Project Work Add Math 2009

PROJECT WORK FOR ADDITIONAL MATHEMATICS 2009 SEKOLAH MENENGAH KEBANGSAAN SUNGAI PUSU, KM 11 GOMBAK, KUALA LUMPUR CIRCLE IN OUR DAILY LIFE

NAME; NURUS SYAMILAH BINTI AMIR HAMZAH CLASS; 5 IBNU SINA TEACHER; EN. ABDUL SAMAT BIN ISMAIL

CONTENT

NO. 1 2 3 4

CONTENTS Acknowledgement Research method Introduction Task specification

PAGE 3 4 5 - 6 8 - 12

5 6 7

Problem statement Conclusion References

13 - 43 44 45

Page 2 of 47

ACKNOWLEDGEMENT

In the name of Allah, the most Gracious and the most merciful.

Alhamdulillah, praise to be Allah for sucsess this Additional Mathematics’s project work. To dearest teacher, Encik Abdul Samat Bin Ismail, I would like to take this opportunity to thank you for being so patient with me and giving me all the guidance necessary to complete this project work. Thank you teacher. I would also like to thank all the 5 Ibnu Sina’s members who gaves me idea or guidance to complete this project work. Thank you.

May Allah bless all of you.

Page 3 of 47

REASEARCH’S METHOD

1. Discussion - Discussion between me and friends. We had discussed about solution of this project work.

2. Investigation - I searched at in internet about history of Pi or π and others informations.

3. References - I had references to a few books that have information about the given task.

Page 4 of 47

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. Page 5 of 47

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.

•

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

•

1880 – Lindemann proves that π is transcendental, effectively settling the millenniaold problem of squaring the circle.

Page 6 of 47

Aim 1. To apply and adapt a variety of problem-solving strategies to solve problems. 2. To improve thinking skills. 3. To promote effective mathematical skills. 4. To develop mathematical communication. 5. To develop mathematical knowledge through problem solvin in a way that increases student’s interest and confidence. 6. To use the language of mathematical ideas precisely. 7. To provide learning environment that stimulates and enhances effective learning. 8. To develop positive attitude towards mathematics.

Page 7 of 47

TASK SPECIFICATION

Page 8 of 47

PART 1

There are a lot of things around us related to circles or parts of a circle.

(a) collect pictures of 5 such object. You may use camera to take picture around your school compound or get pictures from magazines, newspapers, the internet or any other resources.

(b) Pi or π is a mathematical constant related to circles. Define π and write a brief history π.

PART 2 (a) Diagram 1 shows a semicircle PQR of diameter 10 cm. Semicircles PAB and BCR of diameter d1 and d2 respectively are inscribed in the semicircle PQR such that the sum of d1 and d2 is equal to 10 cm.

Page 9 of 47

Complete Table 1 by using various values of d1 and the corresponding values of d2. Hence, determine the relation between the lengths of arcs PQR, PAB and BCR.

(b) Diagram 2 shows a semicircle PQR of diameter 10 cm. Semicircles PAB, BCD and DER of diameter d1, d2 and d3 respectively are inscribed in the semicircle PQR such that the sum of d1, d2 and d3 is equal to 10 cm.

Page 10 of 47

(i) Using various values of d1 and d2 and the corresponding values of d3 , determine the relation between the lengths of arcs PQR, PAB, BCD and DER. Tabulate your findings. (ii) Based on your findings in (a) and (b), make generalisations about the length of the arc of the outer semicircle and the lengths of arcs of the inner semicircles for n inner semicircles where n = 2, 3, 4.... (c) For different values of diameters of the outer semicircle, show that the generalisations stated in b (ii) is still true.

PART 3

The Mathematics Society is given a task to design a garden to beautify the school by using the design as shown in Diagram 3. The shaded region will be planted with flowers and the two inner semicircles are fish ponds. Page 11 of 47

(a) The area of the flower plot is y m2 and the diameter of one of the fish ponds is x m. Express y in terms of it and x. (b) Find the diameters of the two fish ponds if the area of the flower plot is 16.5 m2. (Use π=22/7 ) (c) Reduce the non-linear equation obtained in (a) to simple linear form and hence, plot a straight line graph. Using the straight line graph, determine the area of the flower plot if the diameter of one of the fish ponds is 4.5 m. (d) The cost of constructing the fish ponds is higher than that of the flower plot. Use two methods to determine the area of the flower plot such that the cost of constructing the garden is minimum. (e) The principal suggested an additional of 12 semicircular flower beds to the design submitted by the Mathematics Society as shown in Diagram 4. The sum of the diameters of the semicircular flower beds is 10 m.

The diameter of the smallest flower bed is 30 cm and the diameter of the flower beds are Page 12 of 47

increased by a constant value successively. Determine the diameter of the remaining flower beds.

Page 13 of 47

PROBLEM STATEMENT

Page 14 of 47

PART 1 A. There are a lot of things around us related to circles or parts of a circles. Example;

car tire

Page 15 of 47

cooking pot

cake

lenses

Page 16 of 47

marble balls

DEFINITION OF Pi?

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. π is one of the most important mathematical and physical constants: many formulae from mathematics, science, and engineering involve π. π is an irrational number, which means that its value cannot be expressed exactly as a fraction m/n, where m and n are integers. Consequently, its decimal representation never ends or repeats. It is also a transcendental number, which means that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can be equal to its value; proving this was a late achievement in mathematical history and a significant result of 19th century Page 17 of 47

German mathematics. Throughout the history of mathematics, there has been much effort to determine π more accurately and to understand its nature; fascination with the number has even carried over into non-mathematical culture. The Greek letter π, often spelled out pi in text, was adopted for the number from the Greek word for perimeter "περίμετρος", first by William Jones in 1707, and popularized by Leonhard Euler in 1737. The constant is occasionally also referred to as the circular constant, Archimedes' constant (not to be confused with an Archimedes number), or Ludolph's number (from a German mathematician whose efforts to calculate more of its digits became famous).

Fundamentals The letter π Main article: pi (letter)

Lower-case π is used to symbolize the constant. The name of the Greek letter π is pi, and this spelling is commonly used in typographical contexts when the Greek letter is not available, or its usage could be problematic. It is not normally capitalised (Π) even at the beginning of a sentence. When referring to this constant, the symbol π is always pronounced like "pie" in English, which is the conventional English pronunciation of the Greek letter. In Greek, the name of this letter is pronounced /pi/.

Page 18 of 47

The constant is named "π" because "π" is the first letter of the Greek words περιφέρεια (periphery) and περίμετρος (perimeter), probably referring to its use in the formula to find the circumference, or perimeter, of a circle. π is Unicode character U+03C0 .

Definition

Circumference = π × diameter 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.

Page 19 of 47

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 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.

Page 20 of 47

Irrationality and transcendence Main article: Proof that π is irrational Being an irrational number, π cannot be written as the ratio of two integers. This was proven in 1761 by Johann Heinrich Lambert. In the 20th century, proofs were found that require no prerequisite knowledge beyond integral calculus. One of those, due to Ivan Niven, is widely known. A somewhat earlier similar proof is by Mary Cartwright. Furthermore, π is also transcendental, as was proven by Ferdinand von Lindemann in 1882. This means that there is no polynomial with rational coefficients of which π is a root. An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle. This is historically significant, for squaring a circle is one of the easily understood elementary geometry problems left to us from antiquity; many amateurs in modern times have attempted to solve each of these problems, and their efforts are sometimes ingenious, but in this case, doomed to failure: a fact not always understood by the amateur involved.

Page 21 of 47

Numerical value

The numerical value of π truncated to 50 decimal places is: 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510

While the value of π has been computed to more than a trillion (1012) digits, elementary applications, such as calculating the circumference of a circle, will rarely require more than a dozen decimal places. For example, a value truncated to 11 decimal places is accurate enough to calculate the circumference of a circle the size of the earth with a precision of a millimeter, and one truncated to 39 decimal places is sufficient to compute the circumference of any circle that fits in the observable universe to a precision comparable to the size of a hydrogen atom. Because π is an irrational number, its decimal expansion never ends and does not repeat. This infinite sequence of digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing more digits and investigating the number's properties. Despite much analytical work, and supercomputer calculations that have determined over 1 trillion digits of π, no simple base-10 pattern in the digits has ever been found. Digits of π are available on many web pages, and there is software for calculating π to billions of digits on any personal computer.

Calculating π Main article: Computing π

Page 22 of 47

π can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter. Another geometry-based approach, due to Archimedes, is to calculate the perimeter, Pn , of a regular polygon with n sides circumscribed around a circle with diameter d. Then

That is, the more sides the polygon has, the closer the approximation approaches π. Archimedes determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon with the perimeter of a regular polygon with the same number of sides inscribed inside the circle. Using a polygon with 96 sides, he computed the fractional range: . π can also be calculated using purely mathematical methods. Most formulae used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in trigonometry and calculus. However, some are quite simple, such as this form of the Gregory-Leibniz series;

While that series is easy to write and calculate, it is not immediately obvious why it yields π. In addition, this series converges so slowly that 300 terms are not sufficient to calculate π correctly to 2 decimal places. However, by computing this series in a somewhat more clever way by taking the midpoints of partial sums, it can be made to converge much faster. Let

and then define;

Page 23 of 47

then computing π10,10 will take similar computation time to computing 150 terms of the original series in a brute-force manner, and

, correct to 9

decimal places. This computation is an example of the Van Wijngaarden transformation.

HISTORY The history of π parallels the development of mathematics as a whole. Some authors divide progress into three periods: the ancient period during which π was studied geometrically, the classical era following the development of calculus in Europe around the 17th century, and the age of digital computers.

Geometrical period That the ratio of the circumference to the diameter of a circle is the same for all circles, and that it is slightly more than 3, was known to ancient Egyptian, Babylonian, Indian and Greek geometers. The earliest known approximations date from around 1900 BC; they Page 24 of 47

are 25/8 (Babylonia) and 256/81 (Egypt), both within 1% of the true value. The Indian text Shatapatha Brahmana gives π as 339/108 ≈ 3.139. The Hebrew Bible appears to suggest, in the Book of Kings, that π = 3, which is notably worse than other estimates available at the time of writing. The interpretation of the passage is disputed, as some believe the ratio of 3:1 is of an interior circumference to an exterior diameter of a thinly walled basin, which could indeed be an accurate ratio, depending on the thickness of the walls. Archimedes was the first to estimate π rigorously. He realized that its magnitude can be bounded from below and above by inscribing circles in regular polygons and calculating the outer and inner polygons' respective perimeters:

Liu Hui's π algorithm By using the equivalent of 96-sided polygons, he proved that 223/71 < π < 22/7. Taking the average of these values yields 3.1419.

Page 25 of 47

In the following centuries further development took place in India and China. Around AD 265, the Wei Kingdom mathematician Liu Hui provided a simple and rigorous iterative algorithm to calculate π to any degree of accuracy. He himself carried through the calculation to a 3072-gon and obtained an approximate value for π of 3.1416.

Later, Liu Hui invented a quick method of calculating π and obtained an approximate value of 3.1416 with only a 96-gon, by taking advantage of the fact that the difference in area of successive polygons forms a geometric series with a factor of 4.

Around 480, the Chinese mathematician Zu Chongzhi demonstrated that π ≈ 355/113, and showed that 3.1415926 < π < 3.1415927 using Liu Hui's algorithm applied to a 12288gon. This value was the most accurate approximation of π available for the next 900 years.

Page 26 of 47

Classical period

Until the second millennium, π was known to fewer than 10 decimal digits. The next major advance in π studies came with the development of calculus, and in particular the discovery of infinite series which in principle permit calculating π to any desired accuracy by adding sufficiently many terms. Around 1400, Madhava of Sangamagrama found the first known such series:

This is now known as the Madhava-Leibniz series or Gregory-Leibniz series since it was rediscovered by James Gregory and Gottfried Leibniz in the 17th century. Unfortunately, the rate of convergence is too slow to calculate many digits in practice; about 4,000 terms must be summed to improve upon Archimedes' estimate. However, by transforming the series into

Madhava was able to calculate π as 3.14159265359, correct to 11 decimal places. The record was beaten in 1424 by the Persian mathematician, Jamshīd al-Kāshī, who determined 16 decimals of π. The first major European contribution since Archimedes was made by the German mathematician Ludolph van Ceulen (1540–1610), who used a geometric method to compute Page 27 of 47

35 decimals of π. He was so proud of the calculation, which required the greater part of his life, that he had the digits engraved into his tombstone. Around the same time, the methods of calculus and determination of infinite series and products for geometrical quantities began to emerge in Europe. The first such representation was the Viète's formula,

found by François Viète in 1593. Another famous result is Wallis' product,

by John Wallis in 1655. Isaac Newton himself derived a series for π and calculated 15 digits, although he later confessed: "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time. In 1706 John Machin was the first to compute 100 decimals of π, using the formula

with

Formulas of this type, now known as Machin-like formulas, were used to set several successive records and remained the best known method for calculating π well into the age of computers. A remarkable record was set by the calculating prodigy Zacharias Dase, who in 1844 employed a Machin-like formula to calculate 200 decimals of π in his head at the behest of Gauss. The best value at the end of the 19th century was due to William Shanks, who took 15 years to calculate π with 707 digits, although due to a mistake only the first 527 were

Page 28 of 47

correct. (To avoid such errors, modern record calculations of any kind are often performed twice, with two different formulas. If the results are the same, they are likely to be correct.) Theoretical advances in the 18th century led to insights about π's nature that could not be achieved through numerical calculation alone. Johann Heinrich Lambert proved the irrationality of π in 1761, and Adrien-Marie Legendre also proved in 1794 π2 to be irrational. When Leonhard Euler in 1735 solved the famous Basel problem – finding the exact value of

which is π2/6, he established a deep connection between π and the prime numbers. Both Legendre and Leonhard Euler speculated that π might be transcendental, which was finally proved in 1882 by Ferdinand von Lindemann. William Jones' book A New Introduction to Mathematics from 1706 is said to be the first use of the Greek letter π for this constant, but the notation became particularly popular after Leonhard Euler adopted it in 1737. He wrote:

“

There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to (16/5 - 4/239) 1/3(16/5^3 - 4/239^3) + ... = 3.14159... = π

”

Computation in the computer age

The advent of digital computers in the 20th century led to an increased rate of new π calculation records. John von Neumann used ENIAC to compute 2037 digits of π in 1949, a calculation that took 70 hours. Additional thousands of decimal places were obtained in the Page 29 of 47

following decades, with the million-digit milestone passed in 1973. Progress was not only due to faster hardware, but also new algorithms. One of the most significant developments was the discovery of the fast Fourier transform (FFT) in the 1960s, which allows computers to perform arithmetic on extremely large numbers quickly. In the beginning of the 20th century, the Indian mathematician Srinivasa Ramanujan found many new formulas for π, some remarkable for their elegance and mathematical depth. One of his formulas is the series,

and the related one found by the Chudnovsky brothers in 1987,

which deliver 14 digits per term. The Chudnovskys used this formula to set several π computing records in the end of the 1980s, including the first calculation of over one billion (1,011,196,691) decimals in 1989. It remains the formula of choice for π calculating software that runs on personal computers, as opposed to the supercomputers used to set modern records. Whereas series typically increase the accuracy with a fixed amount for each added term, there exist iterative algorithms that multiply the number of correct digits at each step, with the downside that each step generally requires an expensive calculation. A breakthrough was made in 1975, when Richard Brent and Eugene Salamin independently discovered the Brent–Salamin algorithm, which uses only arithmetic to double the number of correct digits at each step. The algorithm consists of setting

and iterating Page 30 of 47

until an and bn are close enough. Then the estimate for π is given by

Using this scheme, 25 iterations suffice to reach 45 million correct decimals. A similar algorithm that quadruples the accuracy in each step has been found by Jonathan and Peter Borwein. The methods have been used by Yasumasa Kanada and team to set most of the π calculation records since 1980, up to a calculation of 206,158,430,000 decimals of π in 1999. The current record is 1,241,100,000,000 decimals, set by Kanada and team in 2002. Although most of Kanada's previous records were set using the Brent-Salamin algorithm, the 2002 calculation made use of two Machin-like formulas that were slower but crucially reduced memory consumption. The calculation was performed on a 64-node Hitachi supercomputer with 1 terabyte of main memory, capable of carrying out 2 trillion operations per second. An important recent development was the Bailey–Borwein–Plouffe formula (BBP formula), discovered by Simon Plouffe and named after the authors of the paper in which the formula was first published, David H. Bailey, Peter Borwein, and Plouffe. The formula,

is remarkable because it allows extracting any individual hexadecimal or binary digit of π without calculating all the preceding ones. Between 1998 and 2000, the distributed computing project PiHex used a modification of the BBP formula due to Fabrice Bellard to compute the quadrillionth (1,000,000,000,000,000:th) bit of π, which turned out to be 0. In 2006, Simon Plouffe found a series of beautiful formulas. Let q = eπ, then Page 31 of 47

and others of form,

where q = eπ, k is an odd number, and a,b,c are rational numbers. If k is of the form 4m+3, then the formula has the particularly simple form,

for some rational number p where the denominator is a highly factorable number, though no rigorous proof has yet been given.

Page 32 of 47

PART 2

(a) Diagram 1 shows a semicircle PQR of diameter 10 cm. Semicircles PAB and BCR of diameter d1 and d2 respectively are inscribed in the semicircle PQR such that the sum of d1 d2 and is equal to 10 cm. Complete Table 1 by using various values of d1 and the corresponding values of d2. Hence, determine the relation between the lengths of arcs PQR, PAB and BCR. The length of arc (s) of a circle can be found by using the formula;

Page 33 of 47

where r is the radius.

The result is as below:

d1 ( cm )

d2 ( cm)

1 2 3 4 5 6 7 8 9 10

9

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

8 7 6 5 4 3 2 1 0

5π 5π 5π 5π 5π 5π 5π 5π 5π

Length of arc PAB is in terms of π (cm) 0.5 π

Length of arc BCR is in terms of π (cm) 4.5 π

1.0 π 1.5 π 2.0 π 2.5 π 3.0 π 3.5 π 4.0 π 4.5 π 5.0 π

4.0 π 3.5 π 3.0 π 2.5 π 2.0 π 1.5 π 1.0 π 0.5 π 0.0 π

Therefor;

Length of arc PQR = Length of arc PAB + Length of arc BCR

Page 34 of 47

(b) Diagram 2 shows a semicircle PQR of diameter 10 cm. Semicircles PAB, BCD and DER of diameter d1, d2 and d3 respectively are inscribed in the semicircle PQRsuch that the sum of d1, d2 and d3 is equal to 10 cm. (i) Using various values of d1 and d2 and the corresponding values of d3, determine the relation between the lengths of arcs PQR, PAB, BCD and DER. Tabulate your findings.

we use the same formula to find the length of arc of PQR, PAB, BCD and DER.

The result is as below:

Page 35 of 47

d1 (cm) 1 1 1 1 1 1 1 1 2 2 2 2 2

d2 (cm) 1 2 3 4 5 6 7 8 1 2 3 4 5

d3 (cm) 8 7 6 5 4 3 2 1 7 6 5 4 3

Length of Length of Length of Length of arc PQR is arc PAB is arc BCD is arc DER is in terms of π in terms of π in terms of π in terms of π (cm) (cm) (cm) (cm) 5π 0.5π 0.5 π 4.0 π 5π 0.5π 1.0 π 3.5 π 5π 0.5π 1.5 π 3.0 π 5π 0.5π 2.0 π 2.5 π 5π 0.5π 2.5 π 2.0 π 5π 0.5π 3.0 π 1.5 π 5π 0.5π 3.5 π 1.0 π 5π 0.5π 4.0 π 0.5 π 5π 1.0π 0.5 π 3.5 π 5π 1.0π 1.0 π 3.0 π 5π 1.0π 1.5 π 2.5 π 5π 1.0π 2.0 π 2.0 π 5π 1.0π 2.5 π 1.5 π Page 36 of 47

2 2 3 3 3 3 3 3 4 4 4 4 4 5 5 5 5 6 6 6 7 7 8

6 7 1 2 3 4 5 6 1 2 3 4 5 1 2 3 4 1 2 3 1 2 1

2 1 6 5 4 3 2 1 5 4 3 2 1 4 3 2 1 3 2 1 2 1 1

5π 5π 5π 5π 5π 5π 5π 5π 5π 5π 5π 5π 5π 5π 5π 5π 5π 5π 5π 5π 5π 5π 5π

1.0π 1.0π 1.5π 1.5π 1.5π 1.5π 1.5π 1.5π 2.0π 2.0π 2.0π 2.0π 2.0π 2.5π 2.5π 2.5π 2.5π 3.0π 3.0π 3.0π 3.5π 3.5π 4.0π

3.0 π 3.5 π 0.5 π 1.0 π 1.5 π 2.0 π 2.5 π 3.0 π 0.5 π 1.0 π 1.5 π 2.0 π 2.5 π 0.5 π 1.0 π 1.5 π 2.0 π 0.5 π 1.0 π 1.5 π 0.5 π 1.0 π 0.5 π

1.0 π 0.5 π 3.0 π 2.5 π 2.0 π 1.5 π 1.0 π 0.5 π 2.5 π 2.0 π 1.5 π 1.0 π 0.5 π 2.0 π 1.5 π 1.0 π 0.5 π 1.5 π 1.0 π 0.5 π 1.0 π 0.5 π 0.5 π

Therefore; Length of arc PQR = Length of arc PAB + Length of arc BCD + Length of arc DER

(ii) Based on your findings in (a) and (b), make generalisations about the length of the arc of the outer semicircle and the lengths of arcs of the inner semicircles for n inner semicircles where n = 2, 3, 4.... So: The length of the arc of the outer semicircle is equal to the sum of the length of arcs of any number of the inner semicircles.

Page 37 of 47

The length of arc of the outer semicircle

The sum of the length of arcs of the inner semicircles

Then,

where d is any positive real number.

So, we can see that

(c) For different values of diameters of the outer semicircle, show that the generalisations stated in b (ii) is still true. Therefore; Page 38 of 47

The length of the arc of the outer semicircle is equal to the sum of the length of arcs of any number of the inner semicircles. This is true for any value of the diameter of the semicircle. In other words, for different values of diameters of the outer semicircle, show that the generalisations stated in b (ii) is still true.

PART 3

(a) The area of the flower plot is y m2 and the diameter of one of the fish ponds is x m. Express y in terms of it and x. Area ADC = 1/2π [10/2]² Page 39 of 47

= 25/2π Area AEB = 1/2 π[x²/2]² = 1/2π[x²/4] = x²/8π Area of BFC =1/2π [5 – x/2]² = 1/2π [25 – 5x + x²/4] =25/2π – 5x/2π + x²8π Area of shaded region = 25/2π – [x²/8π + (25/2π – 5x/2π + x²/8π)] = 25/2π – [x²/8π + 25/2π – 5x/2π + x²/8π] = 25/2π - x²/8π – 25/2π + 5x/2π + x²/8π = x²/4π + 5x/2π Therefore; Y = x²/4π + 5x/2π

(b) Find the diameters of the two fish ponds if the area of the flower plot is 16.5 m2. (Use π =22/7 ) 16.5 = - x²/4π + 5x/2π = - x²/4 (22/7) + 5x/2 (22/7) 16.5/22/7 = - x²/4 + 5x/2 5.25 = 5x/2 - x²/4 Page 40 of 47

21 = 10x - x² x² - 10x + 21 = 0 (x – 7) (x -3) = 0 x = 7 or x = 3

(c) Reduce the non-linear equation obtained in (a) to simple linear form and hence, plot a straight line graph. Using the straight line graph, determine the area of the flower plot if the diameter of one of the fish ponds is 4.5 m. Y = - x²/4π + 5x/2π Y = mX + c

x

1

2

3

4

5

6

7

y/x

7.1

6.3

5.5

4.7

3.9

3.1

2.4 Page 41 of 47

Y/x

8.0

7.0

6.0

5.0

4.0

3.0

2.0 0

1

2

3

4

5

6

7

X

Page 42 of 47

(d) The cost of constructing the fish ponds is higher than that of the flower plot. Use two methods to determine the area of the flower plot such that the cost of constructing the garden is minimum.

Differentiation,

Y= - x²/4x + 5x/2π

dy/dx = - π x/2 + 5/2π

d²y/dx²= - π/2 minimum value

at maximum value, d²y/dx²= 0 -π x/2 + 5/2π = 0 -π x/2 = 5/2π x = 5m

maximum value y = - π5²/4 + 5π5/2 = 6.25π m² Page 43 of 47

Compleating the square,

Y= - x²/4π + 5x/2 π

= - π/4 ( x² - 10x)

= - π/4 (x² - 10x + 25 - 25)

= - π/4 [(x - 5)² - 25]

= - π/4 (x - 5)² + 2/5/4 π

Y is a “n” shape graph as a = - π/4

X= 5

and maximum value is 6.25πm²

Page 44 of 47

(e) The principal suggested an additional of 12 semicircular flower beds to the design submitted by the Mathematics Society as shown in Diagram 4. The sum of the diameters of the semicircular flower beds is 10 m. The diameter of the smallest flower bed is 30 cm and the diameter of the flower beds are increased by a constant value successively. Determine the diameter of the remaining flower beds.

a = 30 cm = 0.3 m n = 12 s12 = n/2 [(2a + (n – d)] 10 = 12/2 [2 (0.3) + (12 – 1) (d)] = 6 (0.6 + 11) d = 3.6 + 66d 66d = 6.4 d = 16/165

Sequence of arithmetic progression: 3/10, 131/330, 163/330, 13/22, 227/330, 259/330, 97/110, 323/330, 71/66, 129/110, 419/330,

41/30

Page 45 of 47

CONCLUSION

When we finished Additional Mathematics’s project work, we can conclude,the history of π parallels the development of mathematics as a whole. Some authors divide progress into three periods: the ancient period during which π was studied geometrically, the classical era following the development of calculus in Europe around the 17th century, and the age of digital computers.

Page 46 of 47

REFERENCES

1. Yong Kien Cheng, additional mathemathics SPM, pearson Malaysia Sdn. Bhd. 2008.

2. Http://en.wikipedia.org/wiki/History_of_pi

Page 47 of 47