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8/3/2019 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
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Additional Mathematics Project 2011 Page 2
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
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Additional Mathematics Project 2011 Page 3
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
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Additional Mathematics Project 2011 Page 4
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.
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Additional Mathematics Project 2011 Page 5
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:
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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
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(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
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Additional Mathematics Project 2011 Page 8
Log h 0 1 2 3 4
Log d 2.19 1.69 1.19 0.69 0.19
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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.
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
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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
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2 – Triangle-shaped base
Cake without cream
Cake with cream (plan view) Cake with cream (side view)
(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
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3 – Regular pentagon-shaped base
Cake without cream
Cake with cream (plan view) Cake with cream (side view)
* 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
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(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):
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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)
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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
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Additional Mathematics Project 2011 Page 16
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.