FORM 5 PROJECT WORK FOR ADDITIONAL MATHEMATHICS 2009 by mizie0o0

8/14/2019 FORM 5 PROJECT WORK FOR ADDITIONAL MATHEMATHICS 2009 by mizie0o0

http://slidepdf.com/reader/full/form-5-project-work-for-additional-mathemathics-2009-by-mizie0o0 1/22

Additional Mathematics 2009

PROJECT WORK FORADDITIONAL MATHEMATHICS

2009

Circles In Our Daily Life

Name : Muhamad Termizi Bin Muhamad

Kelas : 5 Sains Teknologi Kejuruteraan

K/P : 920102-07-5241

Guru Pembimbing : Sir Mohd Nasir Bin

Sekolah : Sekolah Menengah Kebangsaan Seri Samudera

Mizie0o0

1




No. Content Page1. Acknowledgement 22. Objective 33. Introduction 44. Part 1 6

5. Part 2a 136. Part 2b 157. Part 3 17

First of all, I would like to say Alhamdulillah, for giving me the strength

and health to do this project work until it done.

Mizie0o0

2




Not forgotten to my parents for providing everything, such as money,

to buy anything that are related to this project work and their advise, which

is the most needed for this project. Internet, books, computers and all that as

my source to complete this project. They also supported me and encouraged

me to complete this task so that I will not procrastinate in doing it.

Then I would like to thank my teacher, Sir Mohd Nasir Bin …… for

guiding me and my friends throughout this project. We had some difficulties

in doing this task, but he taught us patiently until we knew what to do. He

tried and tried to teach us until we understand what we supposed to do with

the project work.

Last but not least, my friends who were doing this project with me and

sharing our ideas. They were helpful that when we combined and discussed

together, we had this task done.

The aims carrying out this project work are:

Mizie0o0

3




i. To apply and adapt a variety of problem-solving strategies to solve

problems;

ii. To improve thinking skills;

iii. To promote effective mathematical communication;

iv. To develop mathematical knowledge through problem solving in a

way that increases students’ interest and confidence;

v. To use the language of mathematics to express mathematical ideas

precisely;

vi. To provide learning environment that stimulates and enhances

effective learning;

vii.To develop positive attitude towards mathematics.

Mizie0o0

4




A circle is a simple shape of Euclidean geometry consisting of those

points in a plane which are the same distance from a given point called the

centre. The common distance of the points of a circle from its center is called

its radius. A diameter is a line segment whose endpoints lie on the circle and

which passes through the centre of the circle. The length of a diameter is

twice the length of the radius. A circle is never a polygon because it has no

sides or vertices.

Circles are simple closed curves which divide the plane into two

regions, an interior and an exterior. In everyday use the term "circle" may be

used interchangeably to refer to either the boundary of the figure (known as

the perimeter) or to the whole figure including its interior, but in strict

technical usage "circle" refers to the perimeter while the interior of the circle

is called a disk. The circumference of a circle is the perimeter of the circle

(especially when referring to its length).

A circle is a special ellipse in which the two foci are coincident. Circles

are conic sections attained when a right circular cone is intersected with a

plane perpendicular to the axis of the cone.

The circle has been known since before the beginning of recorded

history. It is the basis for the wheel, which, with related inventions such as

gears, makes much of modern civilization possible. In mathematics, the

study of the circle has helped inspire the development of geometry and

calculus.

Mizie0o0

5

http://en.wikipedia.org/wiki/Shape

http://en.wikipedia.org/wiki/Euclidean_geometry

http://en.wikipedia.org/wiki/Point_(geometry)

http://en.wikipedia.org/wiki/Plane_(mathematics)

http://en.wikipedia.org/wiki/Distance

http://en.wikipedia.org/wiki/Centre_(geometry)

http://en.wikipedia.org/wiki/Radius

http://en.wikipedia.org/wiki/Diameter

http://en.wikipedia.org/wiki/Line_segment

http://en.wikipedia.org/wiki/Endpoint

http://en.wikipedia.org/wiki/Polygon

http://en.wikipedia.org/wiki/Vertex_(geometry)

http://en.wikipedia.org/wiki/Curve


http://en.wikipedia.org/wiki/Interior_(topology)

http://en.wikipedia.org/wiki/Perimeter

http://en.wikipedia.org/wiki/Disk_(mathematics)

http://en.wikipedia.org/wiki/Circumference

http://en.wikipedia.org/wiki/Ellipse

http://en.wikipedia.org/wiki/Focus_(geometry)

http://en.wikipedia.org/wiki/Conic_section

http://en.wikipedia.org/wiki/Conical_surface

http://en.wikipedia.org/wiki/Wheel

http://en.wikipedia.org/wiki/Gear












































































Early science, particularly geometry and Astrology and astronomy, was

connected to the divine for most medieval scholars, and many believed that

there was something intrinsically "divine" or "perfect" that could be found in

circles.

Some highlights in the history of the circle are:

• 1700 BC – The Rhind papyrus gives a method to find the area of a

circular field. The result corresponds to 256/81 as an approximate

value of π.

•

300 BC – Book 3 of Euclid's Elements deals with the properties of

circles.

• 1880 – Lindemann proves that π is transcendental, effectively settling

the millennia-old problem of squaring the circle.

Mizie0o0

6

http://en.wikipedia.org/wiki/Science

http://en.wikipedia.org/wiki/Geometry

http://en.wikipedia.org/wiki/Astrology_and_astronomy

http://en.wikipedia.org/wiki/History_of_science_in_the_Middle_Ages

http://en.wikipedia.org/wiki/Rhind_papyrus

http://en.wikipedia.org/wiki/Euclid's_Elements

http://en.wikipedia.org/wiki/Ferdinand_von_Lindemann

http://en.wikipedia.org/wiki/Squaring_the_circle




























Mizie0o0

7




Mizie0o0

8




Definition

In Euclidean plane geometry, π is defined as the ratio of a circle's

circumference to its

diameter:

The ratio C/d is constant, regardless of a circle's size. For example, if a

circle has twice the diameter d of another circle it will also have twice the

circumference C, preserving the ratio C/d .

Area of the circle = π × area of the shaded square:

Alternatively π can be also defined as the ratio of a circle's area (A) to

the area of a square whose side is equal to the radius:

These definitions depend on results of Euclidean geometry, such as the

fact that all circles are similar. This can be considered a problem when π

occurs in areas of mathematics that otherwise do not involve geometry. For

Mizie0o0

9


http://en.wikipedia.org/wiki/Ratio

http://en.wikipedia.org/wiki/Circle



http://en.wikipedia.org/wiki/Area


http://en.wikipedia.org/wiki/Similarity_(geometry)

http://en.wikipedia.org/wiki/File:Circle_Area.svg




























this reason, mathematicians often prefer to define π without reference to

geometry, instead selecting one of its analytic properties as a definition. A

common choice is to define π as twice the smallest positive x for which

cos( x ) = 0. The formulas below illustrate other (equivalent) definitions.

Mizie0o0

10

http://en.wikipedia.org/wiki/Mathematical_analysis

http://en.wikipedia.org/wiki/Trigonometric_function










History

The ancient Babylonians calculated the area of a circle by taking 3

times the square of its radius, which gave a value of pi = 3. One Babylonian

tablet (ca. 1900–1680 BC) indicates a value of 3.125 for pi, which is a closer

approximation.

In the Egyptian Rhind Papyrus (ca.1650 BC), there is evidence that the

Egyptians calculated the area of a circle by a formula that gave the

approximate value of 3.1605 for pi.

The ancient cultures mentioned above found their approximations by

measurement. The first calculation of pi was done by Archimedes of

Syracuse (287–212 BC), one of the greatest mathematicians of the ancient

world. Archimedes approximated the area of a circle by using the

Pythagorean Theorem to find the areas of two regular polygons: the polygon

inscribed within the circle and the polygon within which the circle was

circumscribed. Since the actual area of the circle lies between the areas of

the inscribed and circumscribed polygons, the areas of the polygons gave

upper and lower bounds for the area of the circle. Archimedes knew that he

had not found the value of pi but only an approximation within those limits.

In this way, Archimedes showed that pi is between 3 1/7 and 3 10/71.

Mizie0o0

11

http://www-history.mcs.st-and.ac.uk/~history/Diagrams/Rhind_papyrus.jpeg







A similar approach was used by Zu Chongzhi (429–501), a brilliant

Chinese mathematician and astronomer. Zu Chongzhi would not have been

familiar with Archimedes’ method—but because his book has been lost, little

is known of his work. He calculated the value of the ratio of the

circumference of a circle to its diameter to be 355/113. To compute this

accuracy for pi, he must have started with an inscribed regular 24,576-gon

and performed lengthy calculations involving hundreds of square roots

carried out to 9 decimal places.

Mathematicians began using the Greek letter π in the 1700s.

Introduced by William Jones in 1706, use of the symbol was popularized by

Euler, who adopted it in 1737.

An 18th century French mathematician named Georges Buffon devised a way

to calculate pi based on probability.

Mizie0o0

12




Mizie0o0

13










c : Assume the diameter of outer semicircle is 30cm and 4 semicircles are

inscribed in the outer semicircle such that the sum of d1(APQ), d2(QRS),

d3(STU), d4(UVC) is equal to 30cm.

d1 d2 d3 d4 SABC SAPQ SQRS SSTU SUVC

10 8 6 6 15 π 5 π 4 π 3 π 3 π

12 3 5 10 15 π 6 π 3/2 π 5/2 π 5 π14 8 4 4 15 π 7 π 4 π 2 π 2 π

15 5 3 7 15 π 15/2 π 5/2 π 3/2 π 7/2 π

let d1=10, d2=8, d3=6, d4=6, SABC = SAPQ + SQRS + SSTU +

SUVC

15 π = 5 π + 4 π + 3 π + 3 π

15 π = 15 π

Mizie0o0

17




a. Area of flower plot = y m2

y = (25/2) π - (1/2(x/2)2π + 1/2((10-x )/2)2 π)

= (25/2) π - (1/2(x/2)2π + 1/2((100-20x+x2)/4) π)

= (25/2) π - (x2/8 π + ((100 - 20x + x2)/8) π)

= (25/2) π - (x2π + 100π – 20x π + x2π )/8

= (25/2) π - ( 2x2– 20x + 100)/8) π

Mizie0o0

18




= (25/2) π - (( x2 – 10x + 50)/4)

= (25/2 - (x2 - 10x + 50)/4) π

y = ((10x – x2)/4) π

b. y = 16.5 m2

16.5 = ((10x – x2)/4) π

66 = (10x - x2) 22/7

66(7/22) = 10x – x2

0 = x2 - 10x + 21

0 = (x-7)(x – 3)

x = 7 , x = 3

c. y = ((10x – x2)/4) π

y/x = (10/4 - x/4) π

x 1 2 3 4 5 6 7

y/x 7.1 6.3 5.5 4.7 3.9 3.1 2.4

Mizie0o0

19



3.0

4.0

5.0

6.0

7.0

8.0

0 1 2 3 4 5 6 7

X

Y/x


When x = 4.5 , y/x = 4.3

Area of flower plot = y/x * x

= 4.3 * 4.5

= 19.35m2

d. Differentiation method

dy/dx = ((10x-x

2

)/4) π

= ( 10/4 – 2x/4) π

0 = 5/2 π – x/2 π

5/2 π = x/2 π

x = 5

Mizie0o0

20





Documents

FORM 5 PROJECT WORK FOR ADDITIONAL MATHEMATHICS 2009 by mizie0o0