Additional Mathematics Project Work 2 @5a1

8/6/2019 Additional Mathematics Project Work 2 @5a1

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Additional MathematicsProject Work 2

Various Shapes Of Cakes

Name :

I/C Number :

School :SMJK Keat Hwa

Class :5A1


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TABLE OF CONTENTS

Num. Question

1 Part I

2

Part II

~ Question 1

~ Question 2 (a)

~ Question 2 (b)

~ Question 2 (c)

~ Question 3 (a)

~ Question 3 (b)

~ Question 3 (c)

3 Part III

4 Further Exploration


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Introduction

PART I

History of cake baking and decorating

Although clear examples of the difference between cake and bread are easy to find, the precise classification has always been elusive. For example, banana bread may be properly considered either a quick bread or a cake.The Greeks invented beer as a leavener,frying fritters in olive oil, and cheesecakes using goat's milk. In ancient Rome, basic

bread dough was sometimes enriched with butter, eggs, and honey, which produced asweet and cake-like baked good. Latin poet Ovid refers to the birthday of him and his

brother with party and cake in his first book of exile,Tristia.Early cakes in England werealso essentially bread: the most obvious differences between a "cake" and "bread" werethe round, flat shape of the cakes, and the cooking method, which turned cakes over oncewhile cooking, while bread was left upright throughout the baking process. Sponge cakes,

leavened with beaten eggs, originated during the Renaissance, possibly in Spain.

Cake decorating is one of the sugar arts requiring mathematics that uses icing or frostingand other edible decorative elements to make otherwise plain cakes more visuallyinteresting. Alternatively, cakes can be moulded and sculpted to resemble three-dimensional persons, places and things. In many areas of the world, decorated cakes areoften a focal point of a special celebration such as a birthday, graduation, bridal shower,wedding, or anniversary.

Mathematics are often used to bake and decorate cakes, especially in the followingactions:

y Measurement of Ingredientsy Calculation of Price and Estimated Costy Estimation of Dimensionsy Calculation of Baking Timesy Modification of Recipe according to scale


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Part II (Question)Best Bakery shop received an order from your school to bake a 5 kg of round cake asshown in Diagram 1 for the Teachers Day celebration.

h cm

d cm

Diagram 1

1) If a kilogram of cake has a volume of 3800cm 3, 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 byyour school. [Use = 3.142]

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,hcm, and the corresponding values of diameters of the baking tray to be used, d cm.Tabulate your answers.


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Height, h (cm) Diameter, d (cm)1.02.03.0

:

::

T able 1 (b) Based on the values in your table,

(i) state the range of heights that is NO T suitable for thecakes and explain your answers.

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

(c) (i) Form an equation to represent the linear relation betweenh and d.

Hence, plot a suitable graph based on the equation thatyou have formed. [You may draw your graph with theaid of computer software.]

(ii) (a) If Best Bakery received an order to bake a cake where theheight of the cake is 10.5 cm, use your graph to determine

the diameter of the round cake pan required.(b) If Best Bakery used a 42 cm diameter round cake tray,use your graph to estimate the height of the cake obtained.

3) Best Bakery has been requested to decorate the cake with fresh cream. Thethickness 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 cakeusing the dimensions that you have suggested in 2(b)(ii).

(b) Suggest three other shapes for cake, that will have the sameheight and volume as those suggested in 2(b)(ii). Estimate theamount of fresh cream to be used on each of the cakes.

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


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PART II (Answers)

1)Given 1 kg = 3800 cm 3 V = r 2hLet h = 7 cm 19000 = 3.142 x r 2 x 7

r 2 = 190005 kg = 3800 x 5 3.142 x 7

= 19000 cm 3 r 2 = 863.872

r = 29.392 cmd = 2r d = 58.783 cm

2) Maximum dimensions of cake: l=80cm

w=60cmh=45cm

Dimension of oven

V = r 2hV =3.142( )2h

19000=3.142( )2h

d2 =

d =

a)

b) i) h < 7 cm , h > 45 cmThis is because any heights lower than 7 cm will result in the diameter of the

cake being too big to fit into the baking oven while any heights higher than 45 cmwill cause the cake being too tall to fit into the baking oven.

h/cm d/cm1 155.52625192 109.97366743 89.793123394 77.763125945 69.55345436 63.493326457 58.783397838 54.986833689 51.84208396

10 49.1817191911 46.89292932

12 44.8965616913 43.1352212214 41.5661392315 40.15670556

h/cm d/cm16 38.8815629717 37.7206567118 36.6578891219 35.6801692120 34.7767271521 33.9386105622 33.1583083123 32.4294652824 31.7466632325 31.1052503726 30.50120743

27 29.9310411328 29.3916989129 28.8804999430 28.39507881

h/cm d/cm31 27.9333394432 27.4934168433 27.0736453734 26.6725321535 26.288734736 25.9210419837 25.5683583138 25.229689639 24.9041315840 24.5908595941 24.28911983

42 23.9982216743 23.7175310644 23.4464646645 23.18448477


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Part II (Answers)

2b) ii) I would suggest the dimensions of the cake to be 29 cm in height and

approximately 29 cm in diameter. This is because a cake with these dimensions ismore symmetrical and easier to decorate.2c) i) V = r 2h

V = 19000 cm 3

r = d/2

19000 = 3.142 x ( d/2)2 x h

d2 = 19000 .4 3.142 x (d 2/4)

d2 = 76000 .3.142 x h

d = 155.53 x h -1/2

log 10 d = - 1/2 log 10 h + log 10 155.53

2c) ii) a) When h = 10.5 cm, log 10 h = 1.0212According to the graph, log 10 d = 1.7 when log 10 h = 1.0212Therefore, d = 50.12 cm

Please turn ov er b) When d = 42 cm, log 10 d = 1.6232 f or gr aph answe r

According to the graph, log 10 h = 1.2 when log 10 d = 1.6232Therefore, h = 15.85 cm

log 10 h log10 d 1 1.6918142 1.1918143 0.6918144 0.191814


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2(C)(i)


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Part II Questions: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 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).

(b) Suggest three other shapes for cake, that will have the sameheight and volume as those suggested in 2(b)(ii). Estimate theamount of fresh cream to be used on each of the cakes.

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

Part II (Answers)3) a) h = 29 cm d=29cm

r = 14.5 cm

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

Vcream = R 2H r 2hVcream = 3.142 x 15.5 2 x 30 19000

= 22646 19000= 3646 cm 3

1 cm

R=15.5cm

Diagram 1: Cake without Cream

r=14.5 cm

Diagram 2: Cake with Cream

1 cm

H=30 cm

h=29 cm


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3) b) i) Square shaped cake

l=25.6cm

w=25.6cm

h=29.0cm

Estimated volume of cream used= 30 x 27.6 x 27.6 - 19000= 22852.8 19000= 3852.8 cm 3

b) ii) Triangle shaped cake

Estimated volume of cream used= x 39.7 x 39.7 x 30 19000= 23641.4 19000= 4641.4 cm 3

b) iii) Trapezium shaped cake

Estimated volume of cream used= x (28+42.5) x 22 x 30 - 19000= 23265 19000= 4265 cm 3


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3(b) Shape of base Volume of cream used/cm 3

(i) Square shaped cake 3852.8(ii) Triangle shaped cake 4641.4

(iii) Trapezium shaped cake 4265.0(iv) Round shaped cake 3646.0

* All estimations in the values are based on the assumption that the layer of cream isuniformly thick at 1 cm

c) Based on the values I have obtained, the round-base shaped cake requires the leastamount of fresh cream (3646.0 cm 3)


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Part III (Question) Find the dimension of a 5 kg round cake that requires the minimum amount of freshcream to decorate. Use at least two different methods including Calculus.State whether you would choose to bake a cake of such dimensions. Give reasonsfor your answers.

FUR THE R EX PLORA T ION

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

Diagram2

The height of each cake is 6.0 cm and the radius of the largest cake is 31.0 cm. Theradius of the second cake is 10% less than the radius of the first cake, the radiusof the third cake is 10% 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. Bycomparing all these values, determine whether the volumes of the cakesform a number pattern? Explain and elaborate on the number patterns.

(b) If the total mass of all the cakes should not exceed 15 kg, calculate themaximum number of cakes that the bakery needs to bake. Verify your answer using other methods.


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PAR T III(Answers)

Answer:

Conjecture :The best dimension of the cylindrical cake is that the base diameter is twicethe height as the surface area is the best.

V = 19000 = h = 1

Vcream used,V = - V = - V = V = V = 2

Subt 1 into 2 V = (2r+1)+ ]

V = [ V = +


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Method 1: Table

Vcream used, V=

R V

15

16

17

18

19

20

r V

18.1

18.2

18.3


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r V

18.21

18.22

18.23

Vmin when r=18.22 , d=18.22x2 =36.44cm

h =

= 18.22 cm

Answer:

h = 18.22 cm, d=36.44cm when V is minimum


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Method 2: Calculus(Differentiation)

V = +

= + +

= - - + 2

For max/min value, = 0

- = -2

2 (r + 1) =

2 (r + 1) =

2

2

2

2

r = -1 OR

(rejected)

Ans: When r = 18.22cm , d = 18.22 = 36.44cm ,V is minimum


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= 18.22cm

When r = 18.22 , > 0

Vmin

Thus, Vmin when d = 36.44cm

h = 18.22cm


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Method III Graph

V is minimum when r = 18.22,

d = 2(18.22) = 36.44

h = = 18.22 cm

Answer: r = 18.22cm, d=36.44cm, h =18.22cm when V is minimum.


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FURTHER EXPLORATION

a) Volume of cake 1,V 1 Volume of cake 2,V 2= r 2h = r 2h= 3.142 x 31 x 31 x 6 = 3.142 x (0.9 x 31) 2 x 6

= 18116.772 cm3

= 3.142 x (27.9)2

x 6= 14676.585 cm 3

Volume of cake 3,V 3 Volume of cake 4,V 4= r 2h = r 2h= 3.142 x (0.9 x 0.9 x 31) 2 x 6 = 3.142 x (0.9 x 0.9 x 0.9 x 31) 2 x 6= 3.142 x (25.11) 2 x 6 = 3.142 x (22.599) 2 x 6= 11886.414 cm 3 = 9627.995 cm 3

The values 18116.772, 14676.585, 11886.414, 9627.995 , form a number pattern.The pattern formed is a geometrical progression.This is proven by the fact that there is a common ratio between subsequent numbers, r =

0.81.V2= 14676.585 = 0.81 V 3=11886.414 = 0.81V1 18116.772 V 2 14676.585

V4= . 9627.995 = 0.81V3 11886.414

b) S n = a(1-r n) = 18116.772 ( 1-0.81 n)1-r 1-0.81

15 kg = 57000 cm 3 57000 >S n57000 > 18116.772(1-0.81 n)

0.19

10830 > 18116.772(1-0.81 n)

0.5978 > 1-0.81 n

-0.4022 > - 0.81 n

0.4022 < 0.81 n

log 0.4022< n log 0.81

log 0.4022> nlog 0.81

4.322 > nn = 4 (Max)


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Verification of answer

If n = 4

Total volume of 4 cakes

= 18116.772 cm3

+ 14676.585 cm3

+ 11886.414 cm3

+ 9627.995 cm3

= 54307.766 cm 3 15kg

Total mass of cakes must not exceed 15 kg.Therefore, maximum number of cakes needed to be made = 4


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Reflection

In the process of conducting this project, I have learnt that perseverance pays off ,especially when I obtain a just reward for all my hard work. For me, succeeding incompleting 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 acake, to designing and building monuments. Besides that, I have learned many moralvalues that I practice. This project work had taught me to be more confident when doingsomething especially the homework given by the teacher. I also learned to be a moredisciplined student who is punctual and independent.Other than that ,I also learnt how touse computer and Internet to find information.

T he Addition al M a t hema tics Song

I realized that this subject is compulsory to me. Without it, I can't fulfill my big dreams and wishes....

I always tried to love every part of it

Throughout day and night...

I sacrificed my precious time to have fun....

From..Monday,

Tuesday,Wednesday,

Thursday,Friday And even the weekend that I always looking forward to....

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Additional Mathematics Project Work 2 @5a1