Project Addmath Amir

8/3/2019 Project Addmath Amir

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PART 1

Rene Descartes:

The Greatest French Mathematician of the 17th Century

Paul Shute

April 29, 2008

Abstract

This paper focuses on Rene Descartes, the 17th century mathematician. The paper summarizesimportant facts about his childhood, his education, and his traveling after college. Thispaper also contains information about the three dreams he claimed to have in one night, wherehis philosophy came from. This paper addresses his most important findings in mathematics,including exponential notation, relationships in functions, and Cartesian coordinates. Finally,this paper explains why Descartes thought process and method of examination makes him the

greatest mathematician of his time.


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1. Who is Rene Descartes

Born on March 31st, 1596 in the town of LaHaye Touraine, France, now known as Descartes, France,Rene Descartes will spend his fifty-four years attempting to understand both himself and the worldaround him. Descartes father was a judge in the courts; this allowed him to live a life of privilege.When Descartes was born his mother was sick with a whooping cough, and likely had tuberculosis.A year after his birth, his mother died giving birth to what would have been his younger brother.Descartes never knew about the fate of his brother, and it is not known if any family member evertold him [1]. Author Rodis-Lewis suspects that the fact that his mother died during childbirthcaused the rest of his family to resent him. Following this, it is also suspected that this is whyDescartes formed such an intimate bond with nature, starting from his early childhood years [1].Descartes suffered from a cough much like his mother, and this cough prevented him from leavingthe house during much of his childhood. However, this cough might have also been a gift, becauseinstead of attending local schools, he was taught by his grandmother and private tutors who would

come to his house.

2. Mathematical Legacy

Descartes made advances in many fields of mathematics. While some of his discoveries may seem

trivial to a reader four hundred years later, these discoveries were made during a time when alchemywas still a science. Furthermore, many of Descartes discoveries are used today in fields rangingfrom 4th grade geometry to advanced mathematics in college, and beyond.In his investigation of powers, Descartes discovered that powers could be represented as lines orcurves. In doing so he also invented the notion of exponents. Descartes created the representationof raising a number to another number. For example, ax as a relation of a raised to some number,x [1]. More specifically, 23 being equivalent to 2 _ 2 _ 2 = 8. Descartes also showed that the higherthe degree of the power, the more complex the curve will be [4]. An example of this is given in the

following graphs.


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In examining the graphs of y = x, what Descartes means by a more complex curve is how the graphof y = x1 forms a straight line, and through the increasing of the power, the graph ceases to actas a straight line, and the way it does act becomes more complicated. This is important in notonly solving geometric problems, but also in developing an understanding of some basic behaviorsof graphs. For example, the graph below depicts the functions y = x2, y = x4, and y = x10,

respectively.

In comparing these graphs to that of the even functions, we can see that the extreme values aredifferent. For an odd function, a negative value for x gives a negative value for y. This is a graphicalrepresentation of the fact that for xn, where n is odd, then if x is negative, xn is also negative. Forexample 23 = 8, and (2)3 = 8.Descartes did not only explore the graph of exponents. From his work with exponents, Descartesgeneralized his graphs to allow any arithmetic idea to be expressed geometrically. This notion ofrepresenting algebraic formulas geometrically helped to ultimately give rise to Calculus by Leibnizand Newton [4]. The system is now known as the Cartesian Coordinate system [1]. More commonly

known as rectangular coordinates, the Cartesian Coordinate system places related numbers

in specific places. Cartesian Coordinates uses the notation (x,y) for 2D systems. For example, thepoint (2, 3) has a distinct x coordinate (2) and a distinct y coordinate (3). An example of this

system is displayed below


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As seen in the graph above, in order to plot the point (2, 3), you first move 2 down the x, then 3up the y. Geometry made tremendous progress through the discovery of this process, since nowanyone studying geometry could represent their problems graphically [4]. However, this discoverywas not limited to those studying geometry. The Cartesian Coordinate system opened doors toanyone dealing with sets of possibly related numbers. The ability to visualize any related setsof numbers has endless possibilities, and is one of the greatest breakthroughs in the history of

geometry. Without a system for visualizing sets of numbers, geometry would remain in our heads.The ability to construct actual images out of geometric functions allows us to visualize and interpret

what the function is telling us in a much more conceptual fashion.

For example, imagine if you wondered whether the temperature outside had an effect on the amountof students showing up for class. If you were an instructor, you could record the temperature outsideand the number of students showing up for class each day. Lets say you did so for a month

that had a lot of temperature change, and you obtained the following set of data:

Temp.(F) 90 100 90 80 70 50 50 70 60 90 80 60 70 80 90 100 80 60 90Students 6 2 7 14 13 4 3 15 7 6 12 7 14 7 6 3 10 6 6

While this may just look like a random collection of points, you can find interesting relationshipsif you graph this set of data using the Cartesian Coordinates. Letting temperature be our xcoordinatesand the number of students be our y-coordinates, we obtain the following graph:

From this graph it appears that more students come to class when the temperature is betweenseventy and eighty degrees. Since humans are likely most comfortable between seventy and eightydegrees, one might assume that in that degree range people would be more willing to do things.This graph allows us to see that one of the things students are more willing to do is come to class.Descartes discovery of the Cartesian Coordinates helped people not only dealing with pure mathematics,but also people dealing with real world type situations, such as the one presented above.Descartes also helped many mathematicians by describing a general strategy to solving any geometric

problem. The method involves three steps: naming, equating, and constructing [4]. Stepone involves naming all the lines that seem needed to solve the problem. Step two requires that theperson working the problem finds relationships between the lines and points of interest. Finally, thethird step involves combining all relationships into one equation by using known geometries, suchas forming triangular relationships with the lines [4]. While this is explained in a geometric sense,Descartes is essentially laying the foundation for the first scientific method. Using a structured wayof solving a problem allows the problem to be reproduced by anyone. Descartes first breaks theproblem down to its most fundamental level. From there, he draws relationships between knownand unknown lines and points, which ultimately leads to the formation of one equation, which


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relates all things known to unknown. Throughout his life, Descartes was determined to leave nodoubt in anything. This is shown brilliantly in his description of a method for solving problems. Ifa problem could be broken down and explained logically step-by-step, then the room for error ordoubt would become increasingly small.In doing this Descartes also explains how a locus of points in a function shows the relationshipsthat the two variables have with each other [4]. For example, consider the simple example of a hot

dog vendor. If the vendor records the number of hot dogs sold and the amount of profit he gets,he might come up with a set of points something like this:

Hot dogs sold ($1 each) 10 20 15 13 9 12 14 13 12 10 8 9 10 9Profit ($) 10 20 15 13 9 12 14 13 12 10 8 9 10 9

The graph of this relationship is shown in the following graph:

From this graph you can see that the relationship between hot dogs sold and profit is linear, becausethe locus of points follows in a straight line.

3. ConclusionDescartes did not set out to make as many mathematical discoveries as he could. Beginning withhis childhood, Descartes developed a deep and intimate relationship with nature and the worldaround him. The things he discovered seem to be motivated not by fame, but by a hope to gain

a deeper understanding of not only himself, but the world. It is my belief that, due partly to hisaptitude for philosophy, Descartes approached math in an almost religious sense, which allowedhim to use math as his compass for life.One of Descartes core beliefs in life was to take nothing for granted, and nothing at face value. Notsimply related to geometry or numbers, Descartes believed that if something could be doubted, thenit should be considered false. A great mathematician should be a seeker of truth, both in natureand within themselves. I believe that Descartes embodied this philosophy and saw it through tothe end. Descartes seemed to have an unwavering devotion to mathematics, a field which manyattempt, but few discover.


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PART 2

Malaysia with its warm tropical climate is rich in flora and fauna. Beautiful gardens are found all over

Malaysia. SMK Permata decided to beautify the school compound by getting the students involved in

the planting and maintenance of the greenery in the school compound as shown in Diagram 1. Each

society is allocated a plot of land in various shapes and sizes to nurture throughout the year. The

Mathematics Society, English Language Society and Malay Language Society are allocated the region P,

Q and R respectively as shown in Diagram 1.

Diagram 1

(a) Determine the area of region P, Q and R by using at least three different methods including the use

of calculus. Verify the answers obtained by using computer software.

(Suggestions: GeoGebra, GSP, graphing calculator etc)

Method 1 :

i) Divide area P into 3 segments. 2 triangles and 1 rectangle.Triangle 1 :

= x 2 x 1

= 1

Triangle 2 :

= x 4 x 3

= 6


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Rectangle 1 :

= 2 x 4

= 8

P = (1 + 6 + 8)

P = 15

ii) Divide area Q into 2 segments. 1 triangle and 1 rectangle.Triangle 1 :

= x 4 x 3

= 6

Rectangle 1 :

= 2 x 3

= 6

Q = (6 + 6)

Q = 12

iii) Divide area R into 3 segments. 1 triangle and 2 rectangles.Triangle 1 :

= x 1 x 2

= 1

Rectangle 1 :

= 2 x 3

= 6

Rectangle 2 :

= 4 x 2

= 8

R = (1 + 6 + 8)

R = 15

Method 2 :

Coordinate Geometry method.

Area P :


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| | ||

Area Q :


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| | ||

Area R :


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| | ||

Method 3 :

Integration method.

Area P :


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[ ] []

Area Q :


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[]


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Area R :

[]


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Verification of answer by GeoGebra.

Area P:


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Area Q :

Area R :


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(b) Suppose there is a hedge alongAB. The Mathematics Society wishes to fence up the remaining sides

of the region P. Determine the length of fence required.


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So, from the diagram we will have the following :

= +

= +

= 25

= +

= +

= 5

DE = 2.236m

CD= 3m

BC = (6 4)

BC = 2m

AB = 7m (not used)

AEDCB = (2 + 3 + 2.236 + 5)m

AEDCB = 12.236m

(c) If a meter of fence costs RM25.00, what is the total cost required by the Mathematics Society to

fence up region P? Is it possible for the society to carry out the fencing with an allocation of RM250.00?

Explain your answer.

It is impossible for the society to carry out the fencing with an allocation of RM250.00.

This is because :

= RM250.00 RM25/m

= 10m

RM250.00 can only cover up to 10m of length.

Length of AEDCB = 12.236m

= 12.236m x RM25.00

= RM305.90 is needed to carry out the fencing plan.

(d) During the Mathematics Week, the society was given a single flag chain of length 9.20 meters to be

used completely. The President of the society wishes to tie the flag chain continuously fromA to Eand

then to another point along the hedge AB to create a triangular-shaped area.


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(i) Make a conjecture about the number of points that the flag chain can be tied to alongAB.

Support your conjecture with suitable calculations. Explain your answer.

1 point for the flag chain to be tied at E.

1 point for the flag chain to be tied at a point along the hedge AB.

(ii) Calculate the maximum area of the triangle obtained. Discuss.

= 9.20m AE

= 9.20m 5m

= 4.20m

Divide the triangle obtained into 2 right triangle.

Triangle 1 :

Calculate the length.

Use Pythagoras Theorem.

L =

L =

L = 3m

Area = x 3 x 4

Area = 6

Triangle 2 :

Calculate the length.

Use Pythagoras Theorem.

L =

L =

L = 1.281m

Area = x 4 x 1.281

Area = 2.562

Maximum area of the triangle :

= (6 + 2.562)

= 8.562

Solution of triangle method :

ab Sin C

Angle EAL ( tan-1 4

/3 ) = 53.13

= (5 x (3 + 1.281))(sin 53.13 )

= 8.562


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PART 3

The Mathematics society decided to build a pond in region P as shown in Diagram 2. The pond is in the

shape of a sector with centre E, radius ED and a depth of 1 meter.

Diagram 2

(a) Calculate the angleAED, in radians, by using at least two different methods.

Method 1

By drawing a horizontal and vertical lines on GeoGebra.


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Next, calculate the angle on the left and right side of AED.


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Finally,


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GeoGebras calculations.

So, AED = 63.43Method 2

By using the formula from the solution of triangle.

() () () ()()()


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(b) Determine the volume of water that has to be pumped in to fill up 80% of the pond.

So the depth of the pond is 1m.

= = (63.43 x )= 1.107rad)= 2.767 x 1m= 2.767 x 80= 2.214(c) If the water is pumped into the pond at a constant rate of 0.001 m3 s-1, calculate

(i) the rate of change of depth of the water,

(ii) the depth of water after 10 minutes,

= 0.000361 =


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(iii) the minimum time taken, in minutes, before the water overflows,

So, the maximum depth of the pond is 1m. Ratio.

(iv) the minimum time taken, in minutes, before the water overflows, if the pond is triangular-shaped

AED and has a depth of 2 meters.

Now, the pond is triangular shaped. Solution of triangle formula can be applied.

)

So, ratio might work. First, find the height of water after 10mins.


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Then, ratio.


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FURTHER EXPLORATION

Maps have been used for thousands of years to aid travelers during their journey from one place to

another. Maps can also be used to estimate distance between places. In the year 2004, a recreation park

will be constructed in town marked X on the map of Malaysia in shown Diagra m 3. This town has the

latitude of 5 41 N and has the same longitude as the city of Malacca.

Diagram 3

Explore and find the distance between these two places in kilometer by using,


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i) the map in Diagram 3

Since the scale is 60km per square.

We can approximately find the distance between X and the city of Malacca by

ii) the formula given below :

Distance = x nautical miles1 nautical mile = 1.852 kilometers.

Malaccas latitude = Thus, Yes, there is a different between the answers obtained. This is because the calculation by using the scale

given by a map is just an approximation method. The answer is correct but less accurate compared to

the answer from the calculation based on the formula given. By using the latitudes, the answer is very

accurate and significant.

(a) Surf the Internet and use the Google map to locate the position of your school and two nearby

hospitals/clinics. Print a copy of this Google map and mark the position of these three places.

(i) Solve the triangle obtained.

On the map.

Scale :

(ii) Calculate the shortest distance from your school to the line joining the two hospitals/clinics.

By using a protractor and a ruler, a perpendicular line between the school and the line connecting the

two clinic/hospital was measured.


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REFLECTION While you were conducting the project, what have you learnt? What moral values did you

practise? Represent your opinions or feelings creatively through usage of symbols, illustrations, drawing

or even in a song.

In the making of this project, I have spent countless hours in doing this project. I realized this subject is

very compulsory to me. Without it I cannot fulfill my dream and goals..

TALKING TO ADD MATH :D

I know you're somewhere out there

Somewhere far away

I want you back

I want you back

My neighbors think

I'm crazy

But they don't understand

You're all I have

You're all I have

Chorus:

At night when the stars

light up my room

I sit by myself

Talking to Add Math

Trying to answer You

In hopes you're on

the other side

Giving me full marks

Or am I a fool

who sits alone


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Talking to Add Math

I'm feeling like I'm famous

The talk of the town

They say

I've gone mad

Yeah

I've gone mad

But they don't know

what I know

Cause when the

sun goes down

someone's talking back

Yeah

They're talking back

Chorus:

At night when the stars

light up my room

I sit by myself

Talking to Add Math

Trying to answer You

In hopes you're on


For the answers to me too

Or am I a fool


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who sits alone

Talking to Add Math

Do you ever hear me calling?

Cause every night

I'm talking to Add Math

Still trying to answer you

In hopes you're on


For the answers to me too

Or am I a fool

who sits alone

Talking to the moon

I know you're somewhere out there

Somewhere far away


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FINAL WORDS

Additional Mathematics Work 3/2011 had really worn me out. The tasks given were quite challenging.

Took me the whole school break to finish this but I think it was worth it. From this work, I gained some

different perspectives in additional mathematics, mainly about how to apply it in real life. Alhamdulillah,

its been quite hard to stay up whole night every day but finally its over. Credits to my friends in giving

me support and for being great teammates. Thank you. Wassalamualaikum warahmatullahi

wabarakatuh.


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CURRICULUM DEVELOPMENT

DIVISION MINISTRY OF

EDUCATION MALAYSIA

ADDITIONAL MATHEMATICS

PROJECT WORK 3/2011

BY:

AMIR ARSYAD BIN KHAIRUDDIN

5 JAYA

941124055543


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CONTENT PAGE

INTRODUCTION

PART1

PART2

PART3

FURTHEREXPLORATION

REFLECTION

FINALWORDS


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INTRODUCTION

For the 2 weeks of holiday, I was given the task to complete this Additional Mathematics Project Work

3/2011 by our teacher. I have been doing some research and the results are in this project presented.

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