Additional Mathematics Project 2009 Tikl

SEKOLAH MENENGAH TEKNIK KUALA LUMPUR

ADDITIONAL MATHEMATICS PROJECT WORK 2009

NAME :

MUHAMMAD ARIF BIN ZAINAL ABIDIN

IC NO. :

920323-14-6699

CLASS:

5 AWAM 1

Contents 1. Introduction 2. Part 01 3. Part 02 4. Part 03 5. conclusion

INTRODUCTION 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. 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.[1] • 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.[2]

PART 01 There are a lot of things around us related to circles or parts of circles. (a) Sample of pictures that are related to circles.

(b) Definition of

Pi or π is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. It is approximately equal to 3.14159 in the usual decimal notation (see the table for its representation in some other bases). π is one of the most important mathematical and physical constants: many formulae from mathematics, science, and engineering involve π

History of

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. 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 24,576gon 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.

PART 02 Q C

(a) A P

d

d1

B 10

d₁ (cm)

d₂ (cm)

1 1.2 1.5 2 2.7 3 3.3 3.5 4 4.5

9 8.8 8.5 8 7.3 7 6.7 6.5 6 5.5

Length of arc PQR in terms of (cm) 5 5 5 5 5 5 5 5 5 5

R

d

d2

10cm

Length of arc PAB in terms of (cm) 0.5 0.6 0.75 1.35 1.5 1.65 1.75 2 2.25

Length of arc BCR in terms of (cm) 4.5 4.4 4.25 4 3.65 3.5 3.35 3.25 3 2.75

From the table above, we can conclude that, Length of arc PQR = length of arc PAB + length of arc BCR Length of arc PQR is not affected by the different in d₁ and d₂ as long as the the sum of both length equals to d. Let d₁ = 3 and d₂ = 7,

PQR = PAB + BCR 5 = 1.5 + 3.5 5 =5

(b) i. d₁ (cm )

d₂ (cm )

d₃ (cm )

length of arc PQR in terms of (cm)

length of arc PAB in terms of (cm)

1 2 1 1

2 3 3 4

7 5 6 5

5 5 5 5

0.5

length of arc BCR in terms of (cm)

length of arc DER in terms of (cm)

1.5 1.5 2

3.5 2.5 3 2.5

0.5 0.5

ii. The length of arc of outer semicircle is equals to the sum of length of arc in the inner semi circle. It can be generalise as, Length of arc of the inner semicircles for n where n = 1, 2, 3, 4,...

Souter = S1 + S2 + S3 + S4 ....... (c) Assume the diameter of outer semicircle ABC 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. d

d

d

d

1

2

3

4

1 0 1 2 1 4 1 5

8

6

6

15

5

4

3

3

3

5

15

6

3/2

5/2

5

8

4

1 0 4

15

7

4

2

2

5

3

7

15

15/2

5/2

3/2

7/2

Arc ABC

Arc APQ

Let, d1 = 10, d2 = 8, d3 =6, d4 = 6, and

Arc QRS

= 15

Thus, the generalisation is still true.

Arc UVC

SABC = SAPQ + SQRS + SSTU + SUVC

15 = 5 + 4 + 3 + 3 15

Arc STU

PART 03

(a) area of the flower plot is Y m2 y in terms of

and χ

Y = 52 /2 - ½( χ2/4) - ½((10-χ)2/4) Y = (25/2)

- (χ2/8)

- ((100 - 20χ + χ2)/8)

Y = ((100 - χ2 - 100 + 20χ - χ2)/8) Y = (2(10χ - χ2)/8) Y = ((10χ - χ2)/4)

(b) Y is equal to 16.5 m2. ( = 22/7) 16.5 = ((70χ - 7χ2)(22))/28 462 = (70χ - 7χ2)(22) 462/22 = 70χ - 7χ2 21 = -7χ2 + 70χ

8.0 2 Y/x0 = 7χ - 70χ + 21 0 = χ2 - 10χ + 3 07.0 = (χ - 3)(χ - 7) χ=3,χ=7

6.0 (c) y = ((10χ - χ2)/4)

5.0

y/χ = 10 /4 - χ /4 χ

4.0

y/χ

1

2

3

4

5

6

7

7.1

6.3

5.5

4.7

3.9

3.1

2.4

3.0

2.0 0

1

2

3

4

5

6

7

X

when χ = 4.5 , y/χ = 4.3 thus, area = χ × y/χ area = 4.5 × 4.3 area = 19.35 m2 (d) using differentiation method. y = ((10χ - χ2)/4) dy/dχ = ((10 - 2χ)/4) 0 = 5 /2 – χ /2 χ /2 = 5 /2 χ=5 using completing square method. y = a(χ – b/2(a) )2 –a(b/2(a))2 + c

y = 10χ /4 - χ2 /4 y = - /4(χ + (10 /4)/2(- /4))2 +

/4((10 /4)/2(- /4))2 + 0

y = - /4(χ - 5) - 25 χ=5 (e) n = 12 , a = 30 cm , d = ? S12 = 1000 cm S12 = 12/2 [2(30) + (12 - 1)d] 1000 = 6 (60 + 11d) 1000 = 360 + 66d 640 = 66d d = 9.697 Tn (flower bed) Diamet er (cm)

T

T2

T3

T4

T5

T6

T7

T8

T9

T10

T11

T12

39.6 97

49.3 94

59.0 91

68.7 88

78.4 85

88.1 82

97.8 79

107.5 76

117.2 73

126. 97

136.6 67

1

3 0

conclusion part 01 Not all objects are related to circle. If all the objects are in circles, there will be no balance and stability. In our daily life, there are many objects related to circle. For example, ball, fan, clock etc. we accept pi( ) as 3.142 or 22/7 as the best value. The circumference is directly proportional to the diameter × pi( ). For example, if a circle has length of twice diameter, the circumference is also twice. part 02 The relation between the length of arcs PQR, PAB, and BCR where the semicircle PQR is the outer semicircle and PAB and BCR are the inner semicircles. The length of arc PQR = length of arc PAB + length of arc BCR. The length of arc obtained as the length of arc = ½(2 r). as the conclusion, the outer semicircles are equals to the inner semicircles, Souter = Sinner part 03 The shaded region can be defined as, the shaded region = area of ADC – (area of AEB + area of BFC). When we plot a graph, we can still get a linear graph because Souter = Sinner, where the diameter has a constant value for a semicircle.