additional mathematics project 2010

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Additional Mathematics

Project Work 2011

Name : Ng Ken Jie

Class : 5S5

I.C. Number : 941020-14-6571

School : SMJK Chong Hwa

Angka Giliran : 22


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Content Page

No. Content Page

1. Introduction 1 -2

2. Part I 3 -4

3. Part II 5 - 13

4. Part III 14 - 16

5. Further Exploration 17 18

6. Reflection 19 -20


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Introduction

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. Circles had been used in daily

lives to help people in their living. There are a lot of things around us that are related to

circles or parts of a circle.

A circle is a simple shape ofEuclidean geometry consisting of the set ofpoints in

a plane that are a given distance from a given point, the centre. The distance

between any of the points and the centre is called the radius.

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, or to the whole figure

including its interior; in strict technical usage, the circle is the former and the latter is

called a disk.

A circle is a special ellipse in which the two foci are coincident and the eccentricity is

0. Circles are conic sections attained when a right circular cone is intersected by a

plane perpendicular to the axis of the cone.

Circle illustration showing a radius, a diameter, the centre and the circumference

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/Centre_(geometry)http://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Curvehttp://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Interior_(topology)http://en.wikipedia.org/wiki/Disk_(mathematics)http://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Focus_(geometry)http://en.wikipedia.org/wiki/Eccentricity_(mathematics)http://en.wikipedia.org/wiki/Conic_sectionhttp://en.wikipedia.org/wiki/Conical_surfacehttp://en.wikipedia.org/wiki/Conical_surfacehttp://en.wikipedia.org/wiki/Conic_sectionhttp://en.wikipedia.org/wiki/Eccentricity_(mathematics)http://en.wikipedia.org/wiki/Focus_(geometry)http://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Disk_(mathematics)http://en.wikipedia.org/wiki/Interior_(topology)http://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Curvehttp://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Centre_(geometry)http://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Point_(geometry)http://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Shape


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History

The word "circle" derives from the Greek, kirkos "a circle," from the base ker- which

means to turn or bend. The origins of the words "circus" and "circuit" are closely

related.

The circle has been known since before the beginning of recorded history. Natural

circles would have been observed, such as the Moon, Sun, and a short plant stalk

blowing in the wind on sand, which forms a circle shape in the sand. The circle 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, astronomy, 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 (3.16049...) as an approximate value of.

300 BC Book 3 ofEuclid's Elements deals with the properties of circles. In Plato's Seventh Letter there is a detailed definition and explanation of the

circle. Plato explains the perfect circle, and how it is different from any drawing,

words, definition or explanation.

1880 Lindemann proves that is transcendental, effectively settling themillennia-old problem ofsquaring the circle.

[2]

2
http://en.wikipedia.org/wiki/Circushttp://en.wikipedia.org/wiki/Circuithttp://en.wikipedia.org/wiki/Wheelhttp://en.wikipedia.org/wiki/Gearhttp://en.wikipedia.org/wiki/Astronomyhttp://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%27s_Elementshttp://en.wikipedia.org/wiki/Platohttp://en.wikipedia.org/wiki/Seventh_Letterhttp://en.wikipedia.org/wiki/Ferdinand_von_Lindemannhttp://en.wikipedia.org/wiki/Squaring_the_circlehttp://en.wikipedia.org/wiki/Circle#cite_note-1http://en.wikipedia.org/wiki/Circle#cite_note-1http://en.wikipedia.org/wiki/Circle#cite_note-1http://en.wikipedia.org/wiki/Circle#cite_note-1http://en.wikipedia.org/wiki/Squaring_the_circlehttp://en.wikipedia.org/wiki/Ferdinand_von_Lindemannhttp://en.wikipedia.org/wiki/Seventh_Letterhttp://en.wikipedia.org/wiki/Platohttp://en.wikipedia.org/wiki/Euclid%27s_Elementshttp://en.wikipedia.org/wiki/Rhind_papyrushttp://en.wikipedia.org/wiki/History_of_science_in_the_Middle_Ageshttp://en.wikipedia.org/wiki/Astrology_and_astronomyhttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Astronomyhttp://en.wikipedia.org/wiki/Gearhttp://en.wikipedia.org/wiki/Wheelhttp://en.wikipedia.org/wiki/Circuithttp://en.wikipedia.org/wiki/Circus


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Part 1

Cakes come in variety of forms and flavours and are among favourite desserts served

during special occasions such as birthday parties, Hari Raya, weddings and etc. Cakes

are treasured not only because of their wonderful taste but also in the art of cake

baking and cake decorating. Find out how mathematics is used in cake bakingdecorating and write about your findings

r3

h3

h2

h1

By knowing the value of height, h1 and radius, r, the volume of cake A can be

calculated. This will help the baker to estimate amount of baking powder, flour,

sugar and etc to be used to bake the cake A.

By fixing the r1 value, the baker can identify the types of baking tray can be used

which the diameter of baking tray should be more than 2r1 (the diameter of cake

A)

Therefore, volume of cake B and cake C also can be calculated by using

r2 h2 and

1r32h3 respectively.By calculating the curved surface area at cake A, cake B and cake C by using

formulae 2r1h1 , 2r2h2 and 2r3h3 respectively, the amount of needed cakedecorating can be estimated.

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r1

A

B r2

C


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Q

P

The area of shaded region, P

=

r1

2

r2

2

= (r12 r22)

The area of shaded region, Q

= r22r32

= (r22 r32)

These calculations also can be used to estimate amount of cake decorating needed

or can be used.

4

A

B

C


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Part 2

Best Bakery shop received an order from your school to bake a 5 kg of round cake as

shown in Diagram 1 for the Teachers Day celebration.

1) If a kilogram of cake has a volume of 3800cm , and the height of the cake is tobe 7.0cm, calculate the diameter of the baking tray to be used to fit the 5 kg

cake ordered by your school. [Use =3.142]

Volume of 1kg = 3800 cm

Volume of 5kg = 3800 cm x 5

= 19000 cm

Volume of cake =

r

2h

19000 = ()r2(7)r2 = 863.87197r = 29.3917

d = 2r= 58.7834 cm

2) The cake will be baked in an oven with inner dimensions of 80.0cm in length,

60.0cm in width and 45.0 cm in height.

a) If the volume of cake remains the same, explore by using differentvalues of heights, h cm, and the corresponding values of diameters of

the baking tray to be used, dcm. Tabulate your answers

If h = r2(1) = 19000

()r2 = 19000r

2= 6047.1038

r = 77.7631

d = 2r = 2 x 77.7631

d = 155.53 cm

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Height,h (cm) Diameter,d(cm) 1/d2

1.0 155.53 0.41 x 10-5

2.0 109.97 0.83 x 10-5

3.0 89.79 1.240 x 10-4

4.0 77.76 1.653 x 10-4

5.0 69.55 2.067 x 10-4

6.0 63.49 2.480 x 10-4

7.0 58.78 2.894 x 10-4

8.0 54.99 3.306 x 10-4

9.0 51.84 3.721 x 10-4

10.0 49.18 4.134 x 10-4

11.0 46.89 4.548 x 10-4

12.0 44.89 4.962 x 10-4

13.0 43.13 5.375 x 10

-4

14.0 41.56 5.789 x 10

-4

15.0 40.15 6.203 x 10-4

16.0 38.88 6.615 x 10-4

17.0 37.72 7.028 x 10-4

18.0 36.65 7.444 x 10-4

19.0 35.68 7.855 x 10-4

20.0 34.77 8.271 x 10-4

21.0 33.93 8.686 x 10-4

22.0 33.15 9.099 x 10-4

23.0 32.42 9.514 x 10-4

24.0 31.74 9.926 x 10

-4

25.0 31.10 10.33 x 10-4

26.0 30.50 10.74 x 10-4

27.0 29.93 11.16 x 10-4

28.0 29.39 11.57 x 10-4

29.0 28.88 11.98 x 10-4

30.0 28.39 12.10 x 10-4

31.0 27.93 12.81 x 10-4

32.0 27.49 13.23 x 10-4

33.0 27.07 13.64x 10-4

34.0 26.67 14.05 x 10-4

35.0 26.28 14.47 x 10-4

36.0 25.92 14.88 x 10-4

37.0 25.56 15.30 x 10-4

38.0 25.22 15.72 x 10-4

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39.0 24.90 16.12 x 10-4

40.0 24.59 16.55 x 10-4

41.0 24.28 16.96 x 10-4

42.0 23.99 16.01 x 10-4

43.0 23.71 17.78 x 10-4

44.0 23.44 18.20 x 10-4

45.0 23.18 18.61 x 10-4

b) Based on the values in your table,

(i) state the range of heights that is NOT suitable for the cakes and

explain your answers.

The height of the cake cannot be higher than 45 cm and diameter

cannot be more than 60 cm as the cake will be too large to fit into

the oven.

(ii) suggest the dimensions that you think most suitable for the cake.

Give reasons for your answer.

The most suitable dimensions for the cake will be h = 8 cm and d =

54.99 cm so that the cake can fit into the oven and for easyhandling.

c) (i) Form an equation to represent the linear relation between h and d.

Hence, plot a suitable graph based on the equation that you have

formed. [You may draw your graph with the aid of computer

software]

(Please refer to the graph as attached next page)

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(ii) (a) If Best Bakery received an order to bake a cake where the

height of the cake is 10.5 cm, use your graph to determine the

diameter of the round cake pan required.

When h = 10.5 cm, from the graph

1 = 4.4 x 10-4

cm-2

d2

d2

= 1

4.4 x 10-4

d =

-4

= 47.67 cm

(b) If Best Bakery used a 42 cm diameter round cake tray, used

your graph to estimate the height of the cake obtained.

When d = 42 cm,

1 = 1

d2

422

= 5.67 x 10-4

cm-2

From the graph, h = 13.5 cm

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3) Best Bakery has been requested to decorate the cake with fresh cream. The

thickness of the cream is normally set to a uniform layer of about 1 cm.

(a) Estimate the amount of fresh cream required to decorate the cake using

the dimensions that you have suggested in 2(b)(ii)

h = 8cm , d = 54.99

Amount of fresh cream = Volume of fresh cream needed

(area x height)

Amount of fresh cream = Volume of cream at the top surface +

Volume of cream at the side surface

Amount of fresh cream

at top surface

= Area of top surface x height of cream

(3.142)()2 x 12 = 2375 cm

3

Volume of cream at the side surface

= Area of side surface x height of cream

= (circumference of cake x height of cake) x height of cream

= 2(3.142)(54.99/2)(8) x 1

= 1382.23 cm3

Therefore, the amount of fresh cream need to decorate the cake with h =

8cm and d = 54.99,

2375 cm3 + 1382.23 cm3 = 3757.23 cm3

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(b) Suggest three other shapes for cake, that will have the same height and

volume as those suggested in 2(b)(ii). Estimate the amount of fresh cream

to be used on each of the cakes.

i ) T r i a n g l e - s h a p e d b a s e

Base area x height = 19000

Base area = 2375

1/7 x length x width = 2375 length x width = 4750

By trial and improvement

By trial and improvement

4750 = 95 x 50 (length = 95, width = 50)

Slant length of triangle =(95+ 25)= 98.23

Therefore, volume of cream

= area of rectangular front side surface (height of cream)

+ 2(area of slant rectangular both side surface)(height of

cream) + volume of top surface

(Height of cream) + volume of top surface

= (50 x 8)(1) + 2 (98.23x8)(1)+2375 = 4346.68 cm3

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ii) Rectangle shaped-base (cuboid)


Base area = 19000

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Length x width = 2375

By trial and improvement,

2375 x 50 x 47.5 (length = 50, width = 47.5, height = 8)


= 2 (area of left/right side surface)(height of cream)

+ 2(area of front/back side surface)

(Height of cream) + volume of top surface

= 2(8x50)(1) + 2(8 x 47.5)(1) + 2375 = 3935 cm3

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3 . P e n t a g o n - s h a p e d b a s e


Base area = 2375

= area of 5 similar isosceles

Triangles in a pentagon

Therefore :

5(length x width) = 2375

Length x width = 475

By trial and improvement,

475 x 25 x 19 (length = 25, width = 19)


= 5 (area of one rectangular side surface)(height of cream)

+ volume of tope surface

= 5(8x19) + 2375 = 3135 cm3

(c) Based on the values that you have found which shape requires the least

amount of fresh cream to be used?

The Pentagon-shaped cake requires the least amount of fresh cream since

it requires only 3135 cm3

cream

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Part III

Find the dimension of a 5kg round cake that requires the minimum amount of fresh

cream to decorate. Use at least two different methods including Calculus.State whether you choose to bake a cake of such dimensions. Give reasons for your

answers.

Method 1 : Differentiation

Use two equations for this method: the formula for volume of cake (as in Q2/a), and

the formula for amount (volume) of cream to be used for the round cake (as in

Q3/a).

19000 = (3.142)r2h(1)

v = (3.142)r2+2(3.142)rh(2)

From (1):h

= 19000 (3)(3.142)r

2

Sub. (3) in (2):

V = (3.132)r2 + 2(3.142)r ( ( ())V = (3.142)r

2+ ( )

V = (3.142)r2

+ 38000r-1

( )

= 2(3.142)r -

(

)--minium value,

Therefore( ) = 0

( )= 2(3.142)r

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( ()= r

3

6047.104 = r3

r = 18.22

Therefore, h=18.22 cm, d=2r 2(18.22) = 36.44 cm

Method 2: Quadratic Functions

Use the two same equations as in Method 1, but only the formula for amount of

cream is the main equation used as the quadratic function.

f(r) = volume of cream, r = radius of round cake :

19000 = (3.142)r2h(1)

f(r) = (3.142)r2+2(3.142)hr(2)

From (2) :f(r) = (3.142)(r2)(r

2+2hr)--factorize (3.142)

= (3.142)[ ( )2-( )2]--Completing square,with a = (3.142), b=2h and c=0

= (3.142)[r+h)2-h2]

= (3.142)[r+h)2-(3.142)h

2

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(a = (3.142) (positive indicates min. value), min. value = f(r) =(3.142)h,

corresponding value of x = r = --h)

Sub.r = -h into (1) :

19000 = (3.142)(-h)2h

h3

= 6047.104

h = 18.22

Sub.h = 18.22 into (1) :

19000 = (3.142)r2(18.22)

r2

= 331.894

r = 18.22

therefore, h = 18.22 cm, d = 2r = 2(18.22) = 36.44 cm

I would not bake a cake with such dimensions because the dimensions are

not suitable as the height of the cake will be too high and are difficult to

handle. The cake with this height will not be attractive.

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FURTHER EXPLORATION

(a) Find the volume of the first, the second, the third and the fourth cakes. Bycomparing all these values, determine whether the volumes of the cakes form a

number pattern? Explain and elaborate on the number patterns

r = 31 cm, h = 6 cm

Volume of first cake = r2h= 3.142(31)

2(6)

= 18116.772 cm3

Radius of second cake = 31(0.9)

= 27.9

Volume of second cake = r2h= (3.142)(27.9)

2(6)

= 14674.585 cm2

Radius of third cake = 27.9(0.9)

= 25.11Volume of third cake = r2h= (3.142)(25.11)

2(6)

= 11886.414 cm3

Radius of fourth cake = 25.11 (0.9)

= 22.599

Volume of fourth cake = r2h= (3.142)(22.599)

2(6)

= 9627.995 cm3

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Volume of second cake = 14674.585

Volume of first cake 18116.772

= 0.81

= (0.9)2

Volume of third cake = 11886.414Volume of second cake 14674.585

= 0.81

= (0.9)2

Hence,

Volume of second cake

Volume of third cake

Therefore, the number of pattern is based on geometric progression

(b) If the total mass of all the cakes should not exceed 15 kg, calculate the maximum

number of cakes that the bakery needs to bake. Verify your answer using other

methods.

r2h(1-0.9

n)

Sn 1-0.9

15 x 3800 (31)(6)(1-0.9)

0.1

5700 3.142(31)(6)(1-0.9n)

5700

3.142(31)(6) 1-0.9n

0.314626 1-0.9n

0.9n 0.685374

n log 0.9 log 0.685374-0.045757n -0.164072

n 3.586

n 3

The maximum number of cake that the bakery needs to bake is 3

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REFLECTION

While I conducting the project, I have learn more of the uses of progressions in ourdaily life. Apart from that, I have also learn how to bake a cake with accurate height

and length. Other than that, this project encourages the students to share

knowledge and work together. It also encourage students to gather information,

improve their knowledge of mathematical communication. Last but not least, this

project help me get interested in additional mathematics

After conducting this project, I have managed to learn the techniques of applying

additional mathematics in the real life situations.

Last but not least, I have learned many valuable moral value that I could not learn

from the text books. I have learned to cooperate well, share knowledge and work

together with my group members. I have also learned to gather information,

improve my knowledge of mathematical communication.

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things around us that are relatedto circles or parts of a circle.

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additional mathematics project 2010

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