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Project Work for Additional Mathematics 2009

The

objectives of

Additional Mathematics is to increase student’s capabilities in Mathematics. With this

purpose, which provide an opportunity for students to apply the knowledge and skills learnt

in the classroom into real-life and challenging situations.

Every Form 5 student who wishes to take Additional Mathematics as an elective subject

is required to carry out a project work. The theme is based on the Science and Technology

Package and Social Science Package. Project work can only be carried out in the second

semester, after students have mastered a few topics in the Additional Mathematics. Project

work can be done in groups or individually but each student is encouraged to prepare an

individual report. Submission of report is expected within a three weeks. It is includes the

exploration of mathematical problems in the context of human activities that will in turn

activate my minds, making the learning of mathematics meaningful and beautiful. The aims

of carrying out this project work are to:

Develop mathematical knowledge in a way which increases student’s interest and

confidence

Apply mathematics to everyday situations and begin to understand the part that

mathematics play in the world we live

Improve thinking skills and promote effective mathematical communication

Assist students to develop positive attitude and personalities, intrinsic mathematical

values such as accuracy, confidence, and systematic reasoning

Stimulate learning and enhance effective learning

Encourage the usage of the computer and graphing calculator.

Problem-solving skills and Strategies such as:

Trial and improvement

Drawing diagrams

Tabulating data

Identifying polars

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Project Work for Additional Mathematics 2009

Experimenting

Simulation

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This project work could not have been completed without Pn. Rusilah, who encouraged and challenged me throughout my academic program.  She never accepted less than my best efforts.  Thank you!

What was collected in this project work are materials that I found in articles, in books and definitely through internet.

I would like to acknowledge and extend my heartfelt gratitude to the following persons who have made the completion of this Project Work possible:Azlin and Azwin, for their vital encouragement and support.Khairul Izwan, for his understanding and assistance.My mom, Ropenah Samuri, for the constant reminders and much needed motivation.Ahmad Nor Ariff for the help and inspiration he extended.To all my seniors, for assisting in the collection of the topics for the chapters.

Most especially to my family, and friends: Words alone cannot express what I owe you for your encouragement and patient love, which enabled me to complete this Project work.

And finally to God, who makes all things possible.

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Project Work for Additional Mathematics 2009

In non-mathematical terms, pi is simply the greek letter "p", and is written like this:

Pi is commonly seen in mathematical formulae. The most well known formula that involves pi is:

c = d ,

where c is the circumference of a given circle, and d is the diameter of a given circle.

In basic terms, this is the definition of pi, as the ratio of a circle's circumference to its diameter. It is also the first equation in which pi was ever used.

Another extremely common formula that relies on pi is also to do with a circle, specifically the relationship between a circle's area and radius:

A = r2 ,

where A is a given circle's area, and r is a given circle's radius.

Aside from these two, there are many other formulae in the study of solid geometry which involve pi, and a few of these are detailed below with brief explanations:

S = 4 r2

The formula for finding the surface area of a sphere V= r2h

The formula for finding the volume of a cylinder S=4 2rR

The formula for finding the surface area of a torus.

Aside from formulae, pi is uncommonly useful in lots of ways. One such way would be in measuring angles in radians, where 360 degrees are equal to 2 radians.

Also, it is thought that the value of pi is in some way mysteriously linked to many natural situations. For example, pi is related to the double helix of DNA and even to rainbows. This is very much a similar phenomenon to the natural number e = 2.718..., which is related to nearly all living things in some way.

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Project Work for Additional Mathematics 2009

PART 1

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

(a) Collect pictures of 5 such objects. You may use camera to take pictures 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 of 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.

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.

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Project Work for Additional Mathematics 2009

(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.

(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.

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Project Work for Additional Mathematics 2009

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

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Project Work for Additional Mathematics 2009

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

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Project Work for Additional Mathematics 2009

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A round sundial ancient

method for measuring

time

A round sundial ancient

method for measuring

time

Round shape hand watch

Round shape hand watch

Our round lovely Earth.

Our round lovely Earth.

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Project Work for Additional Mathematics 2009

Sek. Men. Keb. Teluk Gadong

Round stylish

spectacles.

Round stylish

spectacles.

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Project Work for Additional Mathematics 2009

Sek. Men. Keb. Teluk Gadong

My Little Pony

Round Foil Balloon

My Little Pony

Round Foil Balloon

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Project Work for Additional Mathematics 2009

b. i) The definition of pi, π

Pi or π is defined as the ratio of a circle's circumference to its diameter :

Alternatively π can be also defined as the ratio of a circle's area to the area of a square whose side is equal to the radius :

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Circumference = π x diameter

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

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Project Work for Additional Mathematics 2009

ii) The wonderful history of pi, π

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

Geometrical periodThat 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 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 (600 BC). 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 (287-212 BC) 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:

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Project Work for Additional Mathematics 2009

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.

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.

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Liu Hui's π algorithm

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Project Work for Additional Mathematics 2009

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 12288-gon. This value was the most accurate approximation of π available for the next 900 years.

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 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.

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Project Work for Additional Mathematics 2009

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 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

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Project Work for Additional Mathematics 2009

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:

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 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.

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“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

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Project Work for Additional Mathematics 2009

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

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

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Project Work for Additional Mathematics 2009

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 = e π , then

and others of form,

where q = e π , 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.

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Project Work for Additional Mathematics 2009

a. The length of arc (s) of a circle can be found by using the formula

where r is the radius.

The result is as below:

From the table, we can conclude that

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

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Project Work for Additional Mathematics 2009

b. i) I use the same formula to find the length of arc of PQR, PAB, BCD and DER.

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Project Work for Additional Mathematics 2009

Hence,

ii) Base on the findings in the table in(a) and (b) above, I conclude that:

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.

c.

Diagram above shows a big semicircle with n number of small inner circle. From the diagram,

The length of arc of the outer semicircle

The sum of the length of arcs of the inner semicircles

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Length of arc PQR = Length of arc PAB + Length of arc BCD + Length of arc CDR

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Project Work for Additional Mathematics 2009

Factorise π 2

Substitute

I get,

where d is any positive real number.

I can see that

As a result, I can conclude that

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.

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Project Work for Additional Mathematics 2009

A. Area of flower plot = y m2

y = 25 π - 1 x 2 π + 1 (10-x )2 π2 2 2 2 2

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

2 8 4 = 25 π - x2π + 100π – 20x π + x2π

2 8 = 25 π - 2x2 – 20x + 100 π

2 8 = 25 π - x2 – 10x + 50 π

2 4 = 50 - x2 - 10x + 50 π

4 y = 10x – x2 π

4

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2.0

3.0

4.0

5.0

6.0

7.0

8.0

0 1 2 3 4 5 6 7X

Y/x

Project Work for Additional Mathematics 2009

B. y = 10x – x2 π 4

y = 10 - x πx 4 4

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Project Work for Additional Mathematics 2009

Sek. Men. Keb. Teluk Gadong