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ADDITIONAL MATHEMATICS PROJECT WORK 2/2011

( CIRCLES & GEOMETRY )

NAME :SANGHARGANESH A/L PALANIVELLOO

I/C NUMBER : 94101807 5081

CLASS : 5 MURNI

TEACHER : PUAN SARINA BINTI AMIN

SCHOOL : SMK . GEORGETOWN


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INDEX

NUMBER TITLE PAGE

1 APPRECIATION 1

2 OBJECTIVES 2

3 INTRODUCTION 3-16

3.1) MATHEMATICAL FACTS USED 3-9

3.2) HISTORY OF MATHEMATICAL FACTS USED 10-14

3.3) ABOUT THE PROJECT WORK 15-16

4 TASK SPECIFICATION 17

5 PROCEDURE AND FINDINGS 18

4.1) PART 1 19

4.2) PART 2 20-27

4.3) PART 3 28-29

6 FURTHER EXPLORATION 30-32

7 REFLECTION 33-34


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1.APPRECIATION

I would like to express my gratitude towards my Additional

Mathematics teacher, Pn. Sarina for her patient guidance to

bring about the completion of this project. She has provided

invaluable tips and was very willing to give pointers to help us

with the project.

I would also like to thank my parents for their support and

encouragement and my siblings for providing useful

suggestions to improve my work. They gave me moral and

financial support to complete this project successfully.

Furthermore, I am very grateful for the cooperation and

help I received from my friends. We divided tasks amongstourselves to reduce the workload and later combined all the

information gathered.

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2. OBJECTIVES

The aims of carrying out this project are :

To learn the way to apply the formulas of mathematics in our daily

life accurately.

To widen our prospective view on mathematics.

To improve our mastering skills.

To use the correct language to express our mathematical ideas

properly.

To develop positive attitude towards mathematics.

To improve our way of thinking.

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3.INTRODUCTION

3.1) Mathematical Facts Used

1. Volume of geometry

Volume is how much three-dimensional space a substance (solid, liquid, gas, orplasma) or shape occupies or contains, often quantified numerically using the SIderived unit, the cubic metre. The volume of a container is generally understood to bethe capacity of the container, i. e. the amount of fluid (gas or liquid) that the containercould hold, rather than the amount of space the container itself displaces. Threedimensional mathematical shapes are also assigned volumes. Volumes of some

simple shapes, such as regular, straight-edged, and circular shapes can be easilycalculated using arithmetic formulas. The volumes of more complicated shapes canbe calculated by integral calculus if a formula exists for the shape's boundary. One-dimensional figures (such as lines) and twodimensional shapes (such as squares) areassigned zero volume in the three-dimensional space. The volume of a solid (whetherregularly or irregularly shaped) can be determined by fluid displacement.Displacement of liquid can also be used to determine the volume of a gas. Thecombined volume of two substances is usually greater than the volume of one of thesubstances. However, sometimes one substance dissolves in the other and thecombined volume is not additive. In differential geometry, volume is expressed bymeans of the volume form, and is an important global Riemannian invariant. In

thermodynamics, volume is a fundamental parameter, and is a conjugate variable topressure. The formula related to this subject are: Area X (length/ base/ height). Forthis project volume of cylinder is required. In common use a cylinder is taken to meana finite section of a right circular cylinder, i.e., the cylinder with the generating linesperpendicular to the bases, with its ends closed to form two circular surfaces, as in thefigure (right). If the cylinder has a radius r and length (height) h, then its volume isgiven by V = r2h.

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2. Differentiation (calculus)

Differentiation is a method to compute the rate at which a dependent output ychanges with respect to the change in the independent input x. This rate of change iscalled the derivative of y with respect to x. In more precise language, thedependence ofyuponxmeans that yis a function ofx. This functional relationship isoften denoted y= (x), where denotes the function. Ifxand yare real numbers, andif the graph ofyis plotted againstx, the derivative measures the slope of this graph ateach point. The simplest case is when y is a linear function ofx, meaning that thegraph ofyagainstxis a straight line. In this case, y= (x) = mx+ b, for real numbersm and b, and the slope m is given by

where the symbol (the uppercase form of the Greek letter Delta) is an abbreviationfor "change in." This formula is true because

y+ y= (x+ x) = m (x+ x) + b = mx+ b + m x= y+ mx.

It follows that y= m x.

This gives an exact value for the slope of a straight line. If the function is not linear(i.e. its graph is not a straight line), however, then the change in ydivided by the

change in xvaries: differentiation is a method to find an exact value for this rate ofchange at any given value ofx.

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3. Progression( geometric)

Geometric progression, also known as a geometric sequence, is a sequence of numberswhere each term after the first is found by multiplying the previous one by a fixed non-zero

number called the common ratio.

The behaviour of a geometric sequence depends on the value of the common ratio.If the common ratio is:

* Positive, the terms will all be the same sign as the initial term.* Negative, the terms will alternate between positive and negative.

* Greater than 1, there will be exponential growth towards positive infinity.* 1, the progression is a constant sequence.

* Between 1 and 1 but not zero, there will be exponential decay towards zero.* 1, the progression is an alternating sequence (see alternating series)

* Less than 1, for the absolute values there is exponential growth towards positive andnegative infinity (due to the alternating sign).

4. Circles

A circle is a simple shape of Euclidean geometry consisting of the set of points in a planethat are a given distance from a given point, the centre. The distance between any of the points

and the centre is called the radius.

Circles are simple closed curves which divide the plane into two regions: an interior and anexterior. In everyday use, the term "circle" may be used interchangeably to refer to either the

boundary of the figure, or to the whole figure including its interior; in strict technical usage, thecircle is the former and the latter is called a disk.

A circle is a special ellipse in which the two foci are coincident and the eccentricity is 0. Circlesare conic sections attained when a right circular cone is intersected by a plane perpendicular to

the axis of the cone.

A circle's diameter is the length of a line segment whose endpoints lie on the circle and whichpasses through the centre. This is the largest distance between any two points on the circle. The

diameter of a circle is twice the radius, or distance from the centre to the circle's boundary. Theterms "diameter" and "radius" also refer to the line segments which fit these descriptions. The

circumference is the distance around the outside of a circle.

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A chord is a line segment whose endpoints lie on the circle. A diameter is the longest chord in acircle. A tangent to a circle is a straight line that touches the circle at a single point, while a

secant is an extended chord: a straight line cutting the circle at two points.

An arc of a circle is any connected part of the circle's circumference. A sector is a region

bounded by two radii and an arc lying between the radii, and a segment is a region bounded by achord and an arc lying between the chord's endpoints.

The ratio of a circle's circumference to its diameter is (pi), an irrational constant approximately

equal to 3.141592654. Thus the length of the circumference Cis related to the radius rand

diameterdby:

Area enclosed = >

6

Chord, secant, tangent, and diameter.Arc, sector, and segment


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Area of the circle = area of the shaded square

As proved by Archimedes, the area enclosed by a circle is multiplied by the radius squared:

Equivalently, denoting diameter by d,

that is, approximately 79 percent of the circumscribing square (whose side is of length d). Theseresults can be obtained in several ways, including via integration and by considering the circle asa regular polygon with an infinite number of sides.

The circle is the plane curve enclosing the maximum area for a given arc length. This relates the

circle to a problem in the calculus of variations, namely the isoperimetric inequality.

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5. Polygons

In geometry a polygon is traditionally a plane figure that is bounded by a closed path or

circuit, composed of a finite sequence of straight line segments (i.e., by a closed polygonalchain). These segments are called its edges or sides, and the points where two edges meet are thepolygon's vertices (singular: vertex) or corners. An n-gon is a polygon with n sides. The interior

of the polygon is sometimes called its body. A polygon is a 2-dimensional example of the moregeneral polytope in any number of dimensions.

The word "polygon" derives from the Greek (pols) "much", "many" and (gn a)"corner" or "angle". (The word gnu, with a short o, is unrelated and means "knee".) Today

a polygon is more usually understood in terms of sides.

Usually two edges meeting at a corner are required to form an angle that is not straight (180);

otherwise, the line segments will be considered parts of a single edge.

All regular simple polygons (a simple polygon is one which does not intersect itself anywhere)

are convex. Those having the same number of sides are also similar.

An n-sided convex regular polygon is denoted by its Schlfli symbol {n}.

y Henagon or monogon {1}: degenerate in ordinary space (Most authorities do not regard

the monogon as a true polygon, partly because of this, and also because the formulaebelow do not work, and its structure is not that of any abstract polygon).

y Digon {2}: a "double line segment": degenerate in ordinary space (Some authorities do

not regard the digon as a true polygon because of this).y Equilateral triangle {3}y Square (regular tetragon or quadrilateral) {4}

y Regular pentagon {5}y Regular hexagon {6}

y Regular heptagon {7}y Regular octagon {8}

y Regular nonagon or enneagon {9}y Regular decagon {10}

y Relgular polygon with n = 5: pentagon with side s, apothem a, and circumradius r

y The area A of a convex regularn-sided polygon having side s, apothem a, perimeterp, and

circumradius ris given by

:

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y

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3.2) History Of Mathematical Facts Used

1. History of Volume of Geometry

The earliest recorded beginnings of geometry can be traced to early peoples, who

discovered obtuse triangles in the ancient Indus Valley (see Harappan

Mathematics), and ancient Babylonia (see Babylonian mathematics) from around

3000 BC. Early geometry was a collection of empirically discovered principles

concerning lengths, angles, areas, and volumes, which were developed to meet

some practical need in surveying, construction, astronomy, and various crafts.

Among these were some surprisingly sophisticated principles, and a modern

mathematician might be hard put to derive some of them without the use of

calculus. For example, both the Egyptians and the Babylonians were aware ofversions of the Pythagorean theorem about 1500 years before Pythagoras; the

Egyptians had a correct formula for the volume of a frustum of a square pyramid;

the Babylonians had a trigonometry table. The Babylonians may have known the

general rules for measuring areas and volumes. They measured the circumference

of a circle as three times the diameter and the area as one-twelfth the square of

the circumference, which would be correct if is estimated as 3. The volume of a

cylinder was taken as the product of the base and the height, however, the volume

of the frustum of a cone or a square pyramid was incorrectly taken as the product

of the height and half the sum of the bases. The Pythagorean theorem was alsoknown to the Babylonians. Also, there was a recent discovery in which a tablet

used as 3 and 1/8. The Babylonians are also known for the Babylonian mile,

which was a measure of distance equal to about seven miles today. This

measurement for distances eventually was converted to a time-mile used for

measuring the travel of the Sun, therefore, representing time. However, The

formulae for volume were discovered independently by many people over

centuries of time. Not one individual person 'discovered' how to calculate the

volume of a sphere, cube, rhomboid, cylinder and etc. The formulae for calculating

the volume of an object were arrived at many many years ago by many people inmany civilisations. No single person can be regarded as the first to describe how to

calculate volume.

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2.History ofDifferentiation(calculus)

The concept of a derivative in the sense of a tangent line is a very old one, familiar to

Greek geometers such as Euclid (c. 300 BC), Archimedes (c. 287212 BC) andApollonius of Perga (c. 262 190 BC).[1] Archimedes also introduced the use of

infinitesimals, although these were primarily used to study areas and volumes ratherthan derivatives and tangents; see Archimedes' use of infinitesimals. The use of

infinitesimals to study rates of change can be found in Indian mathematics, perhaps asearly as 500 AD, when the astronomer and mathematician Aryabhata (476550) used

infinitesimals to study the motion of the moon.[2]

The use of infinitesimals to computerates of change was developed significantly by Bhskara II (1114-1185); indeed, it has

been argued[3]

that many of the key notions of differential calculus can be found in hiswork, such as "Rolle's theorem".

[4]The Persian mathematician, Sharaf al-Dn al-Ts

(1135-1213), was the first to discover the derivative of cubic polynomials, an important

result in differential calculus;[5]

his Treatise on Equations developed concepts related todifferential calculus, such as the derivative function and the maxima and minima ofcurves, in order to solve cubic equations which may not have positive solutions.The

modern development of calculus is usually credited to Isaac Newton (1643 1727) andGottfried Leibniz (1646 1716), who provided independent

[7]and unified approaches to

differentiation and derivatives. The key insight, however, that earned them this credit,was the fundamental theorem of calculus relating differentiation and integration: this

rendered obsolete most previous methods for computing areas and volumes,[8]

whichhad not been significantly extended since the time of Ibn al-Haytham (Alhazen).

[9]For

their ideas on derivatives, both Newton and Leibniz built on significant earlier work bymathematicians such as Isaac Barrow (1630 1677), Ren Descartes (1596 1650),

Christiaan Huygens (1629 1695), Blaise Pascal (1623 1662) and John Wallis(1616 1703). Isaac Barrow is generallly given credit for the early development of the

derivative.[10] Nevertheless, Newton and Leibniz remain key figures in the history ofdifferentiation, not least because Newton was the first to apply differentiation to

theoretical physics, while Leibniz systematically developed much of the notation stillused today.Since the 17th century many mathematicians have contributed to the theory

of differentiation. In the 19th century, calculus was put on a much more rigorous footingby mathematicians such as Augustin Louis Cauchy (1789 1857), Bernhard Riemann

(1826 1866), and Karl Weierstrass (1815 1897). It was also during this period thatthe differentiation was generalized to Euclidean space and the complex plane.

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3.History ofProgression

Geometric sequences (with common ratio not equal to 1,1 or 0) show exponential

growth or exponential decay, as opposed to the Linear growth (or decline) of an

arithmetic progression such as 4, 15, 26, 37, 48, (with common difference 11). This

result was taken by T.R. Malthus as the mathematical foundation of his Principle of

Population. Note that the two kinds of progression are related: exponentiating each term

of an arithmetic progression yields a geometric progression, while taking the logarithm

of each term in a geometric progression with a positive common ratio yields an

arithmetic progression.

The geometric progression 1, 2, 4, 8, 16, 32, (or, in the binary numeral system, 1, 10,

100, 1000, 10000, 100000, ) is important in number theory. Book IX, Proposition 36

of Elements proves that if the sum of the first n terms of this progression is a prime

number, then this sum times the nth term is a perfect number. For example, the sum ofthe first 5 terms of the series 1 + 2 + 4 + 8 + 16 = 31, which is a prime number. The sum

31 multiplied by 16 (the 5th term in the series) equals 496, which is a perfect number.

Book IX, Proposition 35, proves that in a geometric series if the first term is subtracted

from the second and last term in the sequence then as the excess of the second is to the

first, so will the excess of the last be to all of those before it. (This is a restatement of

our formula for geometric series from above.)

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4.History ofCircles

The word "circle" derives from the Greek, kirkos "a circle," from the base ker- which means toturn or bend. The origins of the words "circus" and "circuit" are closely related.

The circle has been known since before the beginning of recorded history. Natural circles wouldhave been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand,

which forms a circle shape in the sand. The circle is the basis for the wheel, which, with relatedinventions such as gears, makes much of modern civilization possible. In mathematics, the study

of the circle has helped inspire the development of geometry, astronomy, and calculus.

Early science, particularly geometry and astrology and astronomy, was connected to the divinefor most medieval scholars, and many believed that there was something intrinsically "divine" or

"perfect" that could be found in circles.[citationneeded]

The compass in this 13th century manuscript is a

symbol of God's act of Creation. Notice also thecircular shape of the halo

Circles on an old astronomydrawing

Some highlights in the history of the circle are:

y 1700 BC The Rhind papyrus gives a method to find the area of a circular field. The

result corresponds to 256/81 (3.16049...) as an approximate value of .[1]

y 300 BC Book3 of Euclid's Elements deals with the properties of circles.

y In Plato's Seventh Letter there is a detailed definition and explanation of the circle. Plato

explains the perfect circle, and how it is different from any drawing, words, definition orexplanation.y 1880 Lindemann proves that is transcendental, effectively settling the millennia-old

problem of squaring the circle.

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5.History ofPolygons

Polygons have been known since ancient times. The regular polygons were known to the

ancient Greeks, and the pentagram, a non-convex regular polygon (star polygon), appears on thevase of Aristophonus, Caere, dated to the 7th century B.C..[citationneeded]

Non-convex polygons in

general were not systematically studied until the 14th century by Thomas Bredwardine.

historical image of polygons (1699)

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3.2) ABOUT THE PROJECT WORK

Students taking the elective subject Additional Mathematics in Form Five are

required to carry out a project work as required in the syllabus. This year, theCurriculum Development Division of the Ministry of Education Malaysia prepared atotal of four tasks for the students where we are required to choose and completeone of the four based on our area of interest. The title of the project work I haveselected is on statistics. The objective of this project is to improve the skill ofapplying mathematics in the daily lives of students while adapting a variety ofstrategies to solve routine and non-routine problems. The project aims to improvethe thinking skills of students while learning to use the language of mathematics toexpress mathematical ideas correctly and precisely to increase the studentsconfidence. Students are required to undergo this task within groups. Through thegroup effort of carrying out the project, it promotes effective mathematical

communication and at the same time it develops mathematical knowledge throughproblem solving in a way that increases students interest and confidence. Itprovides an ideal learning environment to stimulate and enhance effectivelearning. The students learn to develop a positive attitude towards mathematics aswell. Students will be able to apply knowledge and skills gained for experience inclassroom environments in meaningful ways to solve real-life problems and at thesame time express their mathematical thinking, reasoning and communication. Itwill also prepare them from the demand of their future undertakings and in theirworkplace while learning to be independent leaders. Lastly, students will learn touse technology especially ICT appropriately and effectively.

I am Sangharganesh A/L Palanivelloo, from 5 Murni, SMK. Georgetown.I chosedthe project work 2 because I find it interesting and challenging. I devide the taskwithin each of us and complete it successfully. Besides that, I learn the way toapply the formulas of mathematics in my daily life accurately, widen myprospective view on mathematics and improve my mastering skills

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CAKES

Cake is a form of food, typically a sweet, baked dessert. Cakes normally contain acombination of flour, sugar, eggs, and butter or oil, with some varieties also requiring liquid(typically milk or water) and leavening agents (such as yeast or baking powder). Flavorful

ingredients like fruit pures, nuts or extracts are often added, and numerous substitutions for theprimary ingredients are possible. Cakes are often filled with fruit preserves or dessert sauces

(like pastry cream), iced with buttercream or other icings, and decorated with marzipan, pipedborders or candied fruit.

Cake is often the dessert of choice for meals at ceremonial occasions, particularly weddings,

anniversaries, and birthdays. There are countless cake recipes; some are bread-like, some richand elaborate and many are centuries old. Cake making is no longer a complicated procedure;

while at one time considerable labor went into cake making (particularly the whisking of eggfoams), baking equipment and directions have been simplified that even the most amateur cook

may bake a cake.

Origin Ofcake

Although clear examples of the difference between cake and bread are easy to find, the precise

classification has always been elusive. For example, banana bread may be properly consideredeither a quick bread or a cake.

The Greeks invented beer as a leavener, frying fritters in olive oil, and cheesecakes using goat's

milk.In ancient Rome, basic bread dough was sometimes enriched with butter, eggs, and honey,which produced a sweet and cake-like baked good.

Latin poet Ovid refers to the birthday of him

and his brother with party and cake in his first book of exile, Tristia.

Early cakes in England were also essentially bread: the most obvious differences between a"cake" and "bread" were the round, flat shape of the cakes, and the cooking method, which

turned cakes over once while cooking, while bread was left upright throughout the bakingprocess.

Sponge cakes, leavened with beaten eggs, originated during the Renaissance, possibly in Spain.

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4.TASK SPECIFICATION

In this challenging project there is a few tasks to be completed.

TASK 1

Best Bakery shop received an order from my school to bake a 5kg of round cake. The volume

given for 1kg of the cake is 3800cm. Height of cake is 7.0cm. The dimensions of the ovengiven as of 80.0cm in length, 60.0 cm in width and 45.0cm in heght. Our group need to :

y Find the diameter of the baking tray.

y Tabulate different values of height and the corresponding diameters.

y State and explain the range of heights that is not suitable for the cakes.

y Suggest the most suitable dimensions for the cake.

y Represent the linear relation between height and diameters by plotting a graph based onthe equation representing the relation.

y Use the graph to determine the diameter of the round cake pan required to bake a cake of10.5cm in height.

y Use the graph to estimate the height of cake obtained for a a cake of diameter of 42 cm.

TASK2

Best Bakery has been requested to decorate the cake with fresh cream. The thickness of thecake is set uniform layer of 1cm.Our group need to:

y Estimate the amount of fresh cream needed to decorate the cake.

y Suggest three suitable shapes for the cake that will have the same height and volume.

y Estimate the amount of fresh cream needed for each cake.

y Determine the shape of cake that requires the least amount of fresh cream.

TASK3

In PART 3 our group is required to :

y Find the dimension of a 5kg round cake that requires the minimum amount offresh cream to decorate.

y Use two different methods including calculus.

y State and explain whether we would choose to bake a cake with such dimensions.

TASK 4We were required to do a further exploration. We helped the Best Bakery to find the volume foreach of the multi storey cake, and explain the number patterns formed by the volumes. We alsohave calculate the maximum number of cakes need to be baked if the total mass should not

exceed 15kg.17


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4 .

PROCEDURE

AND

FINDING

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4.1) PART 1

Cakes come in a variety of forms and flavours and are among favourite desserts served

during special occasions. Cakes are also liked by many people because the beautiful

art of cake baking and decorating. Mathematic knowledge is used widely in many

industries.It is also important and used in cake baking and decorating. Those are my

findings.

1)Geometry:-

Geometrical knowledge is highly required in cake baking and decorating. Geometry isused to determine a suitable dimensions and shapes for the cake. Besides that, thisknowledge is useful to assist one in designing and decorating cakes that comes in many

attractive shapes and designs. An estimation of volume is needed before the cake isproduced. The measurement of volume can help in determining the size of the tray andthe oven needed.

2)Differentiation (calculus) :-

Differentiation is important to determine the maximum or minimum amount ofingredients and to estimate minimum or maximum amount of cream for cake baking anddecorating. By differentiation method we can also able to estimate minimum or

maximum size of cake produced.

3) Progression :-

Progression is very useful to determine total weight or volume of multi-storey cakeswith proportional dimensions, to estimate total ingredients needed for cake-baking, toestimate total amount of cream for decoration and to estimate the number of cakes canbe baked with certain amount of ingredients or the total mass of the cake needed as thelimiting factor.

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4.2) PART 2

SOLUTION

1) I want to help Best Bakery shop to determine the diameter of the baking tray tobe used to fit 5 kg cake ordered by my school by using the mathematical facts

and information.

It is given that 1 kg of cake has a volume of 3800and a height of 7.0cm.

Volume of 5kg cake = Base area of cake x Height of cake

3800 x 5 = (3.142)(

) x 7

(3.142) = (

)

863.872 = (

)

= 29.392

d = 58.784 cm

2) (a) The oven that will be used to bake the cake has inner dimensions of 80.0cm inlength, 60.0 cm in width and 45.0cm in heght.First, form the formula ford in terms ofh by using the above formula for volume ofcake, V = 19000, that is:

W,60.0cm

19000 = (3.142)(d/2)h

=

h, 45.0cm

= d L, 80.0cm

DIAGRAM 1

d =

cm

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I also explore with different values of heights, h cm and the corresponding values of diameters

of baking tray, d cm to be used. I also wanted to help the Best Bakery shop to determine the most

suitable dimensions of the cake.

Height,h (cm) Diameter,d(cm)

1.0 155.53

2.0 109.98

3.0 89.80

4.0 77.77

5.0 68.56

6.0 63.49

7.0 58.78

8.0 54.99

9.0 51.84

10.0 49.18

2) (B)

i. h< 7cm is NOT suitable, because the resulting diameter produced is too large to fitinto the oven. Furthermore, the cake would be too short and too wide, making it lessattractive.

This is due to the width of the oven, 60.0 cm.

ii. h = 8.0cm d = 54.99cm , These are the dimensions that are the most suitable for

the cake because it can fit into the oven, and the size is suitable for easy

handling.

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2) C Equation And Graph of The Linear Relation Between Height And Diameter.

i. 19000 = (3.142)(

)h

19000/(3.142)h =

= d

d =

d =

log d =

log d =

log h + log 155.53, an equation to represent the linear relation between

height, h and diameter, d .

Log h 0 1 2 3 4

Log d 2.19 1.69 1.19 0.69 0.19

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Graph of log d against log h.

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2)(c)

ii. a) From the graph, when the height of cake is 10.5cm the diameter is determined that

h = 10.5cm, log h = 1.021, log d = 1.680, d = 47.86cm

b) From the graph, when a 42.0cm diameter round cake tray is used the height is estimated,

d = 42cm, log d = 1.623, log h = 1.140, h = 13.80cm

3)(a) I wanted to help the Best Bakery shop to decorate the cake with fresh cream. The thickness

of the cream is normally set to a uniform layer of 1 cm and we estimated the amount of fresh

cream needed to decorate the cake of the suitable dimensions we suggested.

h = 8cm, d = 54.99cmAmount of fresh cream = VOLUME of fresh cream needed (area x height)Amount of fresh cream = Vol. of cream at the top surface + Vol. of cream at the sidesurface

Vol. of cream at the top surface= Area of top surface x Height of cream

= (3.142)(

) x 1

= 2375 cm

Vol. of cream at the side surface= Area of side surface x Height of cream= (Circumference of cake x Height of cake) x Height of cream= 2(3.142)(54.99/2)(8) x 1= 1382.23 cm

Therefore, amount of fresh cream = 2375 + 1382.23 = 3757.23 cm

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3)(b) I would also show our generosity by helping Best Bakery shop by suggesting three other

shapes of cake with same height and volume and estimating the amount of fresh cream needed.

I made an early prediction on the shapes of cakes that will need the least amount of fresh cream.The shapes suggested are rectangle-shaped base of cuboid, semi circle-shaped base of semi

cylinder and pentagon-shaped base. The cake that will require the least amount of cream is

pentagon.

1 Rectangle-shaped base (CUBOID)

19000 = base area x heightExample of cuboid cake

base area =

length x width = 2375By trial and improvement, 2375 = 50 x 47.5 (length = 50, width = 47.5, height = 8)

Therefore, volume of cream= 2(Area of left/right side surface)(Height of cream) + 2(Area of front/back sidesurface)(Height of cream) + Vol. of top surface= 2(8 x 50)(1) + 2(8 x 47.5)(1) + 2375 = 3935 cm

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2 Semi circle-shaped base (SEMI CYLINDER)

Example of semi cylindrical cake

19000 = base area x height

base area =

2375 = ( 3.142 x r ) / 21511.78 = r

r = 38.88 cm, Since r = width of the cake, radius is the width.width = 38.88 cmd = 77.76 cm, Since d = length of the cake, diameter is the length.length = 77.76 cm

Therefore, volume of cream(Area of rectangular surface) + (Area of the curved surface) + (Top surface area)

=(Length x height) + (Length of sector of the semicircle x height) + (Area of semircle)= ( d x 8 ) + ( 3.142r x 8 ) + ( 3.142r / 2)= (77.76 x 8) + (3.142 x 38.88 x 8) + (3.142 x 38.88 / 2)= 622.08 + 977.29 + 2374.81

= 3974.18 cm

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3 Pentagon-shaped base

Example of pentagon cake

19000 = base area x heightbase area = 2375 = area of 5 similar isosceles triangles in a pentagontherefore:2375 = 5(length x width)475 = length x widthBy trial and improvement, 475 = 25 x 19 (length = 25, width = 19)

Therefore, amount of cream= 5(area of one rectangular side surface)(height of cream) + vol. of top surface= 5(8 x 19) + 2375 = 3135 cm

3)(c)We also determined the shape of the cake that requires the least amount of fresh cream.

Pentagon-shaped cake requires the least amount of fresh cream, since itrequires only 3135 cm of cream to be used.

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4.3) PART 3

We are curious to find the dimension of the 5kg cake that requires the minimum amount of

fresh cream to be decorated with by using two different methods of additional

mathematics.That is Differentiation(calculus)and Quadratic Functions.

Method 1: Differentiation

Use two equations for this method: the formula for volume of cake (as in Q2/a), and theformula for amount (volume) of cream to be used for the round cake (as in Q3/a).19000 = (3.142)rh (1)V = (3.142)r + 2(3.142)rh (2)

From (1): h =

(3)

Sub. (3) into (2):

V = (3.142)r + 2(3.142)r(

)

V = (3.142)r + (

)

V = (3.142)r + 38000r-1

(

) = 2(3.142)r (

)

0 = 2(3.142)r (

) -->> minimum value, therefore

= 0

= 2(3.142)r

= r

6047.104 = rr = 18.22

Sub. r = 18.22 into (3):

h =

h = 18.22

Therefore, h = 18.22cm, d = 2r = 2(18.22) = 36.44cm

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Method 2: Quadratic FunctionsUse the two same equations as in Method 1, but only the formula for amount of creamis the main equation used as the quadratic function.Let f(r) = volume of cream, r = radius of round cake:19000 = (3.142)rh (1)f(r) = (3.142)r + 2(3.142)hr (2)From (2):f(r) = (3.142)(r + 2hr) -->> factorize (3.142)

= (3.142)[ (r +

) (

) ] -->> completing square, with a = (3.142), b = 2h and c = 0

= (3.142)[ (r + h) h ]= (3.142)(r + h) (3.142)h(a = (3.142) (positive indicates min. value), min. value = f(r) = (3.142)h, correspondingvalue of x = r = --h)

Sub. r = --h into (1):19000 = (3.142)(--h)hh = 6047.104h = 18.22

Sub. h = 18.22 into (1):19000 = (3.142)r(18.22)r = 331.894r = 18.22therefore, h = 18.22 cm, d = 2r = 2(18.22) = 36.44 cm

I would choose not to bake a cake with such dimensions because its dimensionsare not suitable (the height is too high) and therefore less attractive. Furthermore,such cakes are difficult to handle properly and easily.

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5) FURTHER EXPLORATION

5.1 EXPLORATION

In this further exploration, our group wants to study the various size of the cake in term of differentvolume and different radius by using mathematical facts. As the Best Bakery received an order to bake a

multi-storey cake for Merdeka celebration as in the diagram.

Diagram.

Therefore, we are here to do further exploration to help Best Bakery shop to determine the volumes of the

first, second, third and forth cakes. Besides that, we also determine the number patterns formed by the

volumes with mathematical explanation.

3(a) height, h of each cake = 6cm

radius of largest cake = 31cmradius of 2nd cake = 10% smaller than 1st cakeradius of 3rd cake = 10% smaller than 2nd cake

31, 27.9, 25.11, 22.599a = 31, r =

, V = (3.142)rh

Radius of 1st cake = 31, volume of 1st cake = (3.142)(31)(6) = 18116.772Radius of 2nd cake = 27.9, vol. of 2nd cake = 14674.585Radius of 3rd cake = 25.11, vol. of 3rd cake = 11886.414

Radius of4th

cake = 22.599, vol. of4th

cake = 9627.995

18116.772, 14674.585, 11886.414, 9627.995, a = 18116.772, ratio, r = T2/T1 = T3/T2 = = 0.81

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The number pattern formed by the volume is 18116.772, 14674.585, 11886.414, 9627.995, By determining the ratio of the number pattern it is shown that it is a geometric progression.R = T2/T1 = T3 /T2 = = 0.81

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of

numbers where each term after the first is found by multiplying the previous one by a fixed non-zeronumber called the common ratio. For example, the sequence 2, 6, 18, 54, is a geometric progression

with common ratio 3. Similarly 10, 5, 2.5, 1.25, is a geometric sequence with common ratio 1/2. The

sum of the terms of a geometric progression is known as a geometric series.

Thus, the general form of a geometric sequence is

where r 0 is the common ratio and a is a scale factor, equal to the sequences start value.

3(b) We also explore on the effect of the total mass of the cakes on the number of cakes needed.

If the total mass of all cakes should not exceed 15kg, the maximum number of the cakes are

as the follows:

Sn, sum of volumes of 15kg of the cakes are 15kg X 3800= 57000.

Sn =

Sn = 57000, a = 18116.772 and r = 0.81

57000 =

1 0.81n = 0.59779

0.40221 = 0.81n

log0.81 0.40221 = n

n =

n = 4.322 therefore, n 4.

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Our group also verify the answer by using the try and error method. Therefore we are able todetermine the suitable number of cakes that can be made with 15 kg.

VERIFYING ANSWER

S5 = (18116.772(1 (0.81)5)) / (1 0.81) = 62104.443 > 57000 (Sn > 57000, n = 5 is not

suitable)

When n = 4:

S4 = (18116.772(1 (0.81)4)) / (1 0.81) = 54305.767 < 57000 (Sn < 57000, n = 4 is suitable)

Hence, Maximum number of cakes can be baked = 4

5.2 Conclusion

By doing this further exploration we learn that the knowledge of differentiation,

progression and volume calculations can be used to solve daily problems .Through

applying all these formulas, I can save materials used in all sorts of production.

Besides that, we also able to estimate the number of production with certain

limited factors.

For example, when there is only 15kg of total mass of the cakes preferred. Byusing these mathematical formula we are able to estimate the maximum number of

cakes can be baked that is 4.

Last but not least, I know the history of these formulas are also able to determine

the correct formula to be used in certain circumstances. Now, I can finish any work

associated with mathematics at a very fast and accurate rate.

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6. REFLECTION

By doing this paperwork, I now learn that the knowledge of differentiation and

area and even volume calculations can be used to solve daily problems .Through applying all

these formulas, I can save materials used in all sorts of production.

At the same time, the cost will be reduced. Not only that, I know the history of theseformulas and also able to determine the correct formula to be used in certain

circumstances. From this paperwork, I have learnt the correct way to display the workings

correctly and arrange the calculations and topics systematically.

Through this, I have successfully completed this project along with my friends. At thesame time, I also have learnt that co-operation is very important for a team. Only through

teamwork, we can finish any task given by all means. Through teamwork, I also have learnt

the correct way to listen to other members and to judge all the ideas we got afterbrainstorming through comparison based on calculations.

After completing this paper, I have learnt how to co-operate with my teammates to

complete this paperwork. By doing this paperwork, I now learn that the knowledge of

differentiation and area and even volume calculations can be used to solve daily problems.

Through applying all these formulas, I can save materials used in all sorts of production. At

the same time, the cost will be reduced.

Not only that, I know the history of these formulas and also able to determine the

correct formula to be used in certain circumstances. Now, I can finish any work associated

with mathematics at a very fast and accurate rate.

Throughout day and night

I sacrificed my precious time to have fun

From..

Monday,Tuesday,Wednesday,Thursday,Friday

And even the weekend that I always looking forward to the poem NUMBERS.

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NUMBERS

Numeration is a murderer,7 ate 9,

When he got into court,

He said, `I had to dine!

Subtraction is like ice cream,

They both disappear,

I know someone who likes them,And he is a peer,

Subtraction is like geometry,

They use line segments,

Line segments are used a lot,

Subtraction is a casino,You never come out with more,

When you do get some cash,Youll use it at a store.

Addition is a birthday party,

You always get more,

You get a bit of money,And presents galore,

Divisions is like friends,

You have to share with both,Both are essential,

For your childhood growth.

Multiplication is a herd of animals,

Its always getting bigger,But when one set hits another,

I think theyll merge together.

Operations are really cool,Its one of the things I like,

Math and fishing are also great,We measured my caught pike.

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