8/14/2019 +math project 2009 (:

1/138

PROJECT WORK FOR

ADDITIONAL MATHEMATHICS 2009

SMK TTDI JAYA

Circles In Our Daily Life

Name : Siti Nurul Idayu bt Abdullah Sapian

Class : 5 Firasat

Teacher : Madam Lee

8/14/2019 +math project 2009 (:

2/138

Content

No. Contents Page1 Introduction 3

2 Part 1 4 - 6

3 Part 2a 7 - 8

4 Part 2b 9 - 10

5 Part 3 11 - 14

8/14/2019 +math project 2009 (:

3/138

Introduction

A circle is a simple shape ofEuclidean geometry consisting of thosepoints in aplane 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 diameteris 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

apolygon because it has no sides orvertices.

Circles are simple closed curves which divide the plane into two regions, an interiorand an exterior. In

everyday use the term "circle" may be used interchangeably to refer to either the boundary of the figure

(known as theperimeter) 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.

Part 1

http://en.wikipedia.org/wiki/Shapehttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Point_(geometry)http://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Distancehttp://en.wikipedia.org/wiki/Centre_(geometry)http://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Line_segmenthttp://en.wikipedia.org/wiki/Endpointhttp://en.wikipedia.org/wiki/Polygonhttp://en.wikipedia.org/wiki/Vertex_(geometry)http://en.wikipedia.org/wiki/Curvehttp://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Interior_(topology)http://en.wikipedia.org/wiki/Perimeterhttp://en.wikipedia.org/wiki/Perimeterhttp://en.wikipedia.org/wiki/Disk_(mathematics)http://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Focus_(geometry)http://en.wikipedia.org/wiki/Conic_sectionhttp://en.wikipedia.org/wiki/Conical_surfacehttp://en.wikipedia.org/wiki/Shapehttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Point_(geometry)http://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Distancehttp://en.wikipedia.org/wiki/Centre_(geometry)http://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Line_segmenthttp://en.wikipedia.org/wiki/Endpointhttp://en.wikipedia.org/wiki/Polygonhttp://en.wikipedia.org/wiki/Vertex_(geometry)http://en.wikipedia.org/wiki/Curvehttp://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Interior_(topology)http://en.wikipedia.org/wiki/Perimeterhttp://en.wikipedia.org/wiki/Disk_(mathematics)http://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Focus_(geometry)http://en.wikipedia.org/wiki/Conic_sectionhttp://en.wikipedia.org/wiki/Conical_surface

8/14/2019 +math project 2009 (:

4/138

There are a lot of things around us related to circles or parts of a circles. We need to play with circles in

order to complete some of the problems involving circles.

amazing circle chives in a circle circle (people)

circle of love circle purple center

Before I continue the task, first, we do have to know what dopi() related to a circle.

Definition

In Euclidean plane geometry, is defined as theratio of a circle'scircumferenceto its

http://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Ratiohttp://en.wikipedia.org/wiki/Circlehttp://en.wikipedia.org/wiki/Circumference

8/14/2019 +math project 2009 (:

5/138

diameter:

The ratio

C

/d is constant, regardless of a circle's size. For example, if a circle has twice the diameterdof

another circle it will also have twice the circumference C, preserving the ratio C/d.

Area of the circle = area of the shaded square

Alternatively can be also defined as the ratio of a circle'sarea (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 aresimilar.

This can be considered a problem when occurs in areas of mathematics

that otherwise do not involve geometry. For this reason, mathematicians

often prefer to define without reference to geometry, instead selecting

one of its analytic properties as a definition. A common choice is to

define as twice the smallest positivex for whichcos(x) = 0. The

formulas below illustrate other (equivalent) definitions.

History

http://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Similarity_(geometry)http://en.wikipedia.org/wiki/Similarity_(geometry)http://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Trigonometric_functionhttp://en.wikipedia.org/wiki/Trigonometric_functionhttp://en.wikipedia.org/wiki/Diameterhttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Similarity_(geometry)http://en.wikipedia.org/wiki/Mathematical_analysishttp://en.wikipedia.org/wiki/Trigonometric_function

8/14/2019 +math project 2009 (:

6/138

d1 d2

10 cm

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.

Earlyscience, 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 .

300 BC Book 3 ofEuclid's Elementsdeals with the properties of circles.

1880 Lindemann proves that is transcendental, effectively settling the millennia-old problem

ofsquaring the circle.

http://en.wikipedia.org/wiki/Wheelhttp://en.wikipedia.org/wiki/Gearhttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Astrology_and_astronomyhttp://en.wikipedia.org/wiki/History_of_science_in_the_Middle_Ageshttp://en.wikipedia.org/wiki/Rhind_papyrushttp://en.wikipedia.org/wiki/Euclid's_Elementshttp://en.wikipedia.org/wiki/Euclid's_Elementshttp://en.wikipedia.org/wiki/Ferdinand_von_Lindemannhttp://en.wikipedia.org/wiki/Squaring_the_circlehttp://en.wikipedia.org/wiki/Squaring_the_circlehttp://en.wikipedia.org/wiki/Wheelhttp://en.wikipedia.org/wiki/Gearhttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Astrology_and_astronomyhttp://en.wikipedia.org/wiki/History_of_science_in_the_Middle_Ageshttp://en.wikipedia.org/wiki/Rhind_papyrushttp://en.wikipedia.org/wiki/Euclid's_Elementshttp://en.wikipedia.org/wiki/Ferdinand_von_Lindemannhttp://en.wikipedia.org/wiki/Squaring_the_circle

8/14/2019 +math project 2009 (:

7/138

Part 2 (a)

Diagram 1 shows a semicirclePQR of diameter 10cm. SemicirclesPAB andBCR of diameter d1 and d2

respectively are inscribed inPQR such that the sum of d1 and d2 is equal to 10cm. By using various

values of d1 and corresponding values of d2, I determine the relation between length of arcPQR,PAB, and

BCR.

Using formula: Arc of semicircle = d

d1 (cm) d2 (cm) Length of arcPQR in

terms of (cm)

Length of arcPAB in

terms of (cm)

Length of arcBCR in

terms of (cm)

1 9 5 9/2

2 8 5 4 3 7 5 3/2 7/2

4 6 5 2 3

5 5 5 5/2 5/2

6 4 5 3 2

7 3 5 7/2 3/2

8 2 5 4

9 1 5 9/2

Table 1

From the Table 1 we know that the length of arcPQR is not affected by the different in d1 and d2 inPAB

andBCR respectively. The relation between the length of arcsPQR ,PAB andBCR is that the length of

arcPQR is equal to the sum of the length of arcsPAB andBCR, which is we can get the equation:

SPQR = SPAB + SBCR

Let d1= 3, and d2 =7 SPQR = SPAB + SBCR

8/14/2019 +math project 2009 (:

8/138

d1 d2

10

d3D

E

5 = (3) + (7)

5 = 3/2 + 7/2

5 = 10/2

5 = 5

2 (b) i)

8/14/2019 +math project 2009 (:

9/138

d1 d2 d3 SPQR SPAB SBCD SDER1 2 7 5 1/2 7/2

2 2 6 5 3

2 3 5 5 3/2 5/2

2 4 4 5 2 2

2 5 3 5 5/2 3/2

SPQR = SPAB + SBCD + SDER

Let d1 = 2, d2 = 5, d3 = 3 SPQR = SPAB + SBCD + SDER

5 = + 5/2 + 3/2

5 = 5

(b) ii The length of arc of outer semicircle is equal to the sum of the length of arc of inner semicircle for

n = 1,2,3,4,.

Souter = S1 + S2 + S3 + S4 + S5

(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 SUVC10 8 6 6 15 5 4 3 3

12 3 5 10 15 6 3/2 5/2 5

14 8 4 4 15 7 4 2 2

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

8/14/2019 +math project 2009 (:

10/138

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

15 = 5 + 4 + 3 + 3

15 = 15

Part 3

(a). Area of flower plot = y m2

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

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

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

= (25/2) - (x

2

+ 100 20x + x

2

)/8

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

= (25/2) - (( x2 10x + 50)/4)

= (25/2 - (x2 - 10x + 50)/4)

y = ((10x x2)/4)

8/14/2019 +math project 2009 (:

11/138

3.0

4.0

5.0

6.0

7.0

8.0

0 1 2 3 4 5 6 7

X

Y/x

(b). y = 16.5 m2

16.5 = ((10x x2)/4)

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)

y/x = (10/4 - x/4)

x 1 2 3 4 5 6 7

y/x 7.1 6.3 5.5 4.7 3.9 3.1 2.4

8/14/2019 +math project 2009 (:

12/138

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-x2)/4)

= ( 10/4 2x/4)

0 = 5/2 x/2

5/2 = x/2

x = 5

Completing square method

y = ((10x x2)/4)

= 5/2 - x2/4

8/14/2019 +math project 2009 (:

13/13

= -1/4 (x2 10x)

y+ 52 = -1/4 (x 5)2

y = -1/4 (x - 5)2 - 25

x 5 = 0

x = 5

(e). n = 12, a = 30cm, S12 = 1000cm

S12 = n/2 (2a + (n 1)d

1000 = 12/2 ( 2(30) + (12 1)d)

1000 = 6 ( 60 + 11d)

1000 = 360 + 66d

1000 360 = 66d

640 = 66d

d = 9.697

Tn (flower bed) Diameter

(cm)

T1 30

T2 39.697

T3 49.394

T4 59.091

T5 68.788

T6 78.485

T7 88.182

T8 97.879

T9 107.576

T10 117.273

T11 126.97

T12 136.667