Add Maths Project Work 1

8/6/2019 Add Maths Project Work 1

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Additional MathematicsProject Work 2011

Work 2

Noor Atikah Binti Haris

5 Muslim

940917 ² 12 -5228

SMK Agama Kota Kinabalu

Madam Nurhidayati Binti Judin





ObjectivesWe students taking Additional Mathematics are required to carry out a project work while we are in

Form 5. This year the Curriculum Development Division, Ministry of Education prepared three tasks

for us. We are to choose and complete only one task based on our area of interest. This project can

be done in groups or individually, but each of us is expected to submit an individually written report.

Upon completion of the Additional Mathematics Project Work, we are gain valuable experiences and

able to:

y Apply and adapt a variety of problem solving strategies to solve routine and non

routine problems;

y Experience classroom environments which are challenging, interesting and

meaningful and hence improve their thinking skills;

y Experience classroom environments where knowledge and skills are applied in

meaningful ways in solving real life problems;

y Experience classroom environments where expressing ones mathematical thinking,

reasoning and communication are highly encouraged and expected;

y Experience classroom environments that stimulate and enhance effective learning;

y Acquire effective mathematical communication through oral and writing, and to use

the language of mathematics to express mathematical ideas correctly and precisely;

y Enhance acquisition of mathematical knowledge and skills through problem-solving

in ways that increase interest and confidence;

y Prepare ourselves for the demand of our future undertakings and in workplace;

y Realise that mathematics is an important and powerful tool in solving real -lifeproblems and hence develop positive attitude towards mathematics;

y Train ourselves not only to be independent learners but also to collaborate, to

cooperate, and share knowledge is an engaging and healthy environment;

y Use technology especially the ICT appropriately and effectively;

y Train ourselves to appreciate the intrinsic values of mathematics and to become

more creative and innovative;

y Realize the importance and the beauty of mathematics

We are expected to submit the project work within three weeks from the first day the

task is being administered to us. Failure to submit the written report will result in us not

receiving certificate.



I ntroduction

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 samedistance from a given point called the centre. The common distance of the points of a circle

from its center is called its 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.



Definition

Pi, has the value of 3.14159265. In Euclidean plane geometry, 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 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.



History of Pi �

Pi or 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 acircle's area to the square of its radius. It is approximately equal to 3.14159 in the usual

decimal notation. is one of the most important mathematical and physical constants:

many formulae from mathematics, science, and engineering involve .

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 German mathematics. Throughout the history of mathematics, there has beenmuch effort to determine more accurately and to understand its nature; fascination with

the number has even carried over into non-mathematical culture.

The Greek letter , often spelled out pi in text, was adopted for the number from the

Greek word for perimeter "", 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 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 capitalized () even at the beginning of a sentence. When

referring to this constant, the symbol 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/.

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

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

the circumference, or perimeter, of a circle. is Unicode character U+03C0 ("Greek small

letter pi").



PART I

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

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.

QUESTION

Find out how mathematics is used in cake baking and cake decorating and write about

your findings.

Answer:

I. 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.

y The wedding cake must feed between 200 and 250 people.

y 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

y Each layer of cake must remain a cylinder.

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

y Each layer of cake will be cut into sectors that have a top area of exactly 50 cm2

y You may have some left-over cake from a layer

Example of 50 cm2

Top area of sector

y One sector feeds one person.

y 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

y Decide on how many layers of each si ze of cake you will need for your cake.

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

area of 50 cm

2

each, in order to feed between 200 and 250 people.y Complete the ingredients list by identifying the quantities needed for each

ingredient in the cake.

II. 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 m ath skills, it helps you become a more flexible andresourceful baker.

y 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.

y Use as few measuring cups as possible. For example, instead of using a 3/4 cup, use a

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

y Determine what time it will be when the oven timer goes off. For example, if yourcake has to bake for 30 minutes and you set the timer at 3:40, the timer will go off at

4:10.

y 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.

y 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 by 3 inches.

y 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



III. Initial draft of the cake

r = 10 cm

r = 15 cm

r = 20 cm h = 20 cm

r = 25 cm



PART II

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.

h cm

d cm

Diagram 1

QUESTION

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

�

�

��



2) 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.

QUESTION

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:

�

=

²

= 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



QUESTION

(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.

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.

QUESTION

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

reasons for your answer.

Answer:

h = 8cm, d = 54.99cm, because it can fit into the oven, and the size is suitable for easy

handling.

QUESTION

(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 drawyour graph with the air of computer software.]

Answer:

�

�

�

�

��



��

log d =

log h + log 155 53

¡

og h 0 1 2 3 4

Log d 2 19 1 69 1 19 0 69 0 19





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

QUESTION

(b) Suggest three other shapes for cake, which 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 (cuboids)

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)

Therefore, 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


base area = 2375

x length x width = 2375

length x width = 4750

By trial and improvement, 4750 = 95 x 50 (length = 95, width = 50)

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

Therefore, 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


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)Therefore, amount of cream

= 5(area of one rectangular side surface) (height of cream) + vol. of top surface

= 5(8 x 19) + 2375 = 3135 cm³



QUESTION

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

fresh cream to be used?

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



PART III

QUESTION

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 youwould 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 = ²

(3)

Sub. (3) into (2):

V = (3.142) r² + 2(3.142) r (

²)

V = (3.142) r² + (

)

V = (3.142) r² + 38000r -1

(

) = 2(3.142) r (

²

)

0 = 2(3.142) r (

²) -->> minimum value, therefore

= 0

²= 2(3.142) r

= r³

6047.104 = r³

r = 18.22

Sub. r = 18.22 into (3):

h = ²

h = 18.22

therefore, 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

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 choose not to bake a cake with such dimensions because its dimensions are not

suitable (the height is too high) and therefore less attractive. Furthermore, such cakes are

difficult to handle easily.



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.

QUESTION

(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 pattern?

Explain and elaborate on the number patterns.

Answer:

height, h of each cake = 6cm

radius of largest cake = 31cm

radius of 2nd cake = 10% smaller than 1st 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 1st cake = 31, volume of 1st 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/T1 = T3 /T2 = = 0.81

QUESTION

(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:

Sn =

Sn = 57000, a = 18116.772 and r = 0.81

57000 =

1 0.81n

= 0.59779

0.40221 = 0.81n

log0.81 0.40221 = n

n =

n = 4.322

Therefore, n 4



REFLECTI ON While I conducting the project, I had learned some moral values I can practice. This

project also taught me to be responsible on the works given to me to be completed. It also

made me felt more confident to do works and not to give up easily when I could not find the

solution for the question. I also learned to be more punctual, which I was given about a

month to complete this project and submit my teacher just in time. I have done many

researches throughout the internet and discussing with a friend who have helped me a lot in

completing this project.

Through the completion of this project, I have learned many skills and techniques.

This project really helps me to understand more about the uses of progressions in our daily

life. This project also helped expose t he techniques of application of Additional M athematics

in real life situations. I have learnt how to bake a wedding tiered cake stands with good

quality and proper height.

Apart from that, this project encourages the student to work together and share

their knowledge. I also enjoyed doing this project during my school holidays a s I spend my

time with friends to complete this project and it had tightened our friendship. It is also

encourage student to gather information from the internet, improve thinking skills and

promote effective mathematical communication.

Last but not least, I proposed this project should be continue because it brings a lot

of moral values to the student and also test the students understanding a lot in Additional

Mathematics.

Documents

Add Maths Project Work 1