Sekolah Menengah Sains Hulu Selangor 1

8/6/2019 Sekolah Menengah Sains Hulu Selangor 1

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2/2011

Muhammad Nasir Bin Hamidon,

Sekolah Menengah Sains Hulu Selangor,

940221-14-5503,

5 Jujur,

Sir Othman Bin Mohamed

Additional Mathematics Project Work



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Acknowledgement

Alhamdulillah, finally I have finished my Additional Mathematics

Project Work. An appreciation goes to my family for their ongoing support

and encouragement.

I also would like to thank En. Othman b. Mohamed for his advises

and guidance when I was carrying out this project work.

Last but not least, thanks to my friends for lending me their hands

and everyone that had helped me to complete this task.



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Introduction of Additional Mathematics

Project Work 2011

The objectives of carrying out this project work are to

enable students to:

Apply and adapt a variety of problem-solving techniques to solve problems.

Develop mathematical knowledge through problem solving in a way that

increases students’ interest and confidence.

Develop positive attitude towards mathematics.

Improve thinking skills and creativity.

Promote efficiency of mathematical communication.

Provide learning environment that stimulates and enhances effective

learning.



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

A circle is a simple shape of Euclidean geometry consisting of those points in a

plane which is the same distance from a given point called the center. The

common distance of the points of a circle from its center is called a 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 (known as the perimeter) or to the whole figure including its interior. However, in strict

technical usage, "circle" refers to the perimeter while the interior of the circle is

called a disk. The circumference of a circle is the perimeter of the circle

(especially when referring to its length).

A circle is a special ellipse in which the two foci are coincident. Circles are

conic sections attained when a right circular cone is intersected with a plane

perpendicular to the axis of the cone.

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

It is the basis for the wheel, which, with related inventions such as gears, makes

much of modern civilization possible. In mathematics, the study of the circle has

helped inspire the development of geometry and calculus. Circles had been used in

daily lives to help people in their living.



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Definition

Pi, π has the value of 3.142. In Euclidean plane geometry, π is

defined as the ratio of a circle's circumference to its diameter.

The ratio

is constant, regardless of a circle's size. For example, if

a circle has twice the diameter of another circle it will also have twice the

circumference, C, preserving the ratio

.Alternatively π can also be defined

as the ratio of a circle's area (A) to the area of a square whose side is

equal to the radius.

.



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History

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

3.142 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/n, where and 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 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 " π", 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).

The name of the Greek letter πis 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 capitalized ( π ) even at the beginning of a

sentence. When referring to this constant, the symbol πis always pronounced like



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"pie" in English, which is the conventional English pronunciation of the

Greek letter. In Greek, the name of this letter is pronounced /pi/. The

constant is named " π" because " π" is the first letter of the Greek words π

(periphery) and π (perimeter), probably referring to its use in the formula

to find the circumference, or perimeter, of a circle. Π is Unicode character

U+03C0 ("Greek small letter pi").



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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 baking and cake decorating and write

about your findings.

Constructing the structure of a cake,

These cakes are made by using different sizes of circular pans, then stacking the

baked cake sections on top of each other.

You are to plan for a cake that will serve between 200 and 250 people.

The wedding cake must feed between 200 and 250 people.

You have 4 different sizes of pans of you can use. ( All pans have the same height )

r = 10 cm r = 15 cm r = 20 cm r = 25 cm

Each layer of cake must remain a cylinder

You can stack layers . Each layer can then be separated and cut

individually.



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Each layer of cake will be cut into sectors that have a top area of exactly

50 cm2

You may have some left-over cake from a layer

Example of 50 cm2

Top area of sector

One sector feeds one person. Your final ingredients list must be proportional to the ingredients list

provided for you.

By using the theory of arithmetic and geometric progressions in Chapter 1 Form

5, the concept can be used to:

Decide on how many layers of each size of cake you will need for your

cake.

Show how you can cut the layers of the cake into equivalent sectors having

a top area of 50 cm2

each, in order to feed between 200 and 250 people.

Complete the ingredients list by identifying the quantities needed for each

ingredient in the cake.



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Work and calculations to determine the ingredients of the cake

Baking a cake offers a tasty way to practice math skills, such as fractions

and ratios, in a real-world context. Many steps of baking a cake, such as

counting ingredients and setting the oven timer, provide basic math practice for

young children. Older children and teenagers can use more sophisticated math to

solve baking dilemmas, such as how to make a cake recipe larger or smaller or

how to determine what size slices you should cut. Practicing math while baking

not only improves your math skills, it helps you become a more flexible and

resourceful baker.

Calculate the proportions of different ingredients. For example, a frosting recipe that calls for 2 cups cream cheese, 2 cups confectioners' sugar and

1/2 cup butter has a cream cheese, sugar and butter ratio of 4:4:1.

Identifying ratios can also help you make recipes larger or smaller.

Use as few measuring cups as possible. For example, instead of using a ¾

cup, use a 1/4 cup three times. This requires you to work with fractions.

Determine what time it will be when the oven timer goes off. For example,

if your cake has to bake for 30 minutes and you set the timer at 3:40,

the timer will go off at 4:10.

Calculate the surface area of the part of the cake that needs frosting. For example, a sheet cake in a pan only needs the top frosted, while a sheet

cake on a tray needs the top and four sides frosted. A round layer cake

requires frosting on the top, on each layer and on the sides.

Determine how large each slice should be if you want to serve a certain

amount of people. For example, an 18 by 13 inch sheet cake designed to

serve 25 people should be cut into slices that measure approximately 3 by3

inches.

http://www.ehow.com/recipes/

http://www.ehow.com/how_11464_make-yellow-cake.html




http://www.ehow.com/how_11464_make-yellow-cake.html




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Add up the cost of your ingredients to find the cost of your cake. Estimate

the cost of partially used ingredients, such as flour, by determining the

fraction of the container used and multiplying that by the cost of the

entire container.



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Initial draft of the cake:

r = 10 cm

r = 15 cm

h=20 cm

r = 20 cm

r = 25 cm



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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 Teacher’s Day celebration.

h cm

d cm

1) If a kilogram of cake is has a volume of 3800 cm2, and the height of the

cake is to be 7.0cm, calculate the diameter of the baking tray to be used

to

fit the 5 kg cake ordered by your school. [ use ]

Answer:

Volume of 5kg cake = Base area of cake x Height of cake

3800 x 5 = (3.142)(

)² x 7

(3.142) = (

)²

863.872 = (

)²

= 29.392

d = 58.784 cm

The cake will be baked in an oven with the inner dimensions of 80.0 cm in

length, 60.0 cm in width and 45.0 cm in height.



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a.) If the volume of cake remains the same, explore by using the different

values of heights, h cm, and the corresponding values of diameters of the

baking tray to be used , d cm. Tabulate your answers.

Answer:

First, form the formula for d in terms of h by using the above formula

for volume of

cake, V = 19000, that is:

19000 = (3.142)(d/2)²h

=

= d²

d =

√

Height,h (cm) Diameter,d(cm)

1.0 155.53

2.0 109.98

3.0 89.80

4.0 77.77

5.0 68.56

6.0 63.49

7.0 58.78

8.0 54.99

9.0 51.84

10.0 49.18



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b.) Based on the values in your table, state the range of heights that is NOT

suitable for the cakes and explain your answers.

Answer:

h < 7cm is NOT suitable, because the resulting diameter

produced is too large to fit into the oven. Furthermore,

the cake would be too short and too wide, making it less

attractive.

i. Suggest the dimensions that you think most suitable for the cake. Give the

reasons for your answer.

h = 8cm, d = 54.99cm, because it can fit into the oven, and

the size is smaller, so it is suitable for easy handling.

ii. Form an equation to represent the linear relation between h and d . Hence,

plot a suitable graph based on the equation that you had

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

software ]

Answer:

19000 = (3.142)(

)²h

19000/(3.142)h =

= d²

d =

√

d =

log d =

log d =

log h + log 155.53



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Log h 0 1 2 3 4

Log d 2.19 1.69 1.19 0.69 0.19



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iii. If Best Bakery received an order to bake a cake where the height of the

cake is 10.5 cm, use your graph based on the equation that

you had formed.

Answer:

h = 10.5cm, log h = 1.021, log d = 1.680, d = 47.86cm

iv. If Best Bakery used a 42 cm diameter round cake tray, use your graph

to estimate the height of the cake obtained.

Answer:

d = 42cm, log d = 1.623, log h = 1.140, h = 13.80cm

v. 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)

Answer:



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h = 8cm, d = 54.99cm

Amount of fresh cream = VOLUME of fresh cream needed (area x

height)

Amount of fresh cream = Vol. of cream at the top surface + Vol. of

cream at the side surface

Vol. of cream at the top surface

= Area of top surface x Height of cream

= (3.142)(

)² x 1

= 2375 cm³

Vol. 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

Thus,amount of fresh cream = 2375 + 1382.23 = 3757.23 cm

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.

Answer:

1. Rectangle shaped-base (cuboid)



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19000 = base area x height

base area =

length x width = 2375

By trial and improvement, 2375 = 50 x 47.5 (length = 50,

width = 47.5, height = 8)

Thus, volume of cream

= 2(Area of left/right side surface)(Height of cream) +

2(Area of

front/back side surface)(Height of cream) + Vol. of top

surface

= 2(8 x 50)(1) + 2(8 x 47.5)(1) + 2375 = 3935 cm³

2. Triangle-shaped base

19000 = base area x height base area = 2375

x length x width = 2375

length x width = 4750

By trial and improvement, 4750 = 95 x50(length =95, width =

50)



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Slant length of triangle = √ (95² + 25²)= 98.23

Thus, amount of cream = Area of rectangular front side

surface(Height of cream) + 2(Area of slant rectangular left/right

side surface)(Height of cream) + Vol. of top surface

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

3. Pentagon-shaped base

19000 = base area x height

base area = 2375 = area of 5 similar isosceles triangles in a

pentagon

therefore:

2375 = 5(length x width)

475 = length x width

By trial and improvement, 475 = 25 x 19 (length = 25, width=19)

Thus, amount of cream= 5(area of one rectangular side surface)(height of cream) + vol.

of top surface

= 5(8 x 19) + 2375 = 3135 cm³



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

amount of fresh cream to be used?

Pentagon-shaped cake, since it requires only 3135 cm³ of cream to be used.



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PART 3

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

amount fresh cream to decorate. Use at least two different methods including

calculus. State whether you would choose to bake a cake of such dimensions.

Give reasons for your answer.

Answer:

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 =

→

(3)

Sub. (3) into (2):

V = (3.142)r² + 2(3.142)r(

)

V = (3.142)r² + (

)

V = (3.142)r² + 38000r -1

(

) = 2(3.142)r – (

)

0 = 2(3.142)r – (

) -->> minimum value, therefore

= 0

= 2(3.142)r

= r³



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6047.104 = r³

r = 18.22

Sub. r = 18.22 into (3):

h =

h = 18.22

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

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.

Let 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 + 2hr) -->> factorize (3.142)

= (3.142)[ (r +

)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)h²

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

(3.142)h², corresponding value of x = r = --h)

Sub. r = --h into (1):

19000 = (3.142)(--h)²h



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h³ = 6047.104

h = 18.22

Sub. h = 18.22 into (1):

19000 = (3.142)r²(18.22)

r² = 331.894

r = 18.22

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

I would choose not to bake a cake with such dimensions because its

dimensions are not suitable (the height is too high) and less attractive.

Furthermore, such cakes are difficult to handle easily.



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Further Exploration

Best Bakery received an order to bake a multi-storey cake for Merdeka Day celebration, as shown in Diagram 2.

Diagram 2:

The height of each cake is 6.0 cm and the radius of the largest cake is

31.0 cm. The radius of the second cake is 10% less than the radius of the

first cake, the radius of the third cake is 10% less than the radius of the

second cake and so one.

a. Find the volume of the first, the second, the third and the fourth cakes.

By comparing all these values, determine whether the volumes of the cakes

form a number patterns? Explain and elaborate on the number patterns.



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Answer:

height, h of each cake = 6cm

Radius of largest cake = 31cm

Radius of 2nd cake = 10% smaller than 1 st cake

Radius of 3rd cake = 10% smaller than 2nd cake

31, 27.9, 25.11, 22.599…

a = 31, r =

V = (3.142)r²h

Radius of 1 st cake = 31, volume of 1 st cake = (3.142)(31)²(6)

= 18116.772

Radius of 2nd cake = 27.9, vol. of 2nd cake = 14674.585

Radius of 3rd cake = 25.11, vol. of 3rd cake = 11886.414

Radius of 4th cake = 22.599, vol. of 4th cake = 9627.995

18116.772, 14674.585, 11886.414, 9627.995, …

a = 18116.772, ratio, r = T 2/T 1 = T 3 /T 2 = … = 0.81

b. If the total mass of all the cakes should not exceed 1.5 kg, calculate the

maximum number of cakes that the bakery needs to bake. Verify your

answer using other methods.

Answer:

S n =

S n = 57000, a = 18116.772 and r = 0.81

57000 = –



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1 – 0.81 n = 0.59779

0.40221 = 0.81 n

og 0.81 0.40221 = n

n =

n = 4.322



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References

1. Wikipedia.com

2. Pelangi Focus Goal Form 5 Additional Mathematics

3. Subject teacher

4. Scribd.com



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Reflection

I found a lot of information while conducting this project. Moreover, this project

encourages the student to think critically to identify and solve problems. It is also encourage

student to gather information using the technologies such as the internet, improve thinking

skills and promote effective mathematical communication. Finally, I proposed this project

should be continue because it brings a lot of advantages to the student and also test

the student’s understanding in Additional Mathematics.

Sekolah Menengah Sains Hulu Selangor 1

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