Additional Mathematics Project Work 2011/2 (Form 5)

8/6/2019 Additional Mathematics Project Work 2011/2 (Form 5)

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ADDITIONALMATHEMATICSPROJECT WORK 2/2011

Name: JOHNNY KANESHIROI/C Num: 570831-67-9413

Class: 5BSchool: SMK METHODIST (ACS) IPOH


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Contents

Num. Question Page

1 Indroduction 1

2 Part I 2

3

Part II Q1 3

Q2 (a) 4

(b) 5

(c) 5 - 7

Q3 (a) 8

(b) 9 11

(c) 11

3 Part III 12 - 13

4 Further Exploration (a) 14

5 Further Exploration (b) 15

6 Reflection 16

7 Marking Scheme 17


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Introduction - History of cake baking and decorating

Basically, cake is a form of food, typically a sweet, baked dessert. Cake is often the dessert of choice for meals at ceremonial occasions, particularly weddings, 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 a complicated procedure; while at one timeconsiderable labour went into cake making (particularly the whisking of egg foams); bakingequipment and directions have been simplified that even the most amateur cook may bake acake. A finished cake is often enhanced by covering it with icing, or frosting, and toppings suchas sprinkles. Sprinkles are small firm pieces of sugar and oils that are coloured with foodcolouring. In the late 20th century, new cake decorating products became available to thepublic. These include several specialized sprinkles and even methods to print pictures andtransfer the image onto a cake.

Nonetheless, even though 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 beproperly 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 doughwas sometimes enriched with butter, eggs, and honey, which produced a sweet and cake-likebaked good. Latin poet Ovid refers to the birthday of him and his brother with party and cake inhis first book of exile, Tristia . Early cakes in England were also essentially bread: the mostobvious differences between a "cake" and "bread" were the round, flat shape of the cakes, andthe cooking method, which turned cakes over once while cooking, while bread was left uprightthroughout the baking process. Sponge cakes, leavened with beaten eggs, originated during theRenaissance, possibly in Spain.

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A chocolate cake decorated with icing,strawberries, and silvery sugar beads or Drages.

A German chocolate cake in the 60 s


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P art I

Cakes come in a variety of forms and flavours and are among favourite desserts served duringspecial 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 yourfindings.

Answer:

Generally, baking a cake offers a tasty way to practice mathematics skills, such as fractionsand ratios, in a real-world context. In this regard, we would probably calculate the proportionsor the amount of ingredients used. It could involve progressions and geometry where we needto calculate the volume measurements. At first, we would have to determine the size of thecake by calculating its radius and height or its cross-sectional area and height. For instance, afrosting recipe that calls for 2 cups cream cheese, 2 cups confectioners' sugar and 1/2 cupbutter has a cream cheese, sugar and butter ratio of 4:4:1. Identifying ratios can also help youmake recipes larger or smaller. Secondly, use as few measuring cups as possible. As such,instead of using a 3/4 cup, use a 1/4 cup three times. Hence, it requires you to work withfractions. Thirdly, calculate the surface area of the part of the cake that needs frosting. Forexample, a sheet cake in a pan only needs the top frosted, while a sheet cake on a tray needsthe top and four sides frosted. In short, all these calculations involve mathematics as it dependson how big the cake is. Furthermore, we can calculate the price index of a cake which involvesthe calculation of composite index and weightage of ingredients used for economical purposes. On the other hand, cake decorating is one of the sugar arts requiring mathematics that usesicing or frosting and other edible decorative elements to make otherwise plain cakes more

visually interesting. Alternatively, cakes can be moulded and sculpted to resemble three-dimensional persons, places and things.

The mathematics skills involve in cake baking and cake decorating are summarised as below:Geometry - To determine suitable dimensions for the cake, to assist in designing and decoratingcakes that comes in many attractive shapes and designs, to estimate volume of cake to beproducedCalculusa) Differentiation - To determine minimum or maximum amount of ingredients for cake-baking,to estimate minimum or maximum amount of cream needed for decorating, to estimateminimum or maximum size of cake produced.b) Integration - To calculate the volume of cakes with curve (or could be parabolic curves)shapesP rogressions - To determine total weight or volume of multi-storey cakes with proportionaldimensions, to estimate total ingredients needed for cake-baking, to estimate total amount of cream for decoration.Index Number - To determine the price index of the cake which involves the calculation of composite index for economical and commercial purposes. 2


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P art II

Best Bakery shop received an order from your school to bake a 5 kg of round cake as shown inDiagram 1 for the Teachers Day celebration.

1) If a kilogram of cake has a volume of 3800 cm 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 by your school.[Use = 3.142]

Answer:

1 kg = 3800 cm 3

h = 7 cm

5 kg = 3800 x 5

= 19000 cm3

V = r 2h 19000 = 3.142 x r 2 x 7

r 2= 863.872r = 29.392 cm

using r = 29.392 cmd = 2r d

= 58.783 cm

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

Given the inner dimensions of the oven is 80.0 cm in length, 60.0 cm in width and 45.0 cm inheight. Hence, maximum dimensions of cake:Maximum diameter, d = 60.0 cmMaximum height, h = 45.0 cmFirst, form a formula for d in terms of h by using the above formula for volume of cake, V =19000, that is:

Hence, a table is drawn.

4

Height,h (cm)

Diameter, d (cm)

1 155.5363

2 109.9808

3 89.7989

4 77.7682

5 69.5579

6 63.4974

7 58.7872

8 54.9904

9 51.845410 49.184911 46.8960

12 44.8995

13 43.1380

14 41.5688

15 40.1593

Height,h (cm)

Diameter,d (cm)

16 38.8841

17 37.7231

18 36.6603

19 35.6825

20 34.7790

21 33.9408

22 33.1605

23 32.4316

24 31.748725 31.1073

26 30.5032

27 29.9330

28 29.3936

29 28.8824

30 28.3969

Height,h (cm)

Diameter,d (cm)

31 27.9351

32 27.4952

33 27.0754

34 26.6743

35 26.2904

36 25.9227

37 25.5700

38 25.2313

39 24.905740 24.5924

41 24.2907

42 23.9998

43 23.7191

44 23.4480

45 23.1860


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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:

< 7 cm , > 45 cm

This is because any heights lower than 7 cm because the resulting diameter produced is toolarge to fit into the oven. Furthermore, the cake would be too short and too wide, making it lessattractive. Any heights higher than 45 cm will cause the cake being too tall to fit into the bakingoven. Besides, such heights will cause the cake will collapse if the ingredients used is notsuitable for such dimensions.

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

Answer:I would suggest the dimensions of the cake to be 10 cm in height and about 49.1849 cm indiameter. This is because a cake with such dimensions is more symmetrical and easier todecorate. Moreover, it is easy to handle.

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:

Using and cm 3, an equation is formed:

(continue on the next page) 5


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in linear form, Hence, a table is drawn to determine the plotting points:

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 2.1918 1.9418 1.6918 1.4418 1.1918 0.9418 0.6918 0.4418 0.1918

A graph of is plotted (on the next page). ii)a) If Best Bakery received an order to bake a cake where the height of the cake is 10.5 cm, useyour graph to determine the diameter of the round cake pan required.

Answer:When

cm,

According to the graph, when Therefore, = 47.86 cmb) If Best Bakery used a 42 cm diameter round cake tray, use your graph to estimate the heightof the cake obtained.

Answer:When = 42 cm, According to the graph, when Therefore, = 15.85 cm

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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 thatyou have suggested in 2(b)(ii).

Answer:h = 10 cmr = 49.1849 cm

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

8

1 cm

25.5925 cm

Diagram 1: Cake without Cream

24.5925 cm

Diagram 2: Cake with Cream

1 cm

11 cm

10 cm


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b) Suggest three other shapes for cake, that will have the same height and volume as thosesuggested in 2(b)(ii). Estimate the amount of fresh cream to be used on each of the cakes.

Answer:

* All estimations in the values are based on the assumption that the layer of cream is uniformly

thick at 1 cm.

A . Rectangular shaped cake (cuboid)

A cake without cream

A cake with cream

Estimated volume of cream used:

9

10 cm

38 cm50 cm

11 cm

52 cm

40 cm

Top view

52 cm

Side view


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B. Triangle shaped cake

Cake without cream

Cake with cream

Estima ted volume of cream used:

10

10 cm

50 cm76 cm

52 cm

78 cm

52 cm

11 cm

Side view Top view


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C. Trapezium shaped cake

Cake without cream

Cake with cream

Estimated volume of cream used:

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

Answer:Based on the values I have obtained, the triangle shaped cake requires the least amount of fresh cream (3308 cm 3). 11

10 cm

45 cm

50 cm

31 cm

11 cm

52 cm 47 cm

52 cm

33 cm

Side view Top view


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P art III Find the dimension of a 5 kg round cake that requires the minimum amount of fresh cream todecorate. Use at least two different methods including Calculus. State whether you wouldchoose to bake a cake of such dimensions. Give reasons for your answers. Answer:Method 1: Using differentiation*Assuming that the surface area of the cake is proportional to the amount of fresh creamneeded to decorate the cake.Use two equations for this method: the formula for volume of cake as in 2(a) and the formulafor the amount (volume) of fresh cream used.* The formula of surface area in contact with cream can also be used instead of amount (volume) of fresh creamused.

19000 = X A = + Y

From X : Z Substitute Z into Y :

The values, when plotted into a graph will form a minimum value (for ) that can be

obtained through differentiation.

minimum value, therefore

Substitute into Z :

Hence, and (continue on the next page) 12


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* The conjecture is proven by the following

When ,- the total surface area of the cake is 4114.2 cm 2 - the amount of fresh cream needed to decorate the cake is 4381.2 cm 3

- the ratio of total surface area of cake to amount of fresh cream needed is 0.94

When ,- the total surface area of the cake is 3156.8 cm 2 - the amount of fresh cream needed to decorate the cake is 3308.5 cm 3

- the ratio of total surface area of cake to amount of fresh cream needed is 0.94Therefore, the above conjecture is proven to be true.

Method 2: Quadratic Functions Use the two same equations as in Method 1, but only the formula for the surface area in contact incream is used as the quadratic function. Let = surface area in contact in cream, = radius of round cake:

19000 = (3.142) X

= (3.142) + 2(3.142) Y Factorise 3.142 in Y :

= (3.142)( + 2 )

completing square, with = 3.142, = 2 and = 0

Noted that , positive indicates minimum value. Hence, minimum value ,corresponding value of .Substitute

into X :

Substitute into X :

but using

Thus, both methods have calculated that and is approximately 18.22 and 18.22 respectively.

In conclusion, the dimensions of the cake that requires the minimum amount of fresh cream to

decorate is approximately 18.22 cm in height and 18.22 cm in radius.In conclusion, I would choose not to bake a cake with such dimensions because its dimensionsare not suitable (the height is too high). Besides, it will look less attractive and could be difficultto handle such a big volume (size).

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FURTHER EXPL ORA TION

Best Bakery received an order to bake a multi-storey cake for Merdeka Day celebration, asshown 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 thesecond cake is 10% less than the radius of the first cake, the radius of the third cake is 10% lessthan 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 allthese values, determine whether the volumes of the cakes form a number pattern? Explain andelaborate on the number patterns.

Answer:Volume of cake 1 Volume of cake 2

cm 3

cm 3

Volume of cake 3 Volume of cake 4

3.142 (25.11) 2 6 3.142 (22.599) 2 611886.414 cm 3 9627.995 cm 3

The values 118116.772, 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, .

, ratio,

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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:

15 kg cm 3

V erification of answer

If = 4, total volume of 4 cakes:

= 18116.772 cm 3 + 14676.585 cm 3 + 11886.414 cm 3 + 9627.995 cm 3

= 54307.766 cm 3

Hence, total mass of cakes = 14.29 kg

If = 5, total volume of 5 cakes:

= 18116.772 cm 3 + 14676.585 cm 3 + 11886.414 cm 3 + 9627.995 cm 3 + 7798.676 cm 3

= 62106.442 cm 3

Hence, total mass of cakes = 16.34 kg

Total mass of cakes must not exceed 15 kg. Therefore, maximum number of cakes needed to bemade is 4.

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Reflection

As the famous saying goes thing ventured, nothing gained . In the process of conducting thisproject, I have learnt that perseverance pays off, especially when you obtain a just reward forall 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 thingslike baking and decorating a cake, to designing and building monuments. Besides that, I havelearned many moral values that I practice. This project work had taught me to be moreconfident when doing something especially the homework given by the teacher. I also learnedto be a more disciplined student who is punctual and independent. Besides that, I would like toshow my gratitude towards my teacher, Mr Chin who has given me guidance throughout thisproject. So from now on, I will do my best in learning Additional Mathematics. Nevertheless, itis not only this subject that I will give my best but for every subject as well.

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END OFPROJECT WORK

Documents

Additional Mathematics Project Work 2011/2 (Form 5)