s q✉❡ ✐♥t❡r❝❡♣t❛♠ ❛s ❞✉❛s r❡❣✐õ❡s ❢♦r♠❛♥❞♦ s❡❣♠❡♥t♦s ✐❣✉❛✐s ❡♠ ❝❛❞❛ ✉♠❛❞❛s r❡❣✐õ❡s ✭♠❡s♠♦s ✈❛❧♦r❡s ❞❡ x ♣❛r❛ ❛s ❞✉❛s r❡❣✐õ❡s ✮✱ ❡♥tã♦ ✈❛♠♦s ✈❡r✐✜❝❛r q✉❡ ❡st❛sr❡❣✐õ❡s R ❡ Q tê♠ ❛ ♠❡s♠❛ ár❡❛✳ ❆ ❞❡♠♦♥str❛çã♦ ❞❡st❡ ❢❛t♦ s❡ ❞á ✉s❛♥❞♦ ✐♥t❡❣r❛✐s✿

❱❛♠♦s ❝❤❛♠❛r ❞❡ cx ♦s ❝♦♠♣r✐♠❡♥t♦s ❞♦s s❡❣♠❡♥t♦s q✉❡ ❝♦rt❛♠ ❛ r❡❣✐ã♦ R✱ ❞❡♣❡♥❞❡♥❞♦❞♦ x ♥♦ q✉❛❧ ❡❧❡ é r❡❧❛❝✐♦♥❛❞♦ ❞♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s ❡ ❞❡ c′x ♦s ❝♦♠♣r✐♠❡♥t♦s ❞♦s s❡❣♠❡♥t♦sq✉❡ ❝♦rt❛♠ ❛ r❡❣✐ã♦ Q ❝♦♠ r❡❧❛çã♦ ❛♦s ♠❡s♠♦s ✈❛❧♦r❡s ❞❡ x✳ ❖ q✉❡ ❞✐ss❡♠♦s ❛❝✐♠❛ é q✉❡s❡ ❛ ❝❛❞❛ ♠❡s♠♦ ✈❛❧♦r ❞❡ xcx = c′x✱ ❡♥tã♦ ❛s r❡❣✐õ❡s R ❡ Q tê♠ ❛ ♠❡s♠❛ ár❡❛✳ ❋❛r❡♠♦s❡♥tã♦ ♦ s❡❣✉✐♥t❡✿ ❈♦♥s✐❞❡r❡♠♦s q✉❡ ❛ r❡❣✐ã♦ R ❡stá ❧✐♠✐t❛❞❛ ♣♦r ❞✉❛s ❢✉♥çõ❡s✱ ❞❡♥♦♠✐♥❛❞❛sf(x) e g(x)✱ ✐st♦ q✉❡ ❞✐③❡r q✉❡ ♦ ❝♦♠♣r✐♠❡♥t♦ cx✱ s❡rá ♥❛ ✈❡r❞❛❞❡ f(x)− g(x) ❡ ❛ss✐♠ ❛ ár❡❛❞❡st❛ r❡❣✐ã♦ s❡rá

∫ b

a[f(x) − g(x)]dx✱ ♦✉ s❡❥❛✱

∫ a

bcxdx✳ ❆❣♦r❛ ✈❛♠♦s ❝♦♥s✐❞❡r❛r ❛ r❡❣✐ã♦ ◗

t❛♠❜é♠ ❧✐♠✐t❛❞❛ ♣♦r ❞✉❛s ❢✉♥çõ❡s ❞❡♥♦♠✐♥❛❞❛s ❞❡ h(x) ❡ k(x)✱ ✈❛♠♦s ❡♥tã♦ ♦❜s❡r✈❛r q✉❡ ❛ár❡❛ ❞❛ ♠❡s♠❛ s❡rá

∫ a

b[h(x)−k(x)]dx ♦ q✉❡ q✉❡r ❞✐③❡r

∫ a

bc′xdx✳ ❈♦♠♦ ♣♦r ❤✐♣ót❡s❡ cx = c′x✱

♣♦❞❡♠♦s ❡♥tã♦ ❞✐③❡r q✉❡∫ a

bcxdx ❂

∫ a

bc′xdx✱ ♦ q✉❡ r❡♣r❡s❡♥t❛ s❡r❡♠ ❛s ❞✉❛s ár❡❛s ✐❣✉❛✐s✳

✽✳✸ ➪r❡❛ ❞❡ ✉♠❛ r❡❣✐ã♦ ❞❡t❡r♠✐♥❛❞❛ ♣♦r ✉♠❛ ❝✉r✈❛ y =

x2 ❡ ♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s✳

✽✳✸✳✶ ❈♦♥❝❡✐t♦s ❡ ✐♥❢♦r♠❛çõ❡s ♣r❡❧✐♠✐♥❛r❡s

❈♦♥❢♦r♠❡ ✈❡♠♦s ❡♠ ❬✻❪✱ ♣♦❞❡♠♦s ❞❡s❝r❡✈❡r ♦ s❡❣✉✐♥t❡✿

❈❛✈❛❧✐❡r✐ ❡st❛❜❡❧❡❝❡✉ ✉♠ t❡♦r❡♠❛ q✉❡ r❡❧❛❝✐♦♥❛ ♦s ✏✐♥❞✐✈✐sí✈❡✐s✑ ❞❡ ✉♠ ♣❛r❛❧❡❧♦❣r❛♠♦ ❝♦♠❛q✉❡❧❡s ♦❜s❡r✈❛❞♦s ❡♠ tr✐â♥❣✉❧♦s ❞❡t❡r♠✐♥❛❞♦s ♣♦r ✉♠❛ ❞❡ s✉❛s ❞✐❛❣♦♥❛✐s✳ ■♥❞✐❝❛r❡♠♦s ♦s♣r✐♠❡✐r♦s ♣♦r a ❡ ♦s s❡❣✉♥❞♦s ♣♦r x✳ ❈❛✈❛❧✐❡r✐ ♣r♦✈♦✉ q✉❡

∑

a = 2∑

x❀∑

a2 = 3∑

x2 . . .

❖❜s❡r✈❡♠♦s q✉❡ ♦s s♦♠❛tór✐♦s ❛♣r❡s❡♥t❛❞♦s tê♠ ♦ s❡♥t✐❞♦ ❞❡ ❛♣r❡s❡♥t❛r ❛ ✐♥✜♥✐❞❛❞❡❞❡st❛s s❡❝çõ❡s✱ ❝♦rr❡s♣♦♥❞❡♥❞♦ à ✐❞❡✐❛ ❞❡ ✐♥t❡❣r❛çã♦ ♣❛r❛ ❛ ❢♦r♠❛çã♦ ❞❛s ✜❣✉r❛s✳

❖❜s❡r✈❡ ♦ q✉❡ ❢♦✐ ❡♥✉♥❝✐❛❞♦ ♥❛ ❋✐❣✉r❛ ✽✳✸ ❛ s❡❣✉✐r✿

✽✳✸✳ ➪r❡❛ ❞❡ ✉♠❛ r❡❣✐ã♦ ❞❡t❡r♠✐♥❛❞❛ ♣♦r ✉♠❛ ❝✉r✈❛ y = x2 ❡ ♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s✳ ✹✶

❋✐❣✉r❛ ✽✳✸✿ ➪r❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦

❙❡ ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ é ✉♠ r❡tâ♥❣✉❧♦ ❞❡ ❛❧t✉r❛ b✱ s✉❛ ár❡❛∑

a é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞❡ ✉♠❞✐✈✐sí✈❡❧ ♣❡❧♦ ✏♥ú♠❡r♦✑ b ❞❡ ✐♥❞✐✈✐sí✈❡✐s✱ ♦✉ s❡❥❛

∑

a = a.b✳

❊ ❛ss✐♠ ✉s❛♥❞♦ ❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❞❛s r❡❧❛çõ❡s ❞❡♠♦♥str❛❞❛s ♣♦r ❈❛✈❛❧✐❡r✐ ❡ q✉❡ ❢♦r❛♠❛♣r❡s❡♥t❛❞❛s ♥♦ ✐♥í❝✐♦ ❞❛ ❞❡♠♦♥str❛çã♦✱ ♦❜t❡♠♦s ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ q✉❡ s❡rá

∑

x =1

2

∑

a =1

2a.b

❆ s❡❣✉♥❞❛ ♣❛rt❡ ❞❛s r❡❧❛çõ❡s ❛♣r❡s❡♥t❛❞❛s ♣❡r♠✐t❡ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❝♦♠♣r❡❡♥❞✐❞❛ ❡♥tr❡ ❛❝✉r✈❛ y = x2 ❡ ♦ ❡✐①♦ x ✭❞❡ ✵ ❛té ❛✮✱ ✈✐❞❡ ❋✐❣✉r❛ ✽✳✹✿

❋✐❣✉r❛ ✽✳✹✿ ➪r❡❛ ❝♦♠♣r❡❡♥❞✐❞❛ ❡♥tr❡ ❛ ❝✉r✈❛ y = x2 ❡ ♦ ❡✐①♦ ①

❙❡❣✉♥❞♦ ❛s ✐❞❡✐❛s ❛♣r❡s❡♥t❛❞❛s ♣♦r ❈❛✈❛❧✐❡r✐✱ ❡ss❛ ár❡❛ ✈❛❧❡∑

x2 ✱ ♣♦✐s ❝❛❞❛ ✉♠ ❞❡ s❡✉s

✽✳✹✳ ❱♦❧✉♠❡ ❞❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ q✉❛❧q✉❡r ✲ ❉❡♠♦♥str❛çã♦ ✹✷

✐♥❞✐✈✐sí✈❡✐s ✈❛❧❡ x2✳ ❊♥tr❡t❛♥t♦ ♣❡❧❛ r❡❧❛çã♦ ❝✐t❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡✱ t❡r❡♠♦s✿

∑

x2 =1

3

∑

a2

❖♥❞❡∑

a2 é ❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦ q✉❡ ❝♦♥té♠ ❛ r❡❣✐ã♦ ❞❛ q✉❛❧ q✉❡r❡♠♦s ❛ ár❡❛✳ P♦ré♠✱♦❜s❡r✈❛♥❞♦ ❛ ❋✐❣✉r❛ ✽✳✹✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ❛ ár❡❛ ❞❡st❡ r❡tâ♥❣✉❧♦ ♣♦❞❡ s❡r ✏❝❛❧❝✉❧❛❞❛✑❢❛③❡♥❞♦ a.a2 = a3✳ ▲♦❣♦✱ ❛ ár❡❛ ♣❡❞✐❞❛ s❡r✐❛ a3

3✳

✽✳✹ ❱♦❧✉♠❡ ❞❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ q✉❛❧q✉❡r ✲ ❉❡♠♦♥str❛✲çã♦

❊st❛ é ✉♠❛ ❞❡♠♦♥str❛çã♦ s✐♠♣❧❡s ❞❡ s❡r ❢❡✐t❛ ❡ t❡rá ❝♦♠♦ ❡♠❜❛s❛♠❡♥t♦ ❢✉♥❞❛♠❡♥t❛❧ ♦Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳

❈♦♥s✐❞❡r❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ P ✱ ❋✐❣✉r❛ ✽✳✺✱ q✉❛❧q✉❡r ♥♦ q✉❛❧ ❛s ❢❛❝❡s sã♦ q✉❛❞r✐❧át❡✲r♦s ❝♦♥❣r✉❡♥t❡s ❞♦✐s ❛ ❞♦✐s ✭♣♦r ❡①❡♠♣❧♦ ♣❛r❛❧❡❧♦❣r❛♠♦s ❝♦♥❣r✉❡♥t❡s ❞♦✐s ❛ ❞♦✐s✮✱ ✈❛♠♦s❡♥tã♦ t♦♠❛r ✉♠❛ ❞❡ss❛s ❢❛❝❡s ❝♦♠♦ s❡♥❞♦ ❛ ❜❛s❡ ❞❡ss❡ ♣❛r❛❧❡❧❡♣í♣❡❞♦✳ ◆♦ ♣❧❛♥♦ ❡♠ q✉❡❡st❛ ❜❛s❡ ❡stá ❛♣♦✐❛❞❛✱ ✈❛♠♦s t♦♠❛r ✭♣♦r ❝♦♥str✉çã♦ ♣♦r ❡①❡♠♣❧♦✮ ✉♠ r❡tâ♥❣✉❧♦ ❝♦♠ ár❡❛♥✉♠❡r✐❝❛♠❡♥t❡ ✐❣✉❛❧ à ár❡❛ ❞♦ q✉❛❞r✐❧át❡r♦ ❜❛s❡ ❞♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ P ✱ ❛♦ ❝♦♥str✉✐r♠♦s ✉♠❜❧♦❝♦ ✭♣r✐s♠❛ r❡t♦✮ ❝♦♠ ❜❛s❡ s❡♥❞♦ ❡st❡ r❡tâ♥❣✉❧♦ ❡ ❛❧t✉r❛ ✐❣✉❛❧ ❛ ❞♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ q✉❛❧✲q✉❡r P ✱ ✈❛♠♦s ❡st❛r ♥❛ ✈❡r❞❛❞❡ ❝♦♥str✉✐♥❞♦ ✉♠ ♣r✐s♠❛ r❡t❛♥❣✉❧❛r r❡t♦✱ ❝✉❥♦ ✈♦❧✉♠❡ ❥á é❝♦♥❤❡❝✐❞♦ ❡ ❝❛❧❝✉❧❛❞♦ ♣❡❧♦ ♣r♦❞✉t♦ ❞❛ ár❡❛ ❞❛ ❜❛s❡ ♣❡❧❛ ❛❧t✉r❛ ❡ ❛ss✐♠ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❡❈❛✈❛❧✐❡r✐ ♣♦❞❡♠♦s ❡♥tã♦ ❛✜r♠❛r q✉❡ ♦ ✈♦❧✉♠❡ ❞♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ P s❡rá ✐❣✉❛❧ ❛♦ ✈♦❧✉♠❡❞❡ss❡ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❝♦♥str✉í❞♦✱ ♦✉ s❡❥❛✱ ♦ ♣r♦❞✉t♦ ❞❛ ár❡❛ ❞❛ ❜❛s❡ ♣❡❧❛ ❛❧t✉r❛✳

❋✐❣✉r❛ ✽✳✺✿ P❛r❛❧❡❧❡♣í♣❡❞♦ P ✲ ❋♦♥t❡✿ ■♥❢♦❡s❝♦❧❛✳❝♦♠

✽✳✺✳ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ ❞❡ ❜❛s❡ tr✐❛♥❣✉❧❛r ✲ ❉❡♠♦♥str❛çã♦ ✹✸

✽✳✺ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ ❞❡ ❜❛s❡ tr✐❛♥❣✉❧❛r ✲ ❉❡✲♠♦♥str❛çã♦

❆♣r❡s❡♥t❛çõ❡s ♣r❡❧✐♠✐♥❛r❡s

❯♠ ♣r✐s♠❛ ❞❡ ❜❛s❡ tr✐❛♥❣✉❧❛r ❞❡❝♦♠♣♦st♦ ❡♠ três ♣✐râ♠✐❞❡s ❞❡ ❜❛s❡s tr✐❛♥❣✉❧❛r❡s ❡ ❝♦♠♠❡s♠♦s ✈♦❧✉♠❡s✳

❋✐❣✉r❛ ✽✳✻✿ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ tr✐❛♥❣✉❧❛r ✲ ❋♦♥t❡✿ ❊❧♦♥ ▲❛❣❡s ▲✐♠❛ ✲ ▼❡❞✐❞❛ ❡ ❋♦r♠❛❡♠ ●❡♦♠❡tr✐❛

◆❛ ❋✐❣✉r❛ ✽✳✻✱ ♣♦❞❡♠♦s ♣❡r❝❡❜❡r ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞♦ ♣r✐s♠❛ tr✐❛♥❣✉❧❛r ❡♠ três ♣✐râ♠✐❞❡s❝✉❥♦s ✈♦❧✉♠❡s s❡rã♦ ✐❣✉❛✐s ❡ ❧♦❣♦ ♣♦❞❡♠♦s ❡♥tã♦ tr❛❜❛❧❤❛r ❝♦♠ ❛ ✐❞❡✐❛ ❞❡ q✉❡ ♦ ✈♦❧✉♠❡ ❞❡✉♠❛ ♣✐râ♠✐❞❡ tr✐❛♥❣✉❧❛r é ✉♠ t❡rç♦ ❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣r✐s♠❛ tr✐❛♥❣✉❧❛r q✉❡ t❡♠ ❛ ♠❡s♠❛❜❛s❡✳ Pr♦✈❡♠♦s ❡♥tã♦ q✉❡ ❛s ♣✐râ♠✐❞❡s ❢♦r♠❛❞❛s tê♠ ♦ ♠❡s♠♦ ✈♦❧✉♠❡✳

❱❛♠♦s ❞❡♠♦♥str❛r q✉❡ ❛s três ♣✐râ♠✐❞❡s sã♦ ❛ ♣ró♣r✐❛ ABCB′✱ ❛ ♣✐râ♠✐❞❡ A′B′C ′A

✭❝♦♠ ❜❛s❡ ❝♦♥❣r✉❡♥t❡ à ❜❛s❡ ❞❡ ABCB′ ❡ ❝♦♠ ♠❡s♠❛ ❛❧t✉r❛✮ ❡ ❛ ♣✐râ♠✐❞❡ ACC ′B′✱ ❝✉❥❛❜❛s❡ ACC ′ é ❝♦♥❣r✉❡♥t❡ à ❜❛s❡ AA′C ′ ❞❛ s❡❣✉♥❞❛ ❡ ❝✉❥❛ ❛❧t✉r❛✱ ❛ ♣❛rt✐r ❞♦ ✈ért✐❝❡ B′✱ é✐❣✉❛❧ à ❛❧t✉r❛ ❞❛ s❡❣✉♥❞❛ ♣✐râ♠✐❞❡✱ AA′C ′B✱ ❛ ♣❛rt✐r ❞♦ ♠❡s♠♦ ✈ért✐❝❡ B′✳

✽✳✻ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ q✉❛❧q✉❡r ✭♥ã♦ s❡♥❞♦ tr✐❛♥✲❣✉❧❛r✮

P❛r❛ ❝❛❧❝✉❧❛r♠♦s ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ ❞❡ ❜❛s❡ ♥ã♦ tr✐❛♥❣✉❧❛r✱ ✈❛♠♦s r❡❝♦rr❡r à❢ór♠✉❧❛ ❞❡❞✉③✐❞❛ ♥❛ ú❧t✐♠❛ s❡çã♦ ❡ ✉s❛r ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ q✉❡ ❞✐③ q✉❡ só❧✐❞♦s ❛♣♦✐❛❞♦s❡♠ ✉♠ ♠❡s♠♦ ♣❧❛♥♦ q✉❡ tê♠ ❛❧t✉r❛s ❝♦♠ ♠❡❞✐❞❛s ✐❣✉❛✐s ❡ ❛s ár❡❛s ❞❡ s❡❝çõ❡s ♦❜t✐❞❛s ❛tr❛✈és❞❛ s❡❝çã♦ ❞❡ ♣❧❛♥♦s ♣❛r❛❧❡❧♦s às ❜❛s❡s ✐❣✉❛✐s tê♠ ✈♦❧✉♠❡s ✐❣✉❛✐s ❡ ❛ss✐♠ ♣♦❞❡♠♦s ❞❡❞✉③✐rq✉❡ ♦ ❝á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡ ❞❛ ♠❡s♠❛ é✿ ✉♠ t❡rç♦ ❞♦ ♣r♦❞✉t♦ ❞❛ ár❡❛ ❞❛ ❜❛s❡ ♣❡❧❛ ❛❧t✉r❛

✽✳✻✳ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ q✉❛❧q✉❡r ✭♥ã♦ s❡♥❞♦ tr✐❛♥❣✉❧❛r✮ ✹✹

❞❡st❡ só❧✐❞♦✳ P♦ré♠✱ ♣♦❞❡♠♦s ❡❢❡t✉❛r ❛ ❞❡♠♦♥str❛çã♦ ❞❡ ✉♠❛ ♦✉tr❛ ♠❛♥❡✐r❛ ❡ ♣❛r❛ ✐ss♦✉t✐❧✐③❛♠♦s✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱ ✉♠❛ ♣✐râ♠✐❞❡ ♣❡♥t❛❣♦♥❛❧ r❡❣✉❧❛r✳

❆ s✉❛ ❜❛s❡ ♣♦❞❡ s❡r ❛ss✐♠✿

❋✐❣✉r❛ ✽✳✼✿ P❡♥tá❣♦♥♦ ❡♠ n− 2 tr✐â♥❣✉❧♦s

◆❛ ❋✐❣✉r❛ ✽✳✼ ♦❜s❡r✈❛♠♦s q✉❡ ♦ ♣❡♥tá❣♦♥♦ ❢♦✐ ❞✐✈✐❞✐❞♦ ❡♠ três tr✐â♥❣✉❧♦s✳ ▲❡♠❜r❡♠♦sq✉❡ ❡♠ ✉♠ ♣♦❧í❣♦♥♦ q✉❛❧q✉❡r ❞❡ n ❧❛❞♦s✱ ♣♦❞❡♠♦s t❡r n− 2 tr✐â♥❣✉❧♦s✱ q✉❛♥❞♦ ❧✐❣❛♠♦s ✉♠✈ért✐❝❡ ❛♦s ♦✉tr♦s n− 3 ♥ã♦ ❛❞❥❛❝❡♥t❡s✳

❈❛❞❛ ✉♠ ❞❡ss❡s tr✐â♥❣✉❧♦s ♣♦❞❡♠ s❡r ❝❤❛♠❛❞♦s ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ tr✐â♥❣✉❧♦ ABE s❡rá❞❡♥♦♠✐♥❛❞♦ B1✱ ♦ tr✐â♥❣✉❧♦ BEC s❡rá ❞❡♥♦♠✐♥❛❞♦ B2 ❡ ♦ tr✐â♥❣✉❧♦ CED s❡rá ❞❡♥♦♠✐♥❛❞♦B3✳

❉❡st❛ ♠❛♥❡✐r❛✱ ❛ ♣✐râ♠✐❞❡ ❝♦♠ ❜❛s❡ ♥❡st❡ ♣♦❧í❣♦♥♦✱ ♣♦❞❡ s❡r ♠♦♥t❛❞❛ ❝♦♠♦ ♥❛ ❋✐❣✉r❛✽✳✽✿

P♦❞❡♠♦s ❡♥tã♦ ✈❡r✐✜❝❛r ❛ ❡①✐stê♥❝✐❛ ❞❡ três ♣✐râ♠✐❞❡s ❝♦♠ ❜❛s❡s tr✐❛♥❣✉❧❛r❡s ❡ ❝♦♠♠❡s♠❛ ❛❧t✉r❛ ❞❛ ♣✐râ♠✐❞❡ ❞✐t❛ ✏♦r✐❣✐♥❛❧✑✳ ❊st❛ ❛❧t✉r❛ ❞❡♥♦♠✐♥❛r❡♠♦s ❞❡ h✳

❖ ✈♦❧✉♠❡ ❞❛ ♣✐râ♠✐❞❡ ABCDEF s❡rá ❞❛❞♦ ♣♦r✿

V = V1 + V2 + V3 ❖♥❞❡ V1✱ V2 ❡ V3 sã♦ ♦s ✈♦❧✉♠❡s ❞❛s três s✉❜♣✐râ♠✐❞❡s✳

❏á s❛❜❡♠♦s q✉❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ tr✐❛♥❣✉❧❛r ♣♦❞❡ s❡r ❞❛❞♦ ♣♦r✿

V =Ab · h3

.

❙❡♥❞♦ Ab ❛ ➪r❡❛ ❞❛ ❇❛s❡ ❡ ❤ ❛ ❆❧t✉r❛ ❞❛ P✐râ♠✐❞❡✳

✽✳✼✳ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛ ✹✺

❋✐❣✉r❛ ✽✳✽✿ P✐râ♠✐❞❡ P❡♥t❛❣♦♥❛❧

❊ ❛ss✐♠ ❡♥tã♦ V = B1.h3

+ B2.h3

+ B3.h3

= 1

3h(B1 +B2 +B3) =

B.h3

❖♥❞❡ B é ❛ ár❡❛ ❞♦ ♣❡♥tá❣♦♥♦ ✭❙♦♠❛♥❞♦ ❛ ár❡❛ ❞♦s três tr✐â♥❣✉❧♦s✱ t❡r❡♠♦s ❛ ár❡❛ ❞♦♣❡♥tá❣♦♥♦✮✳

❈♦♥❝❧✉í♠♦s✱ ❡♥tã♦✱ q✉❡ ♦ ✈♦❧✉♠❡ ❞❛ ♣✐râ♠✐❞❡ ♣❡♥t❛❣♦♥❛❧ t❛♠❜é♠ ♣♦❞❡rá s❡r ❝❛❧❝✉❧❛❞❛♣♦r AREADABASE ·ALTURA

3✳

✽✳✼ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛

P❛r❛ ❞❡t❡r♠✐♥❛r♠♦s ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛✱ ✈❛♠♦s t♦♠❛r ✉♠ ❝✐❧✐♥❞r♦ ❡q✉✐❧át❡r♦ t❡♥❞♦r❛✐♦ ❞❛ ❜❛s❡ r ❡ ❛❧t✉r❛ 2r ❡ ❞♦✐s ❝♦♥❡s ❞❡ ❜❛s❡s ✐❣✉❛✐s às ❜❛s❡s ❞♦ ❝✐❧✐♥❞r♦ ❡ ❛❧t✉r❛ ✐❣✉❛❧ ❛ r✳❯♥✐♥❞♦ ❡st❡s ❞♦✐s só❧✐❞♦s ❛tr❛✈és ❞❡ s❡✉ ✈ért✐❝❡ ❝♦♠✉♠✱ t❡r❡♠♦s ✉♠ ♥♦✈♦ só❧✐❞♦ ❞❡♥♦♠✐♥❛❞♦❞❡ ❝❧❡♣s✐❞r❛ ✭❛ ✉♥✐ã♦ ❞♦s ❞♦✐s ❝♦♥❡s ❝♦♠ ♠❡s♠❛ ❛❧t✉r❛ ❡ ♠❡s♠♦s r❛✐♦s ❞❛ ❜❛s❡✱ ♦✉ s❡❥❛✱ ❞♦✐s❝♦♥❡s ✐❞ê♥t✐❝♦s✮✳ ❱❡❥❛ ❛s ❋✐❣✉r❛s ✽✳✾✿

❋✐❣✉r❛ ✽✳✾✿ ❈❧❡♣s✐❞r❛ ❡ ❆♥t✐❝❧❡♣s✐❞r❛✳ ❋♦♥t❡✿ ♦❜❛r✐❝❡♥tr♦❞❛♠❡♥t❡✳❜❧♦❣s♣♦t✳❝♦♠

❈✐❧✐♥❞r♦ ❞❡ r❛✐♦ r ❡ ❛❧t✉r❛ 2r✱❛ ❝❧❡♣s✐❞r❛ ❡ ❛ ❛♥t✐❝❧❡♣s✐❞r❛✱ q✉❡ é ♦ só❧✐❞♦ r❡s✉❧t❛♥t❡ ❞❛❡①❝❧✉sã♦ ❞❛ ❝❧❡♣s✐❞r❛ ❞❡ ✉♠ ❝✐❧✐♥❞r♦✱ q✉❡ ♥❛ ✈❡r❞❛❞❡ é ♦ só❧✐❞♦ q✉❡ ❡stá ♥♦ ✐♥t❡r✐♦r ❞♦ ❝✐❧✐♥❞r♦❡ é ❡①t❡r✐♦r ❛♦s ❞♦✐s ❝♦♥❡s ❥✉♥t♦s ✭❝❧❡♣s✐❞r❛✮✳

✽✳✼✳ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛ ✹✻

❈♦♥s✐❞❡r❡♠♦s ❛❣♦r❛ ✉♠❛ ❡s❢❡r❛ ❞❡ r❛✐♦ r✱ ✉♠❛ ❛♥t✐❝❧é♣s✐❞r❛ ❝♦♠♦ ❛ q✉❡ ❢♦✐ ❞❡✜♥✐❞❛❛♥t❡r✐♦r♠❡♥t❡ ❡ ✉♠ ♣❧❛♥♦ α q✉❡ s❡❝❝✐♦♥❛ ♦s ❞♦✐s só❧✐❞♦s s✐♠✉❧t❛♥❡❛♠❡♥t❡✳ ❱✐❞❡ ❛ ❋✐❣✉r❛✽✳✶✵✿

❋✐❣✉r❛ ✽✳✶✵✿ ❱♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛✳ ❋♦♥t❡✿ ♦❜❛r✐❝❡♥tr♦❞❛♠❡♥t❡✳❜❧♦❣s♣♦t✳❝♦♠

◗✉❛❧q✉❡r ♣❧❛♥♦ β q✉❡ é s❡❝❛♥t❡ à ❡s❢❡r❛ ❡ ♣❛r❛❧❡❧♦ ❛♦ ♣❧❛♥♦ α ❞✐st❛♥❞♦ h ✉♥✐❞❛❞❡s ❞❡♠❡❞✐❞❛ ❞♦ ❝❡♥tr♦ ❞❛ ❡s❢❡r❛ ❡ ❞♦ ✈ért✐❝❡ ❞❛ ❝❧❡♣s✐❞r❛✱ t❛♠❜é♠ s❡❝❝✐♦♥❛rá ❛ ❛♥t✐❝❧❡♣s✐❞r❛ ❡❛ss✐♠ s❡♥❞♦ t❡r❡♠♦s ❝♦♠♦ r❡s✉❧t❛❞♦ ❛s s❡❣✉✐♥t❡s ❡①♣r❡ssõ❡s ♣❛r❛ ❛s s❡❝çõ❡s✿

➪r❡❛ ❞❛ s❡❝çã♦ ♥❛ ❡s❢❡r❛✿ As❡❝çã♦✴❡s❢❡r❛= Πs2 = Π(r2 − h2✮

▲❡♠❜r❡♠♦s q✉❡ ❡st❛ s❡❝çã♦ é ♥❛ ✈❡r❞❛❞❡ ✉♠ ❝ír❝✉❧♦ ❞❡ r❛✐♦ s✳

➪r❡❛ ❞❛ s❡❝çã♦ ♥❛ ❛♥t✐❝❧❡♣s✐❞r❛✿ As❡❝çã♦✴❛♥t✐❝❧❡♣s✐❞r❛=Πr2 − Πh2 ❂ Π(r2 − h2✮✳

▲❡♠❜r❡♠♦s q✉❡ ❡st❛ s❡❝çã♦ é ♥❛ ✈❡r❞❛❞❡ ✉♠❛ ❝♦r♦❛ ❝✐r❝✉❧❛r ❝♦♠ ❝ír❝✉❧♦s ❞❡ r❛✐♦ ♠❛✐♦rs❡♥❞♦ r ❡ r❛✐♦ ♠❡♥♦r s❡♥❞♦ h✳

❈♦♠♦ ✈❡r✐✜❝❛♠♦s✱ ❛s ár❡❛s ❞❡ s❡❝çõ❡s sã♦ ❛s ♠❡s♠❛s ❡ ❛ss✐♠ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✱❛ ❡s❢❡r❛ ❡ ❛ ❛♥t✐❝❧❡♣s✐❞r❛ tê♠ ✈♦❧✉♠❡s ✐❣✉❛✐s✳

❱ESFERA = VANTICLEPSIDRA

❊ ❛ss✐♠ s❡♥❞♦ t❡r❡♠♦s ❡♥tã♦✿

VESFERA = VCILINDRO − 2 · VCONE = Πr2 · 2r − 2 ·1

3· Πr2 · r = 2 · Πr3 −

2

3· Πr3 =

4

3· Πr3

❈❛♣ít✉❧♦ ✾

Pr♦♣♦st❛ ♣❛r❛ ♦ tr❛❜❛❧❤♦ ❞✐❞át✐❝♦ ❝♦♠

♣r✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦

❈♦♠♦ ❥á ❝✐t❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡✱ ❛ ❛♣r❡s❡♥t❛çã♦ ❞♦ ♣r✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❞❡✈❡ s❡r ❢❡✐t❛♠❛✐s ♣r❡❝✐s❛♠❡♥t❡ ♥❛ s❡❣✉♥❞❛ sér✐❡ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ♣♦✐s é ♥❡ss❛ ❡t❛♣❛ q✉❡ ♦ ❛❧✉♥♦ t❡♠❝♦♥t❛t♦ ♠❛✐♦r ❝♦♠ ❛ ♣❛rt❡ ❞❛ ❣❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧✱ ❝♦♥❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛✳

❆♣r❡s❡♥t❛♠♦s ❛ s❡❣✉✐r ✉♠❛ ♣r♦♣♦st❛ ♣❛r❛ ♦ tr❛❜❛❧❤♦ ❞✐❞át✐❝♦ q✉❡ ❡stá s❡♥❞♦ ✉s❛❞❛ ❝♦♠♦❡①❡♠♣❧♦ ♥♦ ❧✐✈r♦ ✏❆ ♣rát✐❝❛ ❊❞✉❝❛t✐✈❛✿ ❝♦♠♦ ❡♥s✐♥❛r✑✱ ❞❡ ❆♥t♦♥✐ ❩❛❜❛❧❛ ❬✷✶❪✳ ❊ss❛ ♣r♦♣♦st❛❡stá ❡str✉t✉r❛❞❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

■✲ ❆♣r❡s❡♥t❛çã♦ ❞❡ ✉♠❛ s✐t✉❛çã♦ ♣r♦❜❧❡♠❛ ♣♦r ♣❛rt❡ ❞♦ ♣r♦❢❡ss♦r✳

❖ ♣r♦❢❡ss♦r ❡①♣õ❡ ❛♦s ❛❧✉♥♦s ✉♠❛ s✐t✉❛çã♦ q✉❡ ❣❡r❡ ✉♠ ❝♦♥✢✐t♦ ❡ q✉❡ ♣♦❞❡ s❡r r❡s♦❧✈✐❞❛❛tr❛✈és ❞❡ ♠❡✐♦s ♠❛t❡♠át✐❝♦s✳

■■✲ ❇✉s❝❛ ❞❡ s♦❧✉çõ❡s✳ ❖ ♣r♦❢❡ss♦r ♣❡❞❡ ❛♦s ❛❧✉♥♦s q✉❡ ❡①♣❧✐❝✐t❡♠ ❡ ❛♣r❡s❡♥t❡♠ ❞✐❢❡r❡♥t❡s❢♦r♠❛s ❞❡ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❡ ❛ s✐t✉❛çã♦✳

■■■✲ ❊①♣♦s✐çã♦ ❞♦ ❝♦♥❝❡✐t♦ ❡ ♦ ❛❧❣♦r✐t♠♦✳

❖ ♣r♦❢❡ss♦r ❛♣r♦✈❡✐t❛ ❛s ♣r♦♣♦st❛s ❞♦s ❛❧✉♥♦s ♣❛r❛ ❡❧❛❜♦r❛r ✉♠ ♥♦✈♦ ❝♦♥❝❡✐t♦ ❡ ❛í s✐♠❛♣r❡s❡♥t❛r ♦ ♠♦❞❡❧♦ ❢♦r♠❛❧ ❞♦ ❛❧❣♦r✐t♠♦ ♦✉ ❝♦♠♦ ♥♦ ❝❛s♦ ❞❛ ❣❡♦♠❡tr✐❛ ❡s♣❛❝✐❛❧✱ ❛s❢ór♠✉❧❛s ❛ s❡r❡♠ ✉s❛❞❛s ♥♦s ❡①❡r❝í❝✐♦s ❡ s✐t✉❛çõ❡s ❢✉t✉r❛s✳

■❱✲ ●❡♥❡r❛❧✐③❛çã♦✳

❖ ♣r♦❢❡ss♦r ❞❡♠♦♥str❛ ❛ ❢✉♥çã♦ ❞♦ ♠♦❞❡❧♦ ❝♦♥❝❡✐t✉❛❧ ❡ ♦ ❛❧❣♦r✐t♠♦✱ ♦✉ ❢ór♠✉❧❛s✱ ❡♠t♦❞❛s ❛q✉❡❧❛s s✐t✉❛çõ❡s q✉❡ ❝✉♠♣r❡♠ ❞❡t❡r♠✐♥❛❞❛s ❝♦♥❞✐çõ❡s✳

✹✼

✹✽

❱✲ ❆♣❧✐❝❛çã♦✳

❖s ❛❧✉♥♦s✱ ✐♥❞✐✈✐❞✉❛❧♠❡♥t❡ ♦✉ ❡♠ ❣r✉♣♦✱ ❛♣❧✐❝❛♠ ♦ ♠♦❞❡❧♦ ❡♠ ❞✐✈❡rs❛s s✐t✉❛çõ❡s✳

❱■✲ ❊①❡r❝✐t❛çã♦✳

❖s ❛❧✉♥♦s r❡s♦❧✈❡♠ ❛t✐✈✐❞❛❞❡s ✉s❛♥❞♦ ♦s ❛❧❣♦r✐t♠♦s ❡ ♥♦ ❝❛s♦ ❡s♣❡❝í✜❝♦ ❞❛ ❣❡♦♠❡tr✐❛✱✉s❛♠ ❛s ❢ór♠✉❧❛s ❛♣r❡s❡♥t❛❞❛s ❛♥t❡r✐♦r♠❡♥t❡✳

❱■■✲ Pr♦✈❛ ♦✉ ❡①❛♠❡✳

❊♠ ❝❧❛ss❡✱ ♦s ❛❧✉♥♦s r❡s♣♦♥❞❡♠ às q✉❡stõ❡s ❡♠ ✉♠❛ s✐♠✉❧❛çã♦ ♦✉ r❡❛❧✐③❛çã♦ ❡❢❡t✐✈❛❞❡ ✉♠❛ ♣r♦✈❛✳

❱■■■✲ ❆✈❛❧✐❛çã♦✳

❖s r❡s✉❧t❛❞♦s ❛❧❝❛♥ç❛❞♦s ♣❡❧♦s ❛❧✉♥♦s s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ❛♦s ♠❡s♠♦s ♣❡❧♦ ♣r♦❢❡ss♦r✳

❆ s❡q✉ê♥❝✐❛ ♣r♦♣♦st❛ ❛❝✐♠❛ ❝♦♠❡ç❛ s❡♠♣r❡ ❝♦♠ ✉♠❛ ♣r♦❜❧❡♠❛t✐③❛çã♦ s♦❜r❡ ♦ ❛ss✉♥t♦q✉❡ s❡rá ✐♥tr♦❞✉③✐❞♦✳ ❆♦ ❝♦❧❡t❛r ♦s ❝♦♠❡♥tár✐♦s ❝♦♠ ♦s ❛❧✉♥♦s ❝❤❡❣❛r❡♠♦s às s♦❧✉çõ❡s ❞❛sq✉❡stõ❡s ♣r♦♣♦st❛s ❡ ❛ss✐♠ s❡♥❞♦ ❛♣r❡s❡♥t❛r✲s❡✲á ❞❡✜♥✐çõ❡s ❡ ♦r✐❡♥t❛çõ❡s ♣❡rt✐♥❡♥t❡s ❛♦sq✉❡st✐♦♥❛♠❡♥t♦s ❡ ❛ss✉♥t♦s ♣r♦♣♦st♦s ✐♥✐❝✐❛❧♠❡♥t❡✳ ❖ ♣ró①✐♠♦ ♣❛ss♦ s❡rá ♣r♦♣♦r ✉♠❛ ❧✐st❛❞❡ ❛t✐✈✐❞❛❞❡s ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ♣r✐♥❝✐♣❛❧ ❞❡ ✜①❛r ♦ q✉❡ ❢♦✐ ❝♦♠❡♥t❛❞♦ ❡ ❛♣r❡s❡♥t❛❞♦✳ ❚❛✐s❡①❡r❝í❝✐♦s s❡rã♦ s❡❧❡❝✐♦♥❛❞♦s ❞❡ ❧✐✈r♦s ❞✐❞át✐❝♦s q✉❡ ❡stã♦ s❡♥❞♦ ❝✐t❛❞♦s ♥❛ ❜✐❜❧✐♦❣r❛✜❛ ❞❡✈❡st✐❜✉❧❛r❡s ❞❛s ♣r✐♥❝✐♣❛✐s ❯♥✐✈❡rs✐❞❛❞❡s ❞♦ ♣❛ís✱ ❛❧é♠ ❞❡ ❛t✐✈✐❞❛❞❡s ❞♦ ❊①❛♠❡ ◆❛❝✐♦♥❛❧❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ✭❊◆❊▼✮✳ ❊st❛ s❡q✉ê♥❝✐❛ ♥ã♦ ❢♦✐ ❛♣❧✐❝❛❞❛ ❡♠ ♥❡♥❤✉♠❛ ❝❧❛ss❡ ❞♦ ❡♥s✐♥♦r❡❣✉❧❛r ❡ ❛ss✉♠❡✲s❡ ♦ ❝♦♠♣r♦♠✐ss♦ ❞❡ r❡❛❧✐③á✲❧❛ ❛♣r❡s❡♥t❛♥❞♦ ♦ s❡✉ r❡s✉❧t❛❞♦ ❡♠ ✉♠ ♣ró①✐♠♦tr❛❜❛❧❤♦✳

❊①❡r❝í❝✐♦s q✉❡ ♣♦❞❡♠ s❡r tr❛❜❛❧❤❛❞♦s ❡ q✉❡ ❡♥✈♦❧✈❡♠ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✿

✶✳ ❯♠ ♣r✐s♠❛ t❡♠ ♣♦r ❜❛s❡ ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ ❝✉❥♦ ❧❛❞♦ ❡ ❛ ❛❧t✉r❛ ❞❡ss❡ ♣r✐s♠❛ é✐❣✉❛❧ ❛♦ ❞♦❜r♦ ❞❛ ❛❧t✉r❛ ❞♦ tr✐â♥❣✉❧♦ ❞❛ ❜❛s❡✳ ❉❡t❡r♠✐♥❡ ❡♥tã♦ ♦ ✈♦❧✉♠❡ ❞❡ss❡ ♣r✐s♠❛✳

✷✳ ❯♠ ♣r✐s♠❛ r❡t♦ t❡♠ ♣♦r ❜❛s❡ ✉♠ q✉❛❞r❛❞♦ ✐♥s❝r✐t♦ ❡♠ ✉♠ ❝ír❝✉❧♦ ❞❡ ✷ ♠ ❞❡ r❛✐♦ ❡❛❧t✉r❛ ♠❡❞✐♥❞♦ ♦ ❞♦❜r♦ ❞❛ ❞✐❛❣♦♥❛❧ ❞❡st❡ q✉❛❞r❛❞♦✳ ◗✉❛❧ é ♦ s❡✉ ✈♦❧✉♠❡❄

✸✳ ✭❊◆❊▼ ✲ ✷✵✶✵ ✲ ❆❞❛♣t❛❞♦✮ ✲ ❯♠❛ ❢á❜r✐❝❛ ♣r♦❞✉③ ❜❛rr❛s ❞❡ ❝❤♦❝♦❧❛t❡s ♥♦ ❢♦r♠❛t♦ ❞❡♣❛r❛❧❡❧❡♣í♣❡❞♦s ❡ ❞❡ ❝✉❜♦s ❝♦♠ ♦ ♠❡s♠♦ ✈♦❧✉♠❡✳ ❆s ❛r❡st❛s ❞❛ ❜❛rr❛ ❞❡ ❝❤♦❝♦❧❛t❡ ♥♦❢♦r♠❛t♦ ❞❡ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ♠❡❞❡♠ ✸ ❝♠ ❞❡ ❧❛r❣✉r❛✱ ✶✽ ❝♠ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❡ ✹ ❝♠ ❞❡❡s♣❡ss✉r❛✳ ❆♥❛❧✐s❛♥❞♦ ❛s ❝❛r❛❝t❡ríst✐❝❛s ❞❛s ✜❣✉r❛s ❣❡♦♠étr✐❝❛s ❞❡s❝r✐t❛s✱ ❞❡t❡r♠✐♥❡❛ ♠❡❞✐❞❛ ❞❛s ❛r❡st❛s ❞♦s ❝❤♦❝♦❧❛t❡s q✉❡ tê♠ ♦ ❢♦r♠❛t♦ ❞❡ ❝✉❜♦✳

✹✾

❋✐❣✉r❛ ✾✳✶✿ ❊♥❡♠ ✷✵✶✵

✹✳ ✭❊◆❊▼ ✲ ✷✵✶✵✮ ✲ ❉♦♥❛ ▼❛r✐❛✱ ❞✐❛r✐st❛ ♥❛ ❝❛s❛ ❞❛ ❢❛♠í❧✐❛ ❚❡✐①❡✐r❛✱ ♣r❡❝✐s❛ ❢❛③❡r ❝❛❢é♣❛r❛ s❡r✈✐r ❛s ✈✐♥t❡ ♣❡ss♦❛s q✉❡ s❡ ❡♥❝♦♥tr❛♠ ♥✉♠❛ r❡✉♥✐ã♦ ♥❛ s❛❧❛✳ P❛r❛ ❢❛③❡r ♦ ❝❛❢é✱❉♦♥❛ ▼❛r✐❛ ❞✐s♣õ❡ ❞❡ ✉♠❛ ❧❡✐t❡✐r❛ ❝✐❧í♥❞r✐❝❛ ❡ ❝♦♣✐♥❤♦s ♣❧ást✐❝♦s✱ t❛♠❜é♠ ❝✐❧í♥❞r✐❝♦s✳

❈♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ♥ã♦ ❞❡s♣❡r❞✐ç❛r ❝❛❢é✱ ❛ ❞✐❛r✐st❛ ❞❡s❡❥❛ ❝♦❧♦❝❛r ❛ q✉❛♥t✐❞❛❞❡ ♠í♥✐♠❛❞❡ á❣✉❛ ♥❛ ❧❡✐t❡✐r❛ ♣❛r❛ ❡♥❝❤❡r ♦s ✈✐♥t❡ ❝♦♣✐♥❤♦s ♣❡❧❛ ♠❡t❛❞❡✳ P❛r❛ q✉❡ ✐ss♦ ♦❝♦rr❛✱❉♦♥❛ ▼❛r✐❛ ❞❡✈❡rá✿

❛✮ ❡♥❝❤❡r ❛ ❧❡✐t❡✐r❛ ❛té ❛ ♠❡t❛❞❡✱ ♣♦✐s ❡❧❛ t❡♠ ✉♠ ✈♦❧✉♠❡ ✷✵ ✈❡③❡s ♠❛✐♦r q✉❡ ♦ ✈♦❧✉♠❡❞♦ ❝♦♣♦✳❜✮ ❡♥❝❤❡r ❛ ❧❡✐t❡✐r❛ t♦❞❛ ❞❡ á❣✉❛✱ ♣♦✐s ❡❧❛ t❡♠ ✉♠ ✈♦❧✉♠❡ ✷✵ ✈❡③❡s ♠❛✐♦r q✉❡ ♦ ✈♦❧✉♠❡❞♦ ❝♦♣♦✳❝✮ ❡♥❝❤❡r ❛ ❧❡✐t❡✐r❛ t♦❞❛ ❞❡ á❣✉❛✱ ♣♦✐s ❡❧❛ t❡♠ ✉♠ ✈♦❧✉♠❡ ✶✵ ✈❡③❡s ♠❛✐♦r q✉❡ ♦ ✈♦❧✉♠❡❞♦ ❝♦♣♦✳❞✮ ❡♥❝❤❡r ❞✉❛s ❧❡✐t❡✐r❛s ❞❡ á❣✉❛✱ ♣♦✐s ❡❧❛ t❡♠ ✉♠ ✈♦❧✉♠❡ ✶✵ ✈❡③❡s ♠❛✐♦r q✉❡ ♦ ✈♦❧✉♠❡❞♦ ❝♦♣♦✳❡✮ ❡♥❝❤❡r ❝✐♥❝♦ ❧❡✐t❡✐r❛s ❞❡ á❣✉❛✱ ♣♦✐s ❡❧❛ t❡♠ ✉♠ ✈♦❧✉♠❡ ✶✵ ✈❡③❡s ♠❛✐♦r q✉❡ ♦ ✈♦❧✉♠❡❞♦ ❝♦♣♦✳

✺✳ ❆ ❜❛s❡ ❞❡ ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ♦❜❧íq✉♦ é ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ ①✳ ❯♠❛ ❞❛s ❛r❡st❛s❧❛t❡r❛✐s é ② ❡ ❢♦r♠❛ ✉♠ â♥❣✉❧♦ ❞❡ ✻✵➸ ❝♦♠ ♦s ❧❛❞♦s ❛❞❥❛❝❡♥t❡s ❞❛ ❜❛s❡✳ ❉❡t❡r♠✐♥❡ ♦✈♦❧✉♠❡ ❞❡st❡ ♣❛r❛❧❡❧❡♣í♣❡❞♦✳

✻✳ ✭❋●❱ ✲ ❙P✮ ✲ ❯♠❛ ♣✐s❝✐♥❛ t❡♠ ♦ ❢♦r♠❛t♦ ❞❡ ✉♠ ♣r✐s♠❛ ❤❡①❛❣♦♥❛❧ r❡❣✉❧❛r ❝♦♠ ♣r♦✲❢✉♥❞✐❞❛❞❡ ✐❣✉❛❧ ❛ ♠✳ ❈❛❞❛ ❧❛❞♦ ❞♦ ❤❡①á❣♦♥♦ ♠❡❞❡ ✷ ♠✳ ❉❡t❡r♠✐♥❡ ❡♥tã♦ ♦ ✈♦❧✉♠❡ ❞❡á❣✉❛ ♥❡❝❡ssár✐♦ ♣❛r❛ ❡♥❝❤❡r ✽✵✪ ❞♦ ✈♦❧✉♠❡ ❞❛ ♣✐s❝✐♥❛✳

✼✳ ✭❯❋❯ ✲ ▼●✮ ✲ ◆❛ ✜❣✉r❛ ❛ s❡❣✉✐r✱ t❡♠♦s ✉♠ ❝✉❜♦ ❆❇❈❉❊❋●❍ ❞❡ ❛r❡st❛ ❛ ❂ ✻ ❝♠✳❖s ♣♦♥t♦s ■✱ ❏✱ ❑✱ ▲✱ ▼ ❡ ◆ sã♦ ♣♦♥t♦s ♠é❞✐♦s ❞❛s ❛r❡st❛s ❛ q✉❡ ♣❡rt❡♥❝❡♠✳

❉❡t❡r♠✐♥❡ ♦ ✈♦❧✉♠❡ ❞❛ ♣✐râ♠✐❞❡ ❞❡ ❜❛s❡ ❤❡①❛❣♦♥❛❧ IJKLMN ❡ ✈ért✐❝❡ H✳

✺✵

❋✐❣✉r❛ ✾✳✷✿ ❯❋❯ ✲ ▼●

✽✳ ✭❯❋❘❙✮ ✲ ❆ ♣❛rt✐r ❞❡ q✉❛tr♦ ❞♦s ✈ért✐❝❡s ❞❡ ✉♠ ❝✉❜♦ ❞❡ ❛r❡st❛ ✻✱ ❝♦♥str✉í❞♦ ❝♦♠♠❛❞❡✐r❛ ♠❛❝✐ç❛✱ ❢♦r❛♠ r❡❝♦rt❛❞❛s ♣✐râ♠✐❞❡s tr✐❛♥❣✉❧❛r❡s ❝♦♥❣r✉❡♥t❡s✱ ❝❛❞❛ ✉♠❛ t❡♥❞♦três ❛r❡st❛s ❞❡ ♠❡❞✐❞❛ ✸✱ ❝♦♥❢♦r♠❡ r❡♣r❡s❡♥t❛❞♦ ♥❛ ♣r✐♠❡✐r❛ ✐♠❛❣❡♠ ❞❛ ❋✐❣✉r❛ ✾✳✸✳

❖ só❧✐❞♦ ♦❜t✐❞♦ ❛♣ós ❛ r❡t✐r❛❞❛ ❞❛s ♣✐râ♠✐❞❡s ❡stá r❡♣r❡s❡♥t❛❞♦ ♥❛ s❡❣✉♥❞❛ ✐♠❛❣❡♠❞❛ ❋✐❣✉r❛ ✾✳✸✱ ❛ s❡❣✉✐r✳

❋✐❣✉r❛ ✾✳✸✿ ❯❋❘❙

❖ ✈♦❧✉♠❡ ❞♦ só❧✐❞♦ ♦❜t✐❞♦ é

❛✮ ✶✾✽✳ ❜✮ ✷✵✹✳ ❝✮ ✷✵✽✳ ❞✮ ✷✶✷✳ ❡✮ ✷✶✻✳

✺✶

✾✳ ✭❊◆❊▼ ✲ ✷✵✶✵✮ ✲ ❯♠ ♣♦rt❛✲❧á♣✐s ❞❡ ♠❛❞❡✐r❛ ❢♦✐ ❝♦♥str✉í❞♦ ♥♦ ❢♦r♠❛t♦ ❝ú❜✐❝♦✱ s❡❣✉✐♥❞♦♦ ♠♦❞❡❧♦ ✐❧✉str❛❞♦ ❛ s❡❣✉✐r✿

❋✐❣✉r❛ ✾✳✹✿ ❊◆❊▼✱ ✷✵✶✵

❖ ❝✉❜♦ ❞❡ ❞❡♥tr♦ é ✈❛③✐♦✳ ❆ ❛r❡st❛ ❞♦ ❝✉❜♦ ♠❛✐♦r ♠❡❞❡ ✶✷ ❝♠ ❡ ❛ ❞♦ ❝✉❜♦ ♠❡♥♦r✱ q✉❡é ✐♥t❡r♥♦✱ ♠❡❞❡ ✽ ❝♠✳ ❖ ✈♦❧✉♠❡ ❞❡ ♠❛❞❡✐r❛ ✉t✐❧✐③❛❞♦ ♥❛ ❝♦♥❢❡❝çã♦ ❞❡ss❡ ♦❜❥❡t♦ ❢♦✐ ❞❡✿

❛✮ ✶✷ cm3✳❜✮ ✻✹ cm3✳❝✮ ✾✻ cm3✳❞✮ ✶✷✶✻ cm3✳❡✮ ✶✼✷✽ cm3✳

✶✵✳ ❉❡t❡r♠✐♥❛r ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ❝✐❧✐♥❞r♦ ✐♥s❝r✐t♦ ❡♠ ✉♠ ❝✉❜♦ ❞❡ ❛r❡st❛ ♠❡❞✐♥❞♦ ✷✵ ❝♠✳

❋✐❣✉r❛ ✾✳✺✿ ❈✐❧✐♥❞r♦ ✐♥s❝r✐t♦✳ ❋♦♥t❡✿ ✇✇✇✳♠❛❡str♦♣❧♦♠❡r♦✳❝♦♠

✶✶✳ ❯♠ r❡s❡r✈❛tór✐♦ t❡♠ ❛ ❢♦r♠❛ ❞❡ ✉♠ ❤❡♠✐s❢ér✐♦✳ ◗✉❛❧ ♦ ✈♦❧✉♠❡ ♠á①✐♠♦ ❞❡ ❧íq✉✐❞♦ q✉❡❝❛❜❡ ♥❡ss❡ r❡s❡r✈❛tór✐♦ ❡♠ ❧✐tr♦s❄ ❯s❡ Π ❂ ✸✳

❉❛❞♦✿ ❖ ❞✐â♠❡tr♦ ❞❛ s❡♠✐❡s❢❡r❛ ✈❛❧❡ ✺ ♠✳

✺✷

❋✐❣✉r❛ ✾✳✻✿ ❙❡♠✐❡s❢❡r❛✳ ❋♦♥t❡✿ ✇✇✇✳♠❛❡str♦♣❧♦♠❡r♦✳❝♦♠

✶✷✳ ◗✉❛❧ é ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ t❡tr❛❡❞r♦ r❡❣✉❧❛r ❞❡ ❛r❡st❛ a❄

✶✸✳ ◗✉❛♥❞♦ ❞✉❛s ♣✐râ♠✐❞❡s r❡❣✉❧❛r❡s ❞❡ ❜❛s❡ q✉❛❞r❛❞❛s ❡ ❝✉❥❛s ❢❛❝❡s ❧❛t❡r❛✐s sã♦ tr✐â♥❣✉❧♦s❡q✉✐❧át❡r♦s sã♦ ❝♦❧♦❝❛❞❛s ❜❛s❡ ❛ ❜❛s❡✱ ♦ só❧✐❞♦ r❡s✉❧t❛♥t❡ é ❝❤❛♠❛❞♦ ❞❡ ♦❝t❛❡❞r♦ r❡❣✉❧❛r✳❈❛❧❝✉❧❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ♦❝t❛❡❞r♦ r❡❣✉❧❛r ❞❡ ❛r❡st❛ ♠❡❞✐♥❞♦ ✺ ❝♠✳

✶✹✳ ◗✉❛❧ é ❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ✉♠❛ ❧❛t❛ ❞❡ ♠♦❧❤♦ ❞❡ t♦♠❛t❡ q✉❡ t❡♠ ❛ ❢♦r♠❛ ❝✐❧í♥❞r✐❝❛ ❝♦♠✽ ❝♠ ❞❡ ❞✐â♠❡tr♦ ❡ ✶✶ ❝♠ ❞❡ ❛❧t✉r❛❄ ✭❯s❡ Π ❂ ✸✮

❋✐❣✉r❛ ✾✳✼✿ ❊①tr❛t♦ ❞❡ ❚♦♠❛t❡✿ ❘❡❛❧✐❞❛❞❡❀ ❈✐❧✐♥❞r♦✿ ▼♦❞❡❧♦ ▼❛t❡♠át✐❝♦

✶✺✳ ❯♠ t❛♥q✉❡ ❝ô♥✐❝♦ t❡♠ ✹ ♠ ❞❡ ♣r♦❢✉♥❞✐❞❛❞❡ ❡ s❡✉ t♦♣♦ ❝✐r❝✉❧❛r t❡♠ ✻ ♠ ❞❡ ❞✐â♠❡tr♦✳◗✉❛❧ ♦ ✈♦❧✉♠❡ ♠á①✐♠♦✱ ❡♠ ❧✐tr♦s✱ q✉❡ ❡ss❡ t❛♥q✉❡ ♣♦❞❡ ❝♦♥t❡r ❞❡ ❧íq✉✐❞♦❄

✾✳✶✳ ❆t✐✈✐❞❛❞❡s ❛❧t❡r♥❛t✐✈❛s q✉❡ ❡♥✈♦❧✈❡♠ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ✺✸

✾✳✶ ❆t✐✈✐❞❛❞❡s ❛❧t❡r♥❛t✐✈❛s q✉❡ ❡♥✈♦❧✈❡♠ ♦ Pr✐♥❝í♣✐♦ ❞❡❈❛✈❛❧✐❡r✐

✾✳✶✳✶ ❆t✐✈✐❞❛❞❡ ✶

❱❡r✐✜❝❛çã♦ ❡ ✈❛❧✐❞❛❞❡ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳

❖❜❥❡t✐✈♦ ❞❛ ❛t✐✈✐❞❛❞❡

❉❡s♣❡rt❛r ♥♦s ❛❧✉♥♦s ❛ s✉❛ ✐♥t✉✐çã♦ ♣❛r❛ q✉❡ ♣♦ss❛♠ ♣❡r❝❡❜❡r ❛ r❡❛❧ ✈❛❧✐❞❛❞❡ ❞♦ Pr✐♥❝í♣✐♦❞❡ ❈❛✈❛❧✐❡r✐✱ ❛❝❡✐t❛♥❞♦ ❛ss✐♠ ❡st❡ Pr✐♥❝í♣✐♦ ❝♦♠♦ ✉♠ ❛①✐♦♠❛✱ ✈❡r✐✜❝❛♥❞♦ ❛ s✉❛ ✈❛❧✐❞❛❞❡ ❡❝♦♠♣r♦✈❛♥❞♦ ✈✐❛ ❝á❧❝✉❧♦ ❛ ✐❣✉❛❧❞❛❞❡ ❞♦s ✈♦❧✉♠❡s ❛ s❡r❡♠ ❝♦♠♣❛r❛❞♦s✳ ❈♦♠ ♦ ❛✉①í❧✐♦ ❞♦s♦❢t✇❛r❡ ●❡♦❣❡❜r❛ ✺✳✵ ❝♦♠ ✈❡rsã♦ ❞❡ ❥❛♥❡❧❛ ✸❉✱ ♦s ❛❧✉♥♦s t♦♠❛rã♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ só❧✐❞♦s❝♦♠ ♦s q✉❛✐s tr❛❜❛❧❤❛rã♦✳

▼❛t❡r✐❛✐s ♥❡❝❡ssár✐♦s

• ❈♦♠♣✉t❛❞♦r❡s q✉❡ ♣♦ss✉❡♠ ♦ s♦❢t✇❛r❡ ●❡♦❣❡❜r❛ ✺✳✵ ❝♦♠ ✈❡rsã♦ ❞❡ ❥❛♥❡❧❛ ✸❉✳

• ❖ ❛rq✉✐✈♦ ❡❧❡trô♥✐❝♦ ❈❛✈❛❧✐❡r✐■❧✳❣❣❜✳

• ▲✐✈r♦ ❞✐❞át✐❝♦ ❝♦♠ ❛ss✉♥t♦ ❛ r❡s♣❡✐t♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳

❚❡♠♣♦ ♥❡❝❡ssár✐♦

❙❡rã♦ ♥❡❝❡ssár✐♦s ✺✵ ♠✐♥✉t♦s ✭✶ ❛✉❧❛✮ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❛ ❛t✐✈✐❞❛❞❡✱ ♦♥❞❡ ♦s ❛❧✉♥♦s❜❛s✐❝❛♠❡♥t❡ tr❛❜❛❧❤❛rã♦ ❡♠ ❝♦♠♣✉t❛❞♦r❡s ✉s❛♥❞♦ ♦ s♦❢t✇❛r❡✳

Pré✲r❡q✉✐s✐t♦s ❜ás✐❝♦s ♣❛r❛ r❡❛❧✐③❛çã♦ ❞❛ ❛t✐✈✐❞❛❞❡

• ❈♦♥❤❡❝✐♠❡♥t♦ ♠í♥✐♠♦ ❞❡ ✐♥❢♦r♠át✐❝❛ ❜ás✐❝❛ ❡ ❝♦♥s❡q✉❡♥t❡ ♠❛♥✐♣✉❧❛çã♦ ❞♦ s♦❢t✇❛r❡❀

• ❈♦♥❤❡❝✐♠❡♥t♦s ❜ás✐❝♦s ❞❡ ❝á❧❝✉❧♦s ✈♦❧✉♠étr✐❝♦s ❡ tr❛❜❛❧❤♦ ❝♦♠♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s ❡r❛❝✐♦♥❛✐s✳

✾✳✶✳ ❆t✐✈✐❞❛❞❡s ❛❧t❡r♥❛t✐✈❛s q✉❡ ❡♥✈♦❧✈❡♠ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ✺✹

❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❛t✐✈✐❞❛❞❡

■♥✐❝✐❛❧♠❡♥t❡ s❡rá ❛♣r❡s❡♥t❛❞♦ ♦ s♦❢t✇❛r❡ ♣❛r❛ ♦s ❛❧✉♥♦s✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ♣❛r❛ ❛q✉❡❧❡s q✉❡♥ã♦ t❡♥❤❛♠ ❝♦♥❤❡❝✐♠❡♥t♦ s♦❜r❡ ♦ ♠❡s♠♦✳ ▲♦❣♦ ❛♣ós ✐r❡♠♦s ❛❜r✐r ♦ ❛rq✉✐✈♦ ❈❛✈❛❧✐❡r✐■■✳❣❣❜❡ ❡♥tã♦ ✈✐s✉❛❧✐③❛r❡♠♦s ❡♠ s✉❛ ✐♥t❡r❢❛❝❡ ❞♦✐s só❧✐❞♦s q✉❡ ♣♦❞❡rã♦ s❡r ♠❛♥✐♣✉❧❛❞♦s ❝♦♠ ❝♦✲♠❛♥❞♦s ♦❢❡r❡❝✐❞♦s ♥♦ ❛rq✉✐✈♦ ❡ q✉❡ t❛♠❜é♠ ✐rã♦ ❢❛❝✐❧✐t❛r ♦ ❝á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡ ❞♦s ♠❡s♠♦s✱❥✉st✐✜❝❛❞♦ ♣❡❧♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳

✾✳✶✳✷ ❆t✐✈✐❞❛❞❡ ✲ ✷

❈á❧❝✉❧♦ ❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ❝✐❧✐♥❞r♦✱ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ ❡ ❞❡ ✉♠ ❝♦♥❡✳

❖❜❥❡t✐✈♦ ❞❛ ❛t✐✈✐❞❛❞❡

❊st❛ ❛t✐✈✐❞❛❞❡ t❡♠ ❝♦♠♦ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❛♣r❡s❡♥t❛r ❛♦s ❛❧✉♥♦s ✉♠❛ ♠❛♥❡✐r❛ ♣rát✐❝❛ ❞❡❝♦♥str✉✐r ✉♠ ❝✐❧✐♥❞r♦ ❞❡ r❡✈♦❧✉çã♦ ❡ ❝♦♠ s❡✉s ❡❧❡♠❡♥t♦s q✉❡ s❡rã♦ ❛♣r❡s❡♥t❛❞♦s ♥♦ s♦❢t✇❛r❡❝❛❧❝✉❧❛r ♦ s❡✉ ✈♦❧✉♠❡ ❡ ❛s s✉❛s ár❡❛s ✭❞❛ ❜❛s❡ ❡ ❧❛t❡r❛❧✮✳ ❆❧é♠ ❞✐ss♦✱ s❡rá ♣♦ssí✈❡❧ ❝♦♥str✉✐r♣✐râ♠✐❞❡s ❡ ❝♦♥❡s ♣❛r❛ q✉❡ ♣♦ss❛♠♦s t❛♠❜é♠ ❡❢❡t✉❛r ♦ ❝á❧❝✉❧♦ ❞❡ s❡✉ ✈♦❧✉♠❡✱ ✈✐s✉❛❧✐③❛♥❞♦♦s ❡❧❡♠❡♥t♦s ✐♠♣♦rt❛♥t❡s ❞♦s ♠❡s♠♦s✳

▼❛t❡r✐❛✐s ♥❡❝❡ssár✐♦s

• ❈♦♠♣✉t❛❞♦r❡s q✉❡ ♣♦ss✉❡♠ ♦ s♦❢t✇❛r❡ ●❡♦❣❡❜r❛ ✺✳✵ ❝♦♠ ✈❡rsã♦ ❞❡ ❥❛♥❡❧❛ ✸❉✳

• ❖s ❛rq✉✐✈♦s ❡❧❡trô♥✐❝♦s ❝❛✈❛❧✐❡r✐❝♦♥❡♣✐r❛♠✐❞❡✳❣❣❜✱ ❝✐❧✐♥❞r♦ ❡ ❝♦♥❡✳❣❣❜✱ ❝✐❧✐♥❞r♦r❡✈♦❧✳❣❣❜✳

• ▲✐✈r♦ ❞✐❞át✐❝♦ ❝♦♠ ❛ss✉♥t♦ ❛ r❡s♣❡✐t♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✳

• ❈❛❞❡r♥♦ ❡ ❧á♣✐s ♣❛r❛ ❡❢❡t✉❛r ❛♥♦t❛çõ❡s ❡ ♣❡q✉❡♥♦s ❝á❧❝✉❧♦s✳

▼❛t❡r✐❛✐s ♥❡❝❡ssár✐♦s

❙❡rã♦ ♥❡❝❡ssár✐♦s ✺✵ ♠✐♥✉t♦s ✭✶ ❛✉❧❛✮ ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❛ ❛t✐✈✐❞❛❞❡✱ ♦♥❞❡ ♦s ❛❧✉♥♦s❜❛s✐❝❛♠❡♥t❡ tr❛❜❛❧❤❛rã♦ ❡♠ ❝♦♠♣✉t❛❞♦r❡s ✉s❛♥❞♦ ♦ s♦❢t✇❛r❡✳

✾✳✶✳ ❆t✐✈✐❞❛❞❡s ❛❧t❡r♥❛t✐✈❛s q✉❡ ❡♥✈♦❧✈❡♠ ♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ✺✺

Pré✲r❡q✉✐s✐t♦s ❜ás✐❝♦s ♣❛r❛ r❡❛❧✐③❛çã♦ ❞❛ ❛t✐✈✐❞❛❞❡

• ❈♦♥❤❡❝✐♠❡♥t♦ ♠í♥✐♠♦ ❞❡ ✐♥❢♦r♠át✐❝❛ ❜ás✐❝❛ ❡ ❝♦♥s❡q✉❡♥t❡ ♠❛♥✐♣✉❧❛çã♦ ❞♦ s♦❢t✇❛r❡✳

• ❈♦♥❤❡❝✐♠❡♥t♦s ❜ás✐❝♦s ❞❡ ❝á❧❝✉❧♦s ✈♦❧✉♠étr✐❝♦s ❡ tr❛❜❛❧❤♦ ❝♦♠♦ ♥ú♠❡r♦s ✐♥t❡✐r♦s ❡r❛❝✐♦♥❛✐s✳

• ❈♦♥❤❡❝✐♠❡♥t♦ ❜ás✐❝♦ ❛❝❡r❝❛ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ❡ ❞❛ ❝♦♥str✉çã♦ ❞❡ só❧✐❞♦s ❣❡♦✲♠étr✐❝♦s✳

❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❛t✐✈✐❞❛❞❡

❈♦♠ ♦ s♦❢t✇❛r❡ ❞❡✈✐❞❛♠❡♥t❡ ❛♣r❡s❡♥t❛❞♦ ❛♦s ❛❧✉♥♦s ❡ ❡st❡s ❥á ❞❡ ♣♦ss❡ ❞♦s ❛rq✉✐✈♦s✈❛♠♦s ✐♥✐❝✐❛❧♠❡♥t❡ ✐❞❡♥t✐✜❝❛r ❡❧❡♠❡♥t♦s ❞♦s só❧✐❞♦s✱ ❛♥♦t❛♥❞♦ ❡♠ ✉♠ ❝❛❞❡r♥♦ ✈❛❧♦r❡s q✉❡s❡rã♦ ❛tr✐❜✉í❞♦s ❛ ❡st❡s ❡❧❡♠❡♥t♦s ❡ ❛ s❡❣✉✐r ✈❛♠♦s ❡♥tã♦ ✈❡r✐✜❝❛r✱ ♣♦r ♠❡✐♦ ❞❡ ❝á❧❝✉❧♦s ♥♦❝❛❞❡r♥♦✱ ♦s ✈❛❧♦r❡s ❞❡ ✈♦❧✉♠❡ ❞❡st❡s só❧✐❞♦s✱ ❝♦♠♣❛r❛♥❞♦ ❝♦♠ ♦s ♠❡s♠♦s ❛♣r❡s❡♥t❛❞♦s ♥♦s♦❢t✇❛r❡✳

❈❛♣ít✉❧♦ ✶✵

❘❡s♦❧✉çã♦ ❞♦s ❡①❡r❝í❝✐♦s t❡ór✐❝♦s ❞♦

❝❛♣ít✉❧♦ ✾✳

✶✳ ❱❛♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ ❡st❛❜❡❧❡❝❡r ♦ ❞❡s❡♥❤♦ ❞♦ ♣r✐s♠❛ ♣r♦♣♦st♦ ♥❛ ❛t✐✈✐❞❛❞❡✳

❋✐❣✉r❛ ✶✵✳✶✿ Pr✐s♠❛ ❚r✐❛♥❣✉❧❛r

❈♦♠♦ ❛ ❜❛s❡ é ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦✱ t❡♠♦s✿ AB = x2√3

4✳

❊♠ s❡ tr❛t❛♥❞♦ ❞❡ ✉♠ ♣r✐s♠❛ tr✐❛♥❣✉❧❛r✱ ❝♦♥s❡❣✉✐♠♦s✱ ❛tr❛✈és ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✲✈❛❧✐❡r✐✱ ♦❜t❡r q✉❡ ♦ ✈♦❧✉♠❡ ❞❡ q✉❛❧q✉❡r ♣r✐s♠❛ s❡rá V = AB.h✱ ♦♥❞❡ h é ❛ ❛❧t✉r❛ ❞♦♣r✐s♠❛✳ P♦rt❛♥t♦✿

x2√3

4· x

√3 =

3x3

4

✷✳ ❊st❛ s❡r✐❛ ❡♥tã♦ ❛ ✈✐s✉❛❧✐③❛çã♦ ♣❧❛♥✐✜❝❛❞❛ ❞❛ ❜❛s❡ ✐♥s❝r✐t❛ ❡♠ ✉♠ ❝ír❝✉❧♦✳ P♦rt❛♥t♦✱❛ ❞✐❛❣♦♥❛❧ ❞♦ q✉❛❞r❛❞♦ ♠❡❞✐r✐❛ 4m✳ ❈♦♠♦ s❛❜❡♠♦s q✉❡ ❛ ❞✐❛❣♦♥❛❧ ❞❡ ✉♠ q✉❛❞r❛❞♦ é

✺✻

✺✼

❋✐❣✉r❛ ✶✵✳✷✿ ◗✉❛❞r❛❞♦ ✐♥s❝r✐t♦ ❡♠ ✉♠ ❝ír❝✉❧♦

❞❛❞❛ ♣❡❧❛ ❡①♣r❡ssã♦ l√2✱ ♣♦❞❡♠♦s ❡♥tã♦ ❝♦♥❝❧✉✐r q✉❡ ♦ ❧❛❞♦ ❞❡st❡ q✉❛❞r❛❞♦ t♦♠❛❞♦

❝♦♠♦ ❜❛s❡ s❡rá 2√2m ❡ ❛ s✉❛ ❛❧t✉r❛✱ ❝♦♠♦ ❞✐t♦ ♥♦ ❡♥✉♥❝✐❛❞♦ s❡rá ✽ ♠✳ ❆ss✐♠ s❡♥❞♦✱

♣♦❞❡♠♦s ❡♥tã♦ ❝♦♥❝❧✉✐r q✉❡ ♦ ✈♦❧✉♠❡ ❞♦ ♣r✐s♠❛ s❡rá✿

V = AB.h✱ ♦♥❞❡ ❛ ❜❛s❡ s❡rá ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ 2√2

V = 8 · 8 = 64m3

✸✳ ❚❡♠♦s ❞♦✐s só❧✐❞♦s ❝♦♠ ♠❡s♠♦ ✈♦❧✉♠❡✱ ♣♦ré♠ ❝♦♠ ✏❢♦r♠❛t♦s✑ ❞✐❢❡r❡♥t❡s✳

❋✐❣✉r❛ ✶✵✳✸✿ P❛r❛❧❡❧❡♣í♣❡❞♦ ❡ ❝✉❜♦

❖ ❢♦r♠❛t♦ ❞❡ ♣❛r❛❧❡❧❡♣í♣❡❞♦ t❡♠ ❛s s❡❣✉✐♥t❡s ❞✐♠❡♥sõ❡s✿ ✸✱ ✶✽ ❡ ✹ ❝♠✱ ♣♦rt❛♥t♦✱ ♦ s❡✉✈♦❧✉♠❡ ♣♦❞❡rá s❡r ❝❛❧❝✉❧❛❞♦ ❡❢❡t✉❛♥❞♦ ♦ ♣r♦❞✉t♦ ❞❛s ♠❡s♠❛s✱ ♦✉ s❡❥❛✱ VPARALELEPPEDO =

3 · 18 · 4 = 216cm3✳

❊st❡ ✈♦❧✉♠❡ s❡rá ♦ ♠❡s♠♦ ♣❛r❛ ♦ ❝✉❜♦✱ ♣♦rt❛♥t♦✱ VCUBO = 216cm3✱ ❡ ❝♦♠♦ s❛❜❡♠♦sq✉❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ❝✉❜♦ ♣♦❞❡ s❡r tr❛❜❛❧❤❛❞♦ ❝♦♠♦ ❛r❡st❛ ❡❧❡✈❛❞♦ à t❡r❝❡✐r❛ ♣♦tê♥❝✐❛✱✈❛♠♦s ♠♦♥t❛r ♦ s❡❣✉✐♥t❡ r❛❝✐♦❝í♥✐♦✿

216 = a3✱ ❧♦❣♦ a = 6✳

❈♦♥❝❧✉í♠♦s ❡♥tã♦ q✉❡ ❛ ❛r❡st❛ ❞♦ ❝✉❜♦ ♠❡❞❡ 6cm✳

✹✳ ❙❡ ❡❧❛ ❞❡✈❡rá ❡♥❝❤❡r ♦s ✷✵ ❝♦♣✐♥❤♦s ♣❡❧❛ ♠❡t❛❞❡✱ ♦❜✈✐❛♠❡♥t❡ ❡❧❛ ❡♥❝❤❡rá ✶✵ ❝♦♣♦s✐♥t❡✐r♦s ❡ ♦ ✈♦❧✉♠❡ ❞❡st❡s ❞❡③ ❝♦♣♦s ❝❤❡✐♦s s❡rá✿

✺✽

10 · (VCOPINHO = ABASE · Altura)

10 · (Πr2 · Altura)

10 · (3, 14 · 4 · 4)

10 · 50, 24 = 502, 4 cm3

❖ ✈♦❧✉♠❡ ❞❛ ❧❡✐t❡✐r❛ é ✿

VLEITEIRA = 3, 14 ·16 ·20 = 1004, 8 cm3✱ q✉❡ é ♦ ❞♦❜r♦ ❞♦ ✈♦❧✉♠❡ ❞♦s ✶✵ ❝♦♣✐♥❤♦s✱ ❧♦❣♦♣r❡❝✐s❛rí❛♠♦s ❡♥❝❤❡r ❛ ❧❡✐t❡✐r❛ ❛té ❛ ♠❡t❛❞❡✱ ♣♦✐s ♦ s❡✉ ✈♦❧✉♠❡ é ✷✵ ✈❡③❡s ♦ ✈♦❧✉♠❡ ❞❡❝❛❞❛ ❝♦♣✐♥❤♦✳

✺✳ ❖❜t❡♥❞♦ ♦ ❞❡s❡♥❤♦✱ t❡r❡♠♦s✿

❋✐❣✉r❛ ✶✵✳✹✿ Pr✐s♠❛ ♦❜❧íq✉♦✳ ❋♦♥t❡✿✇✇✇✳♠s♣s✳❡♥❣✳❜r

❈♦♠♦ ♦s ❧❛❞♦s ❞❛ ❜❛s❡ ♠❡❞❡♠ x ❡ ❡st❛ ❜❛s❡ é ✉♠ q✉❛❞r❛❞♦✱ ♣♦❞❡♠♦s ❡♥tã♦ ❝♦♥❝❧✉✐rq✉❡ s✉❛ ár❡❛ s❡rá x2✳ ❇❛st❛ ❛❣♦r❛ ❡♥❝♦♥tr❛r♠♦s ❛ ❛❧t✉r❛ ❞♦ ♣r✐s♠❛✳ P❛r❛ ❡♥❝♦♥tr❛r♠♦s❡ss❛ ❛❧t✉r❛✱ ✈❛♠♦s tr❛❜❛❧❤❛r ❝♦♠ ❛ ❞❡✜♥✐çã♦ ❞❛s r❛③õ❡s tr✐❣♦♥♦♠étr✐❝❛s ♥♦ tr✐â♥❣✉❧♦r❡tâ♥❣✉❧♦✿

❋✐❣✉r❛ ✶✵✳✺✿ ❚r✐â♥❣✉❧♦ ❘❡tâ♥❣✉❧♦✳ ❋♦♥t❡✿ ❜r❛✐♥❧②✳❝♦♠✳❜r

sen600 = alturay

√32 = hy h = y

√3

2

✺✾

❊ ❛ss✐♠ s❡♥❞♦✱ ♣♦❞❡♠♦s ❡♥tã♦ ❝❛❧❝✉❧❛r ♦ ✈♦❧✉♠❡ ❞♦ só❧✐❞♦✳

VPRISMA = ABASE · h

VPRISMA = x2 · y√3

2

VPRISMA = x2 · y√3

2

✻✳ P❡♥s❛♥❞♦ ❡♠ ✉♠ ♣r✐s♠❛ ❤❡①❛❣♦♥❛❧✱ t❡rí❛♠♦s ❛ s❡❣✉✐♥t❡ ✜❣✉r❛✿

❋✐❣✉r❛ ✶✵✳✻✿ Pr✐s♠❛ ❍❡①❛❣♦♥❛❧✳ ❋♦♥t❡✿ ❜r❛✐♥❧②✳❝♦♠✳❜r

❈❛❧❝✉❧❛♥❞♦ ♦ ✈♦❧✉♠❡✱ t❡r❡♠♦s✿

❱PISCINA = ABASE · ALTURA

❖♥❞❡ ❛ ár❡❛ ❞❛ ❜❛s❡ s❡rá ♥❛ ✈❡r❞❛❞❡ ❛ ár❡❛ ❞❡ ✉♠ ❤❡①á❣♦♥♦ r❡❣✉❧❛r ❞❡ ❧❛❞♦ ✷ ♠✱ ♣♦rt❛♥t♦✿

ABASE = 6 · 4·√3

4= 6

√3m2

❊ ❛ss✐♠ ❡♥tã♦✿

VPISCINA = 6√3 ·

√3

2= 9m3

❈♦♠♦ q✉❡r❡♠♦s ❡♥❝❤❡r ❛♣❡♥❛s ✽✵✪ ❞❡ s✉❛ ❝❛♣❛❝✐❞❛❞❡✱ ❣❛st❛r❡♠♦s 0, 8.9 = 7, 2m3✱ ♦✉s❡❥❛✱ 7200 ❧✐tr♦s✳

❆ ♣✐râ♠✐❞❡ ❢♦r♠❛❞❛ t❡rá ✉♠❛ ❜❛s❡ ❤❡①❛❣♦♥❛❧ ❞❡ ❧❛❞♦ 3√2cm ✭✉s♦ ❞♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱

t♦♠❛♥❞♦ ❛ ♠❡t❛❞❡ ❞❛ ❛r❡st❛ ❞♦ ❝✉❜♦✮✳ P❛r❛ ❡♥❝♦♥tr❛r♠♦s ❛ ❛❧t✉r❛ ❞❡ss❛ ♣✐râ♠✐❞❡✱ ✈❛♠♦s t❡r✉♠ ♣♦✉❝♦ ♠❛✐s ❞❡ tr❛❜❛❧❤♦✳ ❯s❛r❡♠♦s ♦ s❡❣♠❡♥t♦ JH✱ q✉❡ s❡rá ♦❜t✐❞♦ ❛tr❛✈és ❞❛ ❛♣❧✐❝❛çã♦❞♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ♥♦ tr✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ HJD

✻✵

❋✐❣✉r❛ ✶✵✳✼✿ ❚r✐â♥❣✉❧♦ r❡tâ♥❣✉❧♦ ❏❉❍

P♦rt❛♥t♦✱ ♥♦✈❛♠❡♥t❡ ♣♦❞❡♠♦s ✉s❛r ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ♣❛r❛ ❡♥❝♦♥tr❛r♠♦s ❛ ♠❡❞✐❞❛❞❛ ❛❧t✉r❛ ❞❛ ♣✐râ♠✐❞❡ ❛ q✉❛❧ ❝❤❛♠❛r❡♠♦s ❞❡ ❛❧t✉r❛HO✱ ❝♦♥s✐❞❡r❛♥❞♦ O ❝♦♠♦ s❡♥❞♦ ♦ ❝❡♥tr♦❞❛ ❜❛s❡ ❞♦ ❤❡①á❣♦♥♦ q✉❡ ❢♦r♠❛ ❛ ❜❛s❡ ❞❛ ♣✐râ♠✐❞❡✳ ❚❡r❡♠♦s ❡♥tã♦ ♦ s❡❣✉✐♥t❡ tr✐â♥❣✉❧♦r❡tâ♥❣✉❧♦✿

❋✐❣✉r❛ ✶✵✳✽✿ ❚r✐â♥❣✉❧♦ ❘❡tâ♥❣✉❧♦ ❍❏❖

❆ ❛t✉r❛ h✱ s❡rá 3√3cm✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ♦ ✈♦❧✉♠❡ ❞❛ ♣✐râ♠✐❞❡✱

✉s❛♥❞♦ ❛ ❢ór♠✉❧❛ ♦❜t✐❞❛✿ V = AREADABASE ·ALTURA3

✱ ♦♥❞❡ ❛ ár❡❛ ❞❛ ❜❛s❡ s❡rá ❛ ár❡❛ ❞❡✉♠ ❤❡①á❣♦♥♦ r❡❣✉❧❛r ❞❡ ❧❛❞♦ 3

√3cm✳ ❙❡♥❞♦ ❛ss✐♠ t❡r❡♠♦s Ab = 27

√3cm2✳ ❊ ✜♥❛❧♠❡♥t❡ ♦

✈♦❧✉♠❡ ❞❛ ♣✐râ♠✐❞❡ s❡rá ❡♥tã♦ 81cm3✳

❖ ✈♦❧✉♠❡ ❞♦ ❝✉❜♦✱ ❛♥t❡s ❞❛ r❡t✐r❛❞❛ ❞❛s ♣✐râ♠✐❞❡s s❡rá 63 ❂ 216 ✉♥✐❞❛❞❡s ❞❡ ✈♦❧✉♠❡✭❜❛st❛ ❧❡♠❜r❛r♠♦s q✉❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ❝✉❜♦ é ♥❛ ✈❡r❞❛❞❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣r✐s♠❛ ❝♦♠ t♦❞❛s❛s ❛r❡st❛s ❞❡ ♠❡s♠❛ ♠❡❞✐❞❛✱ ♣♦rt❛♥t♦ V = AB ·Altura✳ ❈♦♠♦ ❛s ♣✐râ♠✐❞❡s ✏r❡t✐r❛❞❛s✑ sã♦

✻✶

♥❛ ✈❡r❞❛❞❡ t❡tr❛❡❞r♦s ❝♦♠ três ❛r❡st❛s ♠❡❞✐♥❞♦ ✸ ✉♥✐❞❛❞❡s✱ ♣♦❞❡♠♦s ❡♥tã♦ ❝❛❧❝✉❧❛r ♦ ✈♦❧✉♠❡❞❡ ❝❛❞❛ ✉♠❛✳

▲❡♠❜r❡♠♦s q✉❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ♣✐râ♠✐❞❡ é ❞❛❞♦ ♣❡❧❛ s❡❣✉✐♥t❡ ❢ór♠✉❧❛✿

VPIRAMIDE =AREADABASE · ALTURA

3,

r❡s✉❧t❛❞♦ q✉❡ ❢♦✐ ♦❜t✐❞♦ ✉s❛♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♥❛ ❝♦♠♣❛r❛çã♦ ❝♦♠ ✉♠❛ s✐t✉❛çã♦❡♠ q✉❡ tí♥❤❛♠♦s ✉♠ ♣r✐s♠❛ tr✐❛♥❣✉❧❛r ❞❡❝♦♠♣♦st♦ ❡♠ três ♣✐râ♠✐❞❡s ❝♦♠ ♠❡s♠❛ ❜❛s❡ ❡♠❡s♠❛ ❛❧t✉r❛✳

▲♦❣♦✱ ♦ ✈♦❧✉♠❡ ❞❡ ❝❛❞❛ ✉♠❛ ❞❛s ♣❡q✉❡♥❛s ♣✐râ♠✐❞❡s s❡rá ♦❜t✐❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿

❆ ❜❛s❡ ❞❡st❛s ♣✐râ♠✐❞❡s s❡rã♦ tr✐â♥❣✉❧♦s ✐sós❝❡❧❡s ❡ ❡q✉✐❧át❡r♦s ❝♦♠ ❞♦✐s ❧❛❞♦s ♠❡❞✐♥❞♦✸✱ ❧♦❣♦ ❛ ár❡❛ ❞❡ ❝❛❞❛ ✉♠❛ ❞❡st❛s ❜❛s❡s s❡rá ABASE = 4, 5 ✉♥✐❞❛❞❡s ❞❡ ár❡❛ ❡ ❛ ❛❧t✉r❛ ❞❡❝❛❞❛ ✉♠❛ ❞❛s ♣✐râ♠✐❞❡s s❡rá ❛ ♠❡t❛❞❡ ❞❛ ♠❡❞✐❞❛ ❞❛ ❛r❡st❛ ❞♦ ❝✉❜♦✱ ♦✉ s❡❥❛ ✸✳

❆ss✐♠ ♦ ✈♦❧✉♠❡ ❞❡ ❝❛❞❛ ♣✐râ♠✐❞❡ s❡rá✿

VPIRAMIDE =4, 5 · 3

3= 4, 5 unidades de volume

✳

❊ ❝♦♠♦ sã♦ q✉❛tr♦ ♣✐râ♠✐❞❡s✱ ✈❡r✐✜❝❛r❡♠♦s q✉❡ ♦ ✈♦❧✉♠❡ ❞❡❧❛s s❡rá ✶✽ ❡ ✜♥❛❧♠❡♥t❡✱ ♦✈♦❧✉♠❡ ❞♦ só❧✐❞♦ r❡s✉❧t❛♥t❡ s❡rá✿

VFINAL ❂ ✭❱♦❧✉♠❡ ❞♦ ❝✉❜♦✮ ✲ ✭❱♦❧✉♠❡ ❞❛s q✉❛tr♦ ♣❡q✉❡♥❛s ♣✐râ♠✐❞❡s✮

❱FINAL = ✷✶✻ ✲ ✶✽ ❂ ✶✾✽.

❖ ✈♦❧✉♠❡ ❞❡ ♠❛❞❡✐r❛ ❛ s❡r ✉s❛❞♦ ♥❛ ❝♦♥❢❡❝çã♦ s❡rá ❞❛❞♦ ♣♦r✿

❱MADEIRA = VCUBOMAIOR − VCUBOMENOR

❖♥❞❡ ♦ ✈♦❧✉♠❡ ❞♦ ❝✉❜♦ ♠❛✐♦r t❡♠ ❛r❡st❛ ✶✷❝♠ ❡ ♦ ♠❡♥♦r t❡♠ ❛r❡st❛ ✽❝♠✳❱CUBOMAIOR = 123 = 1728 ❝♠3

❱CUBOMENOR = 83 = 512 ❝♠3

❱❛♠♦s ♠❛✐s ✉♠❛ ✈❡③ ❧❡♠❜r❛r q✉❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ❝✉❜♦ é ❞❛❞♦ ♣❡❧❛ r❡❧❛çã♦ ❡♥tr❡ ❛ ár❡❛ ❞❛❜❛s❡ ❡ ❛ s✉❛ ❛❧t✉r❛✱ ✈❡r✐✜❝❛♥❞♦ q✉❡ s✉❛s ❛r❡st❛s sã♦ t♦❞❛s ✐❣✉❛✐s✱ s❡♥❞♦ ❛ss✐♠ ♦ ✈♦❧✉♠❡ s❡rá❝❛❧❝✉❧❛❞♦ ❝♦♠♦ ♦ ✈❛❧♦r ❞❡st❛ ❛r❡st❛ ❡❧❡✈❛❞♦ à t❡r❝❡✐r❛ ♣♦tê♥❝✐❛✳ ❋✐♥❛❧✐③❛♥❞♦✱ t❡r❡♠♦s ❡♥tã♦❱MADEIRA = 1728− 512 = 1216 ❝♠3✳

❈♦♠♦ ♦ ❝✐❧✐♥❞r♦ ❡stá ✐♥s❝r✐t♦ ♥♦ ❝✉❜♦✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ♦ ❝✐❧✐♥❞r♦ s❡rá ✉♠ ❝✐❧✐♥❞r♦ r❡t♦❡ q✉❡ ❛ s✉❛ ❛❧t✉r❛ t❡rá ❛ ♠❡s♠❛ ♠❡❞✐❞❛ ❞❛ ❛r❡st❛ ❞♦ ❝✉❜♦✱ ❛❧é♠ ❞❡ ✈❡r✐✜❝❛r♠♦s q✉❡ ♦ r❛✐♦

✻✷

❞❛ ❜❛s❡ s❡rá ❛ ♠❡t❛❞❡ ❞❛ ♠❡❞✐❞❛ ❞❡st❛ ❛r❡st❛✱ ♦✉ s❡❥❛✱ ✶✵ ❝♠✳

❆ss✐♠ ♦ ✈♦❧✉♠❡ ❞♦ ❝✐❧✐♥❞r♦ ❞❛❞♦ s❡rá ♦❜t✐❞♦ ✉s❛♥❞♦ ❛ ♠❡s♠❛ r❡❧❛çã♦ ✉s❛❞❛ ♣❛r❛ ♦ ❝á❧❝✉❧♦❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ♣r✐s♠❛✱ ♣♦✐s ♣❡❧♦ ♣r✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐

P❛r❛ ❡♥❝♦♥tr❛r ♦ ✈♦❧✉♠❡ ♠á①✐♠♦ ❞♦ r❡s❡r✈❛tór✐♦✱ ✈❛♠♦s ❧❡♠❜r❛r ❞❛ ❢ór♠✉❧❛ ♦❜t✐❞❛ ♣❛r❛ ❝❛❧✲❝✉❧❛r ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ❡s❢❡r❛✱ t♦♠❛♥❞♦ ♦ ❝✉✐❞❛❞♦ ♣❛r❛ ♥ã♦ s❡ ❡sq✉❡❝❡r ❞❡ q✉❡ ✉♠ ❤❡♠✐s❢ér✐♦♥❛❞❛ ♠❛✐s é ❞♦ q✉❡ ✉♠❛ s❡♠✐❡s❢❡r❛✱ ♦✉ s❡❥❛✿ ❱HEMISFERIO = 4ΠR36.

❯s❛♥❞♦ ♦ r❛✐♦ ✐❣✉❛❧ ❛ ✺ ♠✱ Π ❂ ✸✱ t❡r❡♠♦s✿

VHEMISFERIO ❂ 4.3.1256

= 1500

6= 250m3✳ ❖ q✉❡ ❡q✉✐✈❛❧❡ ❛ ✷✺✵✳✵✵✵ ❧✐tr♦s ❞❡ ❝❛♣❛❝✐❞❛❞❡✳

❙❛❜❡♠♦s q✉❡ ✉♠ t❡tr❛❡❞r♦ r❡❣✉❧❛r t❡♠ ❛s q✉❛tr♦ ❢❛❝❡s s❡♥❞♦ tr✐â♥❣✉❧♦s ❡q✉✐❧át❡r♦s✳ ❈❛❧❝✉✲❧❛♥❞♦ ✐♥✐❝✐❛❧♠❡♥t❡ ❛ ár❡❛ ❞❛ ❜❛s❡ ✭q✉❡ é ♦❜✈✐❛♠❡♥t❡ ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦✮✱ t❡r❡♠♦s✿

❋✐❣✉r❛ ✶✵✳✾✿ P✐râ♠✐❞❡ ❚r✐❛♥❣✉❧❛r

ABASE = a2√3

4✱ q✉❡ é ♥❛ ✈❡r❞❛❞❡ ❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ ❞❡ ❧❛❞♦ a✳

❈❛❧❝✉❧❛♠♦s ❛❣♦r❛ ❛ ❛❧t✉r❛ ❞♦ t❡tr❛❡❞r♦✱ ♦♥❞❡ D é ♦ ❝❡♥tr♦ ❞❛ ❜❛s❡ ABC✳ ❆ ❛❧t✉r❛ ❞♦tr✐â♥❣✉❧♦ ❜❛s❡ s❡rá ❞✐✈✐❞✐❞❛ ♥❛ ❞❡ ♠❛♥❡✐r❛ q✉❡ ♦ s❡❣♠❡♥t♦ BD s❡❥❛ 2

3❞❡ t♦❞❛ ❡st❛ ❛❧t✉r❛ ❡

❝♦♠♦ ❛ ❛❧t✉r❛ ❞❛ ❜❛s❡ é ❛ ❛❧t✉r❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ t❡♠♦s q✉❡ HBASE = a√3

2✱ s❡❣✉❡

❡♥tã♦ q✉❡ ♦ s❡❣♠❡♥t♦ BD ♠❡❞❡ a√3

3❡ ❛ss✐♠ ❛♣❧✐❝❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ♥♦ tr✐â♥❣✉❧♦

BDE✱ ❡♥❝♦♥tr❛r❡♠♦s ❛ ♠❡❞✐❞❛ ❞❛ ❛❧t✉r❛ ❞❛ ♣✐râ♠✐❞❡

BE2 = BD2 + ED2✱ s❡♥❞♦ ED ❛ ♠❡❞✐❞❛ ❞❛ ❛❧t✉r❛ ❞❛ ♣✐râ♠✐❞❡✳

a2 =(

a√3

3

)2

+ h2

✻✸

h = a√6

3❡ s❡♥❞♦ ❞❡st❛ ♠❛♥❡✐r❛✱ ♣♦❞❡r❡♠♦s✱ ❡♥✜♠✱ ❝❛❧❝✉❧❛r ♦ ✈♦❧✉♠❡ ❞❛ ♣✐râ♠✐❞❡ ✉s❛♥❞♦

❛ ❢ór♠✉❧❛ ❞♦ ✈♦❧✉♠❡

V =AREADABASE · ALTURA

3=

3a3√2

36=

a3√2

12

❈❛❧❝✉❧❛r❡♠♦s ♦ ✈♦❧✉♠❡ ❞❛ ♣✐râ♠✐❞❡ q✉❡ ❢♦r♠❛ ♦ ♦❝t❛❡❞r♦✱ t❡r❡♠♦s ❡♥tã♦ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛♣✐râ♠✐❞❡ ❞❡ ❜❛s❡ q✉❛❞r❛❞❛ ❡♠ q✉❡ ❝❛❞❛ ✉♠❛ ❞❛s ❛r❡st❛s ♠❡❞❡♠ ✺ ❝♠✳

❋✐❣✉r❛ ✶✵✳✶✵✿ P✐râ♠✐❞❡ ◗✉❛❞r❛♥❣✉❧❛r

❆ ár❡❛ ❞❛ ❜❛s❡ s❡rá ❛ ár❡❛ ❞❡ ✉♠ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ ♠❡❞✐♥❞♦ ✺ ❝♠✱ ♦✉ s❡❥❛✿

ABASE = 52 = 25 ❝♠2

P❛r❛ ❡♥❝♦♥tr❛r ❛ ♠❡❞✐❞❛ ❞❛ ❛❧t✉r❛ ❞❡st❛ ♣✐râ♠✐❞❡✱ ✈❛♠♦s ✉s❛r ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✱✉s❛♥❞♦ ❛ ♠❡t❛❞❡ ❞❛ ❞✐❛❣♦♥❛❧ ❞❛ ❜❛s❡ ❡ ✉♠❛ ❞❛s ❛r❡st❛s ✭s❡❣♠❡♥t♦ GD✱ ❛r❡st❛ HD ❡ ❛❧t✉r❛HG✮

52 =(

5√2

2

)2

+ h2

h = 5√2

2

❊ ❛ss✐♠✱ ♣♦❞❡♠♦s ❡♥tã♦ ❝❛❧❝✉❧❛r ♦ ✈♦❧✉♠❡ ❞❡st❛ ♣✐râ♠✐❞❡✿

VPIRAMIDE = Ab·h3

=25·5

√

2

2

3= 29, 1 ❝♠3

❈♦♠♦ ♥♦ ♦❝t❛❡❞r♦✱ t❡♠♦s ❞✉❛s ♣✐râ♠✐❞❡s ✐❣✉❛✐s✱ ✈❛♠♦s ♦❜t❡r s❡✉ ✈♦❧✉♠❡ ✜♥❛❧ s❡♥❞♦ ♦s❡❣✉✐♥t❡✿

❱OCTAEDRO =✷ · 29, 1 = 58, 2 ❝♠3

✻✹

❈♦♠♦ ♦ ❞✐â♠❡tr♦ ❞❛ ❜❛s❡ ❞❛ ❧❛t❛ ❝✐❧í♥❞r✐❝❛ ♠❡❞❡ ✽ ❝♠✱ ✈❡r✐✜❝❛r❡♠♦s q✉❡ s❡✉ r❛✐♦ ♠❡❞✐rá ✹❝♠ ❡ ❛ss✐♠ s❡♥❞♦ ❛ ár❡❛ ❞❡st❛ ❜❛s❡ s❡rá ABASE = 16Πcm2✱ ♦✉ s❡❥❛✿ 48cm3 ❛♣r♦①✐♠❛❞❛♠❡♥t❡✳

❖ ✈♦❧✉♠❡ ❞❛ ❧❛t❛ s❡rá✿ VLATA = ABASE ·H✱ ♦♥❞❡ ❍ é ❛ ❛❧t✉r❛ ❞❛ ❧❛t❛✳

✭❇❛st❛ r❡❧❡♠❜r❛r♠♦s ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐✮

❱LATA =✹✽ · 11 = 528 ❝♠3

❈♦♠♦ ♦ ❞✐â♠❡tr♦ ❞♦ t❛♥q✉❡ ♠❡❞❡ ✻ ♠✱ t❡♠♦s q✉❡ ♦ r❛✐♦ ❞❡ s✉❛ ❜❛s❡ ♠❡❞❡ ✸ ♠ ❡ ❝♦♠♦✐♥❢♦r♠❛❞♦ ♥♦ ❡♥✉♥❝✐❛❞♦✱ ❛ s✉❛ ❛❧t✉r❛ ✈❛❧❡ ✹ ♠✳

❱❛♠♦s ❡♥tã♦✱ ✐♥✐❝✐❛❧♠❡♥t❡✱ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❛ ✏❜❛s❡✑ ❞❡st❡ t❛♥q✉❡✳

❆❞♠✐t✐r❡♠♦s q✉❡ ♦ ✈❛❧♦r ❞❡ Π s❡rá ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ✸✱✶✹✳

ABASE = 9Πm2 = 28, 26♠2

❈♦♠♦ ❛ ❛❧t✉r❛ ♠❡❞❡ ✹ ♠✱ ♣❛r❛ ❡♥❝♦♥tr❛r♠♦s ♦ ✈♦❧✉♠❡✱ ✈❛♠♦s ✉s❛r✿

VCONE =AREADABASE · ALTURA

3=

28, 3 · 43

= 37, 68m3

❖ q✉❡ ❡q✉✐✈❛❧❡ ❛ ✸✼✳✻✽✵ ❧✐tr♦s✳

❈♦♥❝❧✉sã♦

◆❡st❡ tr❛❜❛❧❤♦✱ ❛♣r❡s❡♥t❛♠♦s ✐♥❢♦r♠❛çõ❡s ❡ ❛s♣❡❝t♦s ✐♠♣♦rt❛♥t❡s ❛ r❡s♣❡✐t♦ ❞♦ ❝á❧❝✉❧♦✱❞❛ ❞❡❞✉çã♦ ❡ ❞❡ ❝❛r❛❝t❡ríst✐❝❛s r❡❧❡✈❛♥t❡s s♦❜r❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ só❧✐❞♦ ❣❡♦♠étr✐❝♦✱ ❛❧é♠ ❞❛❝♦♥tr✐❜✉✐çã♦ ❞♦ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ♣♦❞❡r ❡❢❡t✉❛r t❛✐s ❝á❧❝✉❧♦s✳ ❈♦♠ ❡st❡ tr❛❜❛❧❤♦♣♦❞❡♠♦s ✈❡r✐✜❝❛r ❛ ♥❛t✉r❡③❛ ❞❛s ❢ór♠✉❧❛s✱ ❝♦♥str✉çõ❡s ❞♦s só❧✐❞♦s ❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡✱ ✉♠❛♠❛♥❡✐r❛ ❛❧t❡r♥❛t✐✈❛ ❞❡ s❡ ❡st❛❜❡❧❡❝❡r ♦s r❡s✉❧t❛❞♦s ❞❡ ❡①❡r❝í❝✐♦s ❡ s✐t✉❛çõ❡s q✉❡ ❡♥✈♦❧✈❡♠♦ ✈♦❧✉♠❡ ❞❡ ✉♠❛ ✜❣✉r❛ ♦✉ ✉♠❛ s✐t✉❛çã♦ ❡♠ q✉❡ s❡ ❢❛③ ♥❡❝❡ssár✐♦ ♦ ✉s♦ ♦✉ ❝á❧❝✉❧♦ ❞❡ t❛❧❣r❛♥❞❡③❛✱ ❢❛③❡♥❞♦ ❝♦♠ q✉❡ ♦s ❥♦✈❡♥s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ✜q✉❡♠ ♠❛✐s ✐♥t❡r❡ss❛❞♦s ❡♠ ❞❡t❡r✲♠✐♥❛❞♦s tó♣✐❝♦s ❞❛ ♠❛t❡♠át✐❝❛ q✉❡ s✉♣♦st❛♠❡♥t❡ s❡r✐❛♠ ♠❛✐s ár✐❞♦s ♣❛r❛ ♦s ♠❡s♠♦s✳ ❖♠❛t❡r✐❛❧ ❝♦♥❝r❡t♦ ❡ ♦s s♦❢t✇❛r❡s q✉❡ ❢❛③❡♠ r❡❢❡rê♥❝✐❛ à ❞✐s❝✐♣❧✐♥❛ ▼❛t❡♠át✐❝❛ sã♦ ❝✐t❛❞♦s♥❡st❡ tr❛❜❛❧❤♦ ❝♦♠♦ ♣❡ç❛s ❢✉♥❞❛♠❡♥t❛✐s q✉❡ ❡stã♦ s❡♥❞♦ ❛♣r❡s❡♥t❛❞❛s ❥✉♥t❛♠❡♥t❡ ❛♦s ❡①❡r✲❝í❝✐♦s ❡ ❛s ❞❡✜♥✐çõ❡s t❡ór✐❝❛s✱ ❢♦r♠❛♥❞♦ ❛ss✐♠✱ ✉♠ ❝♦♥❥✉♥t♦ ✐♠♣♦rt❛♥t❡ ❞❡ ❢❡rr❛♠❡♥t❛s ♣❛r❛❛✉①✐❧✐❛r ❡st✉❞❛♥t❡s ❡ ♣r♦❢❡ss♦r❡s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ♥♦ ❛♣r❡♥❞✐③❛❞♦✱ ✜①❛çã♦ ❡ ❞❡t❛❧❤❛♠❡♥t♦❞❡ ❡str✉t✉r❛s ✐♠♣♦rt❛♥t❡s ❛ s❡r❡♠ ✈✐st❛s✳

✻✺

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

❬✶❪ ❇❊❘▲■◆●❍❖❋❋✱ ❲✳P✳✱ ●❖❯❱✃❆✱ ❋✳◗✳ ✲ ✏❆ ♠❛t❡♠át✐❝❛ ❛tr❛✈és ❞♦s t❡♠♣♦s✿ ❯♠❣✉✐❛ ❢á❝✐❧ ♣❛r❛ ♣r♦❢❡ss♦r❡s ❡ ❡♥t✉s✐❛st❛s✳✑ ✴ ❚r❛❞✉çã♦✿ ❊❧③❛ ❋✳ ●♦♠✐❞❡ ❡ ❍❡❧❡♥❛ ❈❛str♦✳❙ã♦ P❛✉❧♦✿ ❊❞✐t♦r❛ ❇❧✉❝❤❡r

❬✷❪ ❇■❈❯❉❖✱ ▼✳❆✳❱✳ ❋✐❧♦s♦✜❛ ❞❛ ❊❞✉❝❛çã♦ ▼❛t❡♠át✐❝❛ ✭❋❡♥♦♠❡♥♦❧♦❣✐❛✱ ❝♦♥❝❡♣çõ❡s✱

♣♦ss✐❜✐❧✐❞❛❞❡s ❞✐❞át✐❝♦✲♣❡❞❛❣ó❣✐❝❛s✮ ❙ã♦ P❛✉❧♦✿ ❊❞✐t♦r❛ ❞❛ ❯◆❊❙P✱ ✷✵✶✵✳

❬✸❪ ❇❖❨❊❘✱ ❈✳ ❇✳ ❍✐stór✐❛ ❞❛ ♠❛t❡♠át✐❝❛✳ ✷➟ ❊❞✐çã♦✳ ❙ã♦ P❛✉❧♦✳ ❊❞❣❛r❞ ❇❧ü❝❤❡r✱ ✶✾✾✻✳

❬✹❪ ❇❘❆❙■▲✳ ❙❊❈❘❊❚❆❘■❆ ❉❊ ❊❉❯❈❆➬➹❖ ❋❯◆❉❆▼❊◆❚❆▲✳ P❛r❛♠êtr♦s ❝✉r✲

r✐❝✉❧❛r❡s ♥❛❝✐♦♥❛✐s✿ ▼❛t❡♠át✐❝❛✳ ❙❡❝r❡t❛r✐❛ ❞❡ ❊❞✉❝❛çã♦ ❋✉♥❞❛♠❡♥t❛❧ ❙❡❝r❡t❛r✐❛ ❞❡

❊❞✉❝❛çã♦ ❋✉♥❞❛♠❡♥t❛❧✳ ❇r❛sí❧✐❛✿ ▼❊❈✴❙❊❋✱ ✶✾✾✽✳

❬✺❪ ❉❆◆❚❊✱ ▲✳ ❘✳ ▼❛t❡♠át✐❝❛✿ ❈♦♥t❡①t♦ ❡ ❆♣❧✐❝❛çõ❡s✳ ✈✳✷✳ ❊❞✐t♦r❛ ➪t✐❝❛✱ ✷✵✶✸✳

❬✻❪ ❉❖▲❈❊✱ ❖✳✱ P❖▼P❊❖✱ ❏✳ ◆✳ ❋✉♥❞❛♠❡♥t♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r✱ ✶✵✿ ●❡♦♠❡✲

tr✐❛ ❊s♣❛❝✐❛❧✱ P♦s✐çã♦ ❡ ▼étr✐❝❛✳ ❙ã♦ P❛✉❧♦✳ ❊❞✐t♦r❛ ❆t✉❛❧✱ ✷✵✶✸✳ ❬✶✶❪ ▲■▼❆ ✱ ❊✳ ▲✳✱▼❡❞✐❞❛ ❡ ❋♦r♠❛ ❡♠ ●❡♦♠❡tr✐❛✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✱ ❙❇▼✱ ✶✾✾✶✳

❬✼❪ ▲■▼❆ ✱ ❊✳ ▲✳ ▼❡❞✐❞❛ ❡ ❋♦r♠❛ ❡♠ ●❡♦♠❡tr✐❛✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✱ ❙❇▼✱ ✶✾✾✶✳

❬✽❪ ❊❱❊❙✱ ❍❖❲❆❘❉✳ ■♥tr♦❞✉çã♦ à ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✳ ❚r❛❞✳ ❍②❣✐♥♦ ❍✳ ❉♦♠✐♥✲❣✉❡s✳ ❙ã♦ P❛✉❧♦✿ ❊❞✐t♦r❛ ❞❛ ❯◆■❈❆▼P✱ ✷✵✵✹✳

❬✾❪ ❋❊❘◆❆◆❉❊❩✱ P✳ ❏✳▼❡❞✐❞❛ ❡ ■♥t❡❣r❛çã♦✳ ▼❡❞✐❞❛ ❡ ■♥t❡❣r❛çã♦✳ ✷➟ ❊❞✐çã♦✳ ❘✐♦❞❡ ❏❛♥❡✐r♦✳ ❙❇▼✱ ✶✾✻✻✳

❬✶✵❪ ▲■▼❆✱ ❊✳ ▲✳ ❚❡♠❛s ❡ ♣r♦❜❧❡♠❛s ❡❧❡♠❡♥t❛r❡s✳ ✶✷➟ ❊❞✐çã♦✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✳ ❙❇▼✱ ✷✵✵✻✳

❬✶✶❪ ▲■▼❆ ✱ ❊✳ ▲✳ ▼❡❞✐❞❛ ❡ ❋♦r♠❛ ❡♠ ●❡♦♠❡tr✐❛✳ ✈♦❧✉♠❡ ✷ ✲ ✻✳ ❡❞✳ ❘✐♦ ❞❡ ❏❛♥❡✐r♦✿ ❙❇▼✱✷✵✵✻✳

❬✶✷❪ ▲■◆❉◗❯■❙❚✱ ▼✳ ❆♣r❡♥❞❡♥❞♦ ❡ ❡♥s✐♥❛♥❞♦ ❣❡♦♠❡tr✐❛✳ ❙ã♦ P❛✉❧♦✿ ❊❞✐t♦r❛ ❆t✉❛❧✱ ✶✾✾✹

✻✻

❘❊❋❊❘✃◆❈■❆❙ ❇■❇▲■❖●❘➪❋■❈❆❙ ✻✼

❬✶✸❪ ❇❘❆❙■▲✳ ▼■◆■❙❚➱❘■❖ ❉❆ ❊❉❯❈❆➬➹❖✳ P❛r❛♠êtr♦s ❝✉rr✐❝✉❧❛r❡s ♥❛❝✐♦♥❛✐s ♣❛r❛

♦ ❡♥s✐♥♦ ♠é❞✐♦✿ ❈✐ê♥❝✐❛s ❞❛ ◆❛t✉r❡③❛✱ ▼❛t❡♠át✐❝❛ ❡ s✉❛s ❚❡❝♥♦❧♦❣✐❛s✳ ❙❡❝r❡t❛r✐❛ ❞❡

❊❞✉❝❛çã♦ ❋✉♥❞❛♠❡♥t❛❧✳ ❇r❛sí❧✐❛✿ ▼❊❈✴❙❊❋✱ ✷✵✵✻✳

❬✶✹❪ ▼❖■❙❊✱ ❊✳❀ ❉❖❲◆❙✱ ❋✳●❡♦♠❡tr✐❛ ▼♦❞❡r♥❛✳ ❱♦❧✉♠❡ ✷✱ ❙ã♦ P❛✉❧♦✿ ❊❞❣❛r❞ ❇❧ü❝❤❡r✱✶✾✼✺✳

❬✶✺❪ P❆❚❊❘▲■◆■✱ ❘✳ ❘✳ ❖s ✏❚❡♦r❡♠❛s✑ ❞❡ ❈❛✈❛❧✐❡r✐✳ ❘❡✈✐st❛ ❞♦ Pr♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛

no✳ ✼✷✱ 2o q✉❛❞r✐♠❡str❡ ❞❡ ✷✵✶✵✱ ♣á❣s✳ ✹✸ ✲ ✹✼ ❱❡rsã♦ ❛♠♣❧✐❛❞❛ ❝♦♠ ❛s ❞❡♠♦♥str❛çõ❡s❞♦s t❡♦r❡♠❛s✳ ✇✇✇✳❞♠✳✉❢s❝❛r✳❜r✴ ♣t❧✐♥✐✴✳✭ ❆❝❡ss♦ ❡♠ ✷✵✴✶✶✴✷✵✶✹ ✮✳

❬✶✻❪ P■◆❚❖✱ ❆✳ ❆ t❡♦r✐❛ ❞♦s ✐♥❞✐✈✐sí✈❡✐s✿ ❯♠❛ ❝♦♥tr✐❜✉✐çã♦ ❞♦ ♣❛❞r❡ ❇♦♥❛✈❡♥t✉r❛ ❈❛✈❛✲

❧✐❡r✐✳ ❉✐ss❡rt❛çã♦✳ P♦♥t✐❢í❝✐❛ ❯♥✐✈❡rs✐❞❛❞❡ ❈❛tó❧✐❝❛ ❞❡ ❙ã♦ P❛✉❧♦ P❯❈✲ ❙P✳ ❙ã♦ P❛✉❧♦✳✷✵✵✽✳

❬✶✼❪ P■❚❖▼❇❊■❘❆✱ ❏✳ ❇✳❀ ❘❖◗❯❊✱ ❚✳ ▼✳ ❚ó♣✐❝♦s ❞❡ ❍✐stór✐❛ ❞❛ ▼❛t❡♠át✐❝❛✳ ❘✐♦ ❞❡❏❛♥❡✐r♦ ✿ ❙❇▼✱ ✷✵✶✷ ✳✭❈♦❧❡çã♦ P❘❖❋▼❆❚✮✳

❬✶✽❪ P❘■▼❖✳ ▼✳❊✳ ❖ ♣r✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ♣❛r❛ ❝á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✿

❛❧❣✉♠❛s ♣♦ss✐❜✐❧✐❞❛❞❡s✳ ❉✐ss❡rt❛çã♦ ✭▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡◆❛❝✐♦♥❛❧✮✳ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❏✉✐③ ❞❡ ❋♦r❛✱ ✷✵✶✸✳

❬✶✾❪ ❙▼❖▲❊✱ ❑✳❈✳❙✳ ✱ ❉■◆■❩✱ ▼✳■✳❙✳ ▼❛t❡♠át✐❝❛✿ ❡♥s✐♥♦ ♠é❞✐♦ ❱♦❧✉♠❡ ✷✱ ❙ã♦ P❛✉❧♦✿❊❞✐t♦r❛ ❙❛r❛✐✈❛✱ ✷✵✶✵✳

❬✷✵❪ ❙❖❋❚❲❆❘❊ ●❊❖●❊❇❘❆ ✺✳✵ ❏❖●▲✶ ❇❊❚❆ ✸❉ ❉✐s♣♦♥í✈❡❧ ❡♠✿✇✇✇✳❣❡♦❣❡❜r❛✳♦r❣ ✲ ❆❝❡ss❛❞♦ ❡♠ ✶✶✴✵✶✴✷✵✶✺✳

❬✷✶❪ ❩❆❇❆▲❆✱ ❆✳ ❆ ♣rát✐❝❛ ❡❞✉❝❛t✐✈❛✿ ❝♦♠♦ ❡♥s✐♥❛r✳ ❊❞✐t♦r❛ ❆rt❡s ▼é❞✐❝❛s ❙✉❧ ▲t❞❛✱✶✾✾✽✳