Additional Mathematics Project by Wafa Nabilah-5T

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Wafa Nabilah bt Kamal [940530-13-5624][SM007A028]

Additional Mathematics Project 2011 Page 1

Kolej Tun Datu Tuanku Haji Bujang

Jalan Bunga Tanjong 2Tanjong Lobang,

98000, Miri

ADDITIONAL MATHEMATICS

PROJECT WORK 2/2011

Wedding Cake

Name : Wafa Nabilah bt Kamal

Class : 5 Tekun

IC Number : 940530-13-5624

Subject Teacher : Chen Fui Ping





TABLE OF CONTENTS

Num. Question Page

1 Part I 3

2 Part II

Question 1

Question 2 (a)

Question 2 (b)

Question 2 (c)

Question 3 (a)

Question 3 (b)

Question 3 (c)

5

3 Part III 13

4 Further Exploration 14

5 Reflection 16





Introduction

Part I

Cakes come in a 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 notonly 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.

Answer:

According to my findings, there are four Mathematics concept that can be applied in cake

baking and decorating. We can applied Calculus, geometry, progressions and ratio.

The most important aspect of cake baking is the amount of ingredients used. This is

because aside from reasonable price, aspect of taste and beauty should be not left out. The

ingredients should be used wisely and must not be wasted unnecessarily. Therefore, the concept

of Calculus can be applied in cake baking. Second derivative is used widely to allow bakers to

calculate the minimum or maximum amount of ingredients to increase profit and avoid in

wasting of ingredients. Concept of differentiation also allow bakers to estimate minimum or

maximum amount of cream needed for decorating and size of cake produced. This will help in

the efficiency of bakeries in business since everything is under control and there is no under-

order or over-order the ingredients.

The art of the cake baking is the decoration. Most of bakers aim to produce cakes that

fulfill the beauty aspect. For cake decorating, geometry is given priority. With the help of

geometry, cakes can be created in many interesting shapes and sizes. Geometry is used to

determine suitable dimensions for cake to be baked in ovens, to assist in designing and

decorating cakes. Geometry also helps to estimate volume of cake to be produced. Geometry

will determine the price of cake by calculating the surface area and volume of the cake. Then it

will used to determine the price per kilogram and also the area that available for decorating and

writing words such as ‘Happy Birthday’, ‘Congratulation’ and other according to costumer’s

interest.

Progressions is applied in creating more complex design of cakes such as multi-storey or

multilayered cakes. Progressions is used in determine total weight or volume and size of a

subsequent layer of multi-storey cakes with proportional dimensions. Particularly, geometric

progression is to estimate total ingredients needed for cake-baking. Also, it allows bakers to

estimate total amount of cream for decoration.

Last but not least, another mathematics concept that can be used is ratios. Bakers need to

estimate the amount of ingredients used or even substitute the ingredient with another if that

ingredient is not available. In cookbooks, some recipe guided us to use 3 parts of water for 1 part





of flour. By using ratio of 3:1, allow us to bake different size of cakes. Cakes then, can be bake

creatively according to baker’s creativity.





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. (Diagram 11)

1) If a kilogram of cake has a volume of 3800, 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)(

)² x 7

(3.142) = (

)²

863.872 = (

)²

= 29.392

d = 58.784 cm

2) The cake will be baked in an oven with inner dimensions of 80.0 cm in length, 60.0 cm in

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:

19000 = (3.142)(d/2)²h

=

= d²

d =

√

Oven dimension, width:60cm, height:45cm

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

1.0 155.53 16.0 38.89 31.0 27.93

2.0 109.98 17.0 37.72 32.0 27.49

3.0 89.80 18.0 36.66 33.0 27.07

4.0 77.77 19.0 35.68 34.0 26.67

5.0 68.56 20.0 34.78 35.0 26.29

6.0 63.49 21.0 33.94 36.0 25.92

7.0 58.78 22.0 33.16 37.0 25.57

8.0 54.99 23.0 32.43 38.0 25.23

9.0 51.84 24.0 31.75 39.0 24.90

10.0 49.18 25.0 31.11 40.0 24.59

11.0 46.89 26.0 30.50 41.0 24.29

12.0 44.90 27.0 29.93 42.0 24.0

13.0 43.14 28.0 29.40 43.0 23.72

14.0 41.57 29.0 28.88 44.0 23.45

15.0 40.16 30.0 28.40 45.0 23.18





(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:

i) h < 7cm is NOT suitable, because the resulting diameter produced is too large tofit into the oven. Furthermore, the cake would be too short and too wide, making

it less attractive.

ii) h > 45cm NOT suitable, because the cake being too tall to fir into the baking

oven and it less attractive.

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

answer.

Answer:

h = 29cm, d = 29 cm approximately, because a cake with these dimensions is more

symmetrical and easier to decorate

(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)(

)²h

19000/(3.142)h =

= d²

d =

√

d =

log d =

log d =

log h + log 155.53





Log h 0 1 2 3 4

Log d 2.19 1.69 1.19 0.69 0.19





(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 = 1.02, log d = 1.68, d = 47.86cm

(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 = 1.62, log h = 1.14, h = 13.80cm

3) Best Bakery has been requested to decorate the cake with fresh cream. The thickness of thecream 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 = 29 cm

r = 29/2

= 14.44 cm

Diagram 1 : Cake without cream

Diagram 2 : Cake with cream

14.44 cm

15.44 cm

29.00 cm

30.00 cm





To calculate the volume of cream used, the cream is symbolised as the larger cylinder and thecake is symbolized as the smaller cylinder.

Vcream = 3.142 x 15.442

x 30 – 19000

= 22470.98 - 19000= 3470.98 cm3

(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 – Square-shaped base (cuboid)

Cake without cream

Cake with cream (plan view) Cake with cream (side view)

Estimated volume of cream used= 30 x 27.6 x 27.6 -19000

= 22852.80 – 19000

= 3852.80 cm3

29.0 cm

25.6 cm

25.6 cm

27.6 cm

27.6 cm 30.0 cm





2 – Triangle-shaped base

Cake without cream


(Value of 2.5cm obtained from scale drawing)

Estimated volume of cream used

= 1/2 x 39.7 x 39.7 x 30 -19000= 23641.35 – 19000

= 4641.35 cm3

29.0 cm

36.2 cm36.2 cm

39.7 cm

39.7 cm

2.5 cm

2.5 cm

30.0 cm





3 – Regular pentagon-shaped base

Cake without cream


* By trial and improvement, length of each sides of regular pentagon = 13.1

* Length of each sides of cake with cream as added 1 cm approximately

Estimated volume of cream used= 30 x 5 (11 x 14.1 ) – 19000

= 23265 – 19000

= 4265 cm3

29.0 cm

13.1 cm

10.0 cm

14.1 cm

30.0 cm





(c) Based on the values that you have found which shape requires the least amount of freshcream to be used?

Answer:

Square-shaped cake, since it requires only 3852.80 cm³ of cream to be used.

Part III

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:

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)rh → (1)

V = (3.142)r + 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

6047.104 = r³

r = 18.22

Sub. r = 18.22 into (3):





h =

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 themain equation used as the quadratic function.

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

19000 = (3.142)rh → (1)

f(r) = (3.142)r + 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 valueof x = r = --h)

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

19000 = (3.142)(--h)²hh³ = 6047.104

h = 18.22

Sub. h = 18.22 into (1):

19000 = (3.142)r²(18.22)

r² = 331.894r = 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.

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 =

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

(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 =

Sn = 57000, a = 18116.772 and r = 0.81

57000 = –

1 – 0.81n

= 0.59779





0.40221 = 0.81n

og0.81 0.40221 = n

n =

n = 4.322

therefore, n ≈ 4

REFLECTION

I have learnt that Mathematics is applicable and mainly uses in our daily life . Before

conducting this project, I never know that we can applied Mathematics such as differentiation,

ratio, progression and calculus in simple things such as baking cake to complex doing such as

architecture and construction.

Besides, I have learnt that perseverance is important in doing something. Beforehand, I

found that this project is quite difficult. I was very worried about completing this project since I

am busy with other school works and I afraid that I might not complete it. If I have give up

earlier and not persevere enough, I would never complete this project.

Additional Mathematics Project by Wafa Nabilah-5T

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