Add Maths Project Work

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Cake is a form of food,typically a sweet,baked dessert.Cakesnormally contain a combination of flour,sugar,eggs and

butter or oil,wth some varieties also requiring

liquid(typically milk or water) and leavening agent(such as

yeast or baking powder).Flavorful ingredients like fruit

purees,nuts or extract are often added,and numerous

substitutions for the primary ingredients are possible.Cakes are often filled with

fruits preserves or dessert sauces(like pastry cream),iced with buttercream or

other icings,and decorated with marzipan,piped borders or candied fruits.

Cake is often the dessert of choice for meals at ceremonial

occasions,particularly wedding,anniversaries and birthdays.There are countless

cake recipes;some are bread-like,some rich and elaborate and many are

centuries old.Cake making is no longer acomplicated procedure;while at one

time considerable labour went into cake making(particularly the whisking of

egg foams),baking equipments and directions have been simplified that even

the most amateur cook may bake a cake.

Varieties

Cakes are broadly divided into several catogeries based primarily ingredients

and cooking technique:

Yeast cakes are the oldest and are very similar to yeast breads.Such cakes

are very traditional in form,and include such pastries as babka and

stollen.

Cheese cakes,despite their name aren’t really cakes at all.Cheese cakes

are in fact custard pies,with a filling made mostly of some form of

INTRODUCTION



cheese(often cream cheese,mascarpone,ricotta or the like),and have very

little to no flour added,although a flour-based crust may be used.Cheese

cakes are also very old,with evidenceof honey-sweetened cakes dating

back to ancient Greece.Sponge cakes are thought to be the first of the non-yeast-based cakes and

rely primarily on trapped air in a protein matrix(generally of beaten eggs)

to provide leavening,sometimes with a bit of baking powder or other

chemical leaven added as insurance.

Butter cakes,including the pound cake and devil’s food cake,rely on the

combinations of butter,eggs,and sometimes baking powder to provide

both lift and a moist texture.

Beyond these classifications,cakes can be classified based

on their appropriate accompaniment(such as coffee cake)

and contents(e.g. fruit cake or flour less chocolate cake).

Special-purpose cake

Cakes may be classified according to the occasion for which they are

intended.For example,wedding cakes,birthday cakes,and Passover plava (a type

of Jewish spong cake sometimes made with matzomeal)are all identified primarily according to the

celebration they are intended to accompany.

Cake Decorating

A finished cake is often enhanced by covering it with

icing,or frosting,and topping such as sprinkles,which are



also known as ‘jimmies’ in certain parts of United States and ‘hundreds and

thousands’ in the United Kingdom.Frosting is usually made from

powdered(icing)sugar,sometimes a fat of some sort,milk or cream,and often

flavourings such as vanilla extract and cocoa powder.Some decorators userolled fondant icing.Commercial bakeries tend to use lard for the fat,and often

whip the lard to introduce air bubbles.This makes the icing light and

spreadable.Home bakers either use lard,butter,margarine or some combination

thereof.Sprinkles are small firm pieces of sugar and oils that are colored with

food colouring.In the late 20th century,new cake decorating products became

available to the public.This include several specialized sprinkles and even

methods to pictures and transfer the image onto a cake.

Special tools are needed for more complex cake decorating, such as piping

bags or syringes, and various piping tips. To use a piping bag or syringe, a

piping tip is attached to the bag or syringe using a coupler. The bag or syringe is

partially filled with icing which is sometimes colored. Using different piping

tips and various techniques, a cake decorator can make many different designs.Basic decorating tips include open star, closed star, basketweave, round, drop

flower, leaf, multi, petal, and specialty tips.

Royal icing, marzipan (or a less sweet version, known as almond paste), fondant

icing (also known as sugarpaste) and buttercream are used as covering icings

and to create decorations. Floral sugarcraft or wired sugar flowers are an

important part of cake decoration. Cakes for special occasions, such as wedding

cakes, are traditionally rich fruit cakes or occasionally Madeira cakes (also

known as whisked or fatless sponge), that are covered with marzipan and either

iced using royal icing or sugarpaste. They are finished with piped borders (made

with royal icing) and adorned with a piped message, wired sugar flowers, hand-

formed fondant flowers, marzipan fruit, piped flowers, or crystallized fruits or

flowers such as grapes or violetsr.





PART

1





Geometry – To determine suitable dimensions for the cake, to assist in

designing and decorating cakes that comes in many attractive shapes and

designs, to estimate volume of cake to be produced.

Calculus (differentiation) – To determine minimum or maximum amount of

ingredients for cake-baking, to estimate min. or max. amount of cream needed

for decorating, to estimate min. or max. size of cake produced.

Progressions – To determine total weight/volume of multi-storey cakes with

proportional dimensions, to estimate total ingredients needed for cake-baking,

to estimate total amount of cream for decoration.



PAR

T

2



QUESTION

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. (Diagram 11)

1) If a kilogram of cake has a volume of 3800cm3, 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 π = 3.142]

Answer

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

3800 x 5 = (3.142)(d2)² x 7

190007 = (3.142)(d2)²

2714.286=(3.142)(d2)2

863.872 = (d2)²

d2 = 29.392

d = 58.784 cm

QUESTION



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

length, 60.0 cmin width and 45.0 cm in height.

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

V=πr2h

19000 = (3.142)(d/2)²h

19000(3.142)h = d²4

24188.415h = d²

d = 155.53h

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

2.0 109.98

4.0 77.77



6.0 63.49

8.0 54.99

10.0 49.1812.0 44.90

14.0 41.58

16.0 38.88

18.0 36.66

20.0 34.78

22.0 33.16

24.0 33.75

26.0 30.50

28.0 29.39

30.0 28.40

32.0 27.49

34.0 26.67

36.0 25.92

38.0 25.2340.0 24.59

42.0 24.00

44.0 23.45

46.0 22.93

48.0 22.45



50.0 22.00

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.

h>36 cm is NOT suitable,because the resulting diameter produced is too small

and the cake would be too tall so,the cake would be not suitable for theoven.Instead,we can use oven which is more small in width to bake a cake with

the heights.

h>44 cm is NOT suitable,because the height is too big to fit into the oven

because.Futhermore, the cake would be too small in diameter which is less

attractive.

QUESTION

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

reasons for your answer.

ANSWER



h = 10cm, d = 49.18cm, because it can fit into the oven, and the size is suitable

for easy handling.Futhermore,the height and the diameter is just nice to use the

oven.Not only that,the height and the diameter makes the cake to be more

attractive.

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

draw your graph with the aid of computer software.]

ANSWER

19000/(3.142)h = d²4

24188.415h = d²

19000 = (3.142)( d2)²h

d = 155.53√h

d = 155.53h-12



log d = log 155.53h-12

log d = -12 log h + log 155.53

log10h log10d

0.0 2.19

0.5 1.94

1.0 1.691.5 1.44

2.0 1.19

2.5 0.94

3.0 0.69

3.5 0.44

4.0 0.19





QUESTION

(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.

Answer:

h = 10.5cm

log h=log 10.5 cm

log h=1.021 cm

When log h=1.021,log d=1.680

so,d=47.86 cm

QUESTION

(b) 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=log 42

log d=1.623 cm

When log d=1.623,log h=1.140

so.h=13.80 cm

QUESTION



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 1cm.

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

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

ANSWER

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 surfaceVol. of cream at the top surface

= Area of top surface x Height of cream

=πr2h

= (3.142)(54.992)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

= 2πrh×1

= 2(3.142)(54.99/2)(8) x 1= 1382.23 cm³

Therefore, amount of fresh cream

=2375+1382.23



=3757.23 cm3

QUESTION



(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)

19000 = base area x height base area = 1900010

length x width = 1 900By trial and improvement,1900 = 50 x 38 (length = 50, width = 38, height = 10)

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(10 x 50)(1) + 2(10 x 38)(1) + 1900

=1000+760+1900

=3660cm3

2-TRIANGLE-SHAPED BASE



19000 = base area x height

base area = 1900

12 x length x width = 1900

length x width = 3800

By trial and improvement, 3800 = 76 x 50 (length = 76, width = 50)

Slant length of triangle = √(76² + 25²)= 80.00

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 10)(1) + 2(80.00 x 10)(1) + 1900

=500+1600+1900

=4000cm3

3-HEXAGON-SHAPED BASE



19000 = base area x height base area = 1900 = area of 6 similar isosceles triangles in a hexagon

therefore:

1900 = 6(length x width)

316.67 = length x width

By trial and improvement, 316.67 = 20 x 15.83 (length = 20, width = 15.83)

Therefore, amount of cream

= 6(area of one rectangular side surface)(height of cream) + vol. of top surface

= 6(10 x 15.83) + 1900

=2849.8 cm3

QUESTION

(c) Based on the values that you have found which shape requires the leastamount of fresh cream to be used?

ANSWER



Hexagon-shaped cake, since it requires only 2849.8 cm³ of cream to be used.



QUESTION

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

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 answers.

Answer

PART

3



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(3.142)r² → (3)

Sub. (3) into (2):

V = (3.142² + 2(3.142)r(19000(3.142)r²)

V = (3.142)r² + (38000r)

V = (3.142)r² + 38000r -1

(dVdr) = 2(3.142)r – (38000r²)

0 = 2(3.142)r – (38000r²) -->> minimum value, therefore dVdr =

0

38000r² = 2(3.142)r

380002(3.142)= r³

6047.104 = r³

r = 18.22

Sub. r = 18.22 into (3):

h = 19000(3.142)(18.22)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 + 2h2)² – (2h2)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.



QUESTION

FURTHER EXPLORATION

Best Bakery received an order to bake a multi-storey cake for Merdeka Day

celebration, as shown in 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 is10% less than the radius of the second cake and so on.

(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 = 910

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 = T2/T1 = T3 /T2 = … = 0.81



QUESTION

(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.

Answer

Sn = (a(1 - rn)) (1 - r)

Sn = 57000, a = 18116.772 and r = 0.8157000 = (18116.772(1 – (0.81)n))(1 - 0.81)

1 – 0.81n = 0.59779

0.40221 = 0.81n

log0.81 0.40221 = n

n = log 0.40221 log 0.81

n = 4.322

therefore, n ≈ 4

REFLECTION



In the making of this project, I have spent countless hours doing this

project.I realized that this subject is a compulsory to me. Without it, I can’t

fulfill my big dreams and wishes.

In the process of conducting this project,I have learnt that perseverance pays off,especially when you obtain a just reward for all your hard work.For

me,succeeding in completing this project work has been reward enough.I have

also learnt that mathematics is used everywhere in daily life,from the most

simple things like baking and decorating cake to designing and building

monuments.Besides that,I have learnt many moral values that we practice.This

project me had taught me to be more confident when doing something

especially the homework given by teacher.I also learned to be more disciplined

student who is punctual and independent.

During the period of completing the project work, I had discovered that

additional mathematics have multi-uses especially in pastry industry. The

inventions of different mathematics theorem are to solve daily problems and

even make our works more efficient. Furthermore, this project also encourages

knowledge transfer as students practice what they had learnt and found out the

information they don’t know. I also had experienced team work, patience and

creative thinking while conducting the project. Personally, Calculus used to be a

very confusing and difficult topic for me. But after applying it in this project, I

have built a deeper understanding toward Differentiation and Integration.

The poem below is what I feel about Additional Mathematics :-



ADD MATH,

A well known taught subject.

Perhaps there’s a concept,

That I wouldn’t able to accept.

Addition, Subtraction, Multiplication and Division,

Miraculously transform to Differentiation,

Then followed by Integration.

Yet, I never fail to figure out the solution,

But it always ends up with correction!

To my surprise,

Cake baking requires geometry to apply,

In determine its shape and size.

If I have been wise,

And listen to Calculus’s advice,

Baking will never fail once or twice...

Add Math used to be my pain,

But if I do practice again and again,

Works will never be in vain...



INTRODUCTIONTO

CAKE



REFLECTION

CONTENT !!!



ACKNOWLEDGEMENT

‘THANK YOU’



CONCLUSION





NO. CONTENT PAGE

01 ACKNOWLEDGEMENT

02 OBJECTIVE03 INTRODUCTION

04 PART 1

05 PART 2

06 PART 3

07 FUTHER EXPLORATION

CONTENT PAGE....



08 CONCLUSION

09 REFLECTION

First of all, I would like to thank God because I have managed to

complete this project work successfully without any problems. I am also very

thankful to my add maths teacher, Cik Suzana bt Ayob for being helpful and as

guidance to me throughout my project work.I am also thankful to all my friends

and others who where helpful directly and indirectly in my effort to complete

the project work.My parents should also be thanked because they provided

moral support and encouragement throughout the time.

Thank you Joseph Carl Robnett Licklider the person which invented the

internet, without the help of internet I will not be able to finish this project. I’d

also like to express my thanks to all my fellow friends especially the 5

ACKNOWLEDGEMENT….



Cemerlang students. They are my classmates who involved in this project as we

work together as a secondary family. We do practice teamwork.

Thank you.

In conclusion, cake baking required a lot of mathematical knowledge

including geometry, calculus, ratios and progressions in order to produce a cake

in a systematic way. Those skills are important especially for pastry industry so

that the cake produced can be unified with the same quality. Without it, the

outlook of the cakes will be less attractive. So, I should be thankful of the

people who contribute in the idea of geometry, calculus, ratios and progressions.

Those mathematical theories make daily problems easier to be solved.

From this project I have learned the importance of perseverance as time

will be invested to ensure the completion and excellence of this project.

Similarly, I have learned the virtue of working together as I have help and

receive help from my fellow peers in the production of this project as sharing

knowledge is vital achieving a single goal. Also, I have learned to be thankful

CONCLUSION…



and appreciative. This is because I have been able to apply my mathematical

knowledge in daily life and appreciate the beauty of mathematics.

This project will serve as a training stage for me to prepare myself for the

demands for my future undertakings in the university and work life. I will apply

mathematics in everyday situations and situations and appreciate the importance

and beauty of mathematics. I aim to improve my problem solving skills,

thinking skills, reasoning and mathematical communication skills. This project

shall stimulate a learning environment which enhances effective learning,

inquiry-based and teamwork. This project will develop my mathematical

knowledge such that it increases my interest and confidence.

OBJECTIVE…



One very important aspect of cake baking is the amount of ingredients

required. This is because a cake should be of reasonable pricing, at the same

time, beauty is also a priority. In cake baking, ingredients must not be wasted

unnecessarily. Therefore, Calculus can be applied in cake baking. Particularly,

the second derivative is greatly used. This is because the second derivative

allows bakeries to calculate the maximum or minimum amount of ingredients

needed to increase profit and efficiency. This allows positive growth in

business. At the same time, the bakeries can avoid under-order, or over-order

the ingredients.

Cake decorating is an art. Many bakeries strive to produce cakes of

beauty and of substance. In art, geometry is given importance. Geometry allows

cakes of many interesting shapes and sizes to be created. Geometry also defines

the ideal dimensions of a cake to be baked in ovens. Geometry will determine

the popularity of the cake besides the price. We calculate the surface area and



volume of the cake to determine the price per kilogram and also the area

available for decorating and writing words.

In the baking of more complex cakes, such as multi-storey cakes or

multilayered cakes, progressions are applied. Progressions allow us to calculate

the size or volume of a subsequent layer. Also, it allows us the estimate the

quantity of ingredients needed. Usually, geometric progressions are used.

Lastly, ratios are used in cake baking. More often than not bakers need to

estimate the amount of ingredients used or substitute the ingredient with another

if that ingredient is not available. For example, we often use cookbooks guiding

us to use 3 parts of water for 1 part of flour. This is ratio of water to flour 3:1,

allows us to bake a cake of different sizes. Although we may bake a smaller or

larger cake, the flour and water used still obeys the proportion set. We are then

allowed to creatively bake cakes.

OBJECTIVE



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Add Maths Project Work