additional mathematics project work 2/2011

8/6/2019 Additional Mathematics Project Work 2/2011

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ADDITIONAL

MATHEMATICS

PROJECT WORK

2/2011

NAME:

IC. NUMBER:

CLASS:

TEACHER:


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Contents

No.

Tittles Pg.

1 Acknowledgement

s2 Objective3 Introduction4 Part I

5 Part II6 Part III7 Further

Exploration8 Reflection


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Acknowledgeme

nts

First of all, I would like to express my

gratitude to my Additional Mathematic teachers,

Puan Fatimah binti Hashim. Without her advice

and guidance, this project would not have been

completed.

Special thanks to my friends, who was very

helpful with their invaluable suggestions and

assistance on this project.

Last but not least, I would like to say thank

you to both of my parents for providing financialsupport on making of this project.


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Objectives

Appreciate the importance of mathematics in

everyday lives.

Improve problem-solving skills and thinking

skills.

Develop positive attitude and personalities

such as confidence.

Develop mathematical knowledge


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Introduction


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CAKE HISTORY

Cakes are made from various combinations of refined flour, some form of shortening,

sweetening, eggs, milk, leavening agent, and flavouring. There are literally thousands of


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cakes recipes (some are bread-like and some rich and elaborate) and many are centuries

old. Cake making is no longer a complicated procedure.

Baking utensils and directions have been so perfected and simplified that even theamateur cook can easily become and expert baker. There are five basic types ofcake, depending on the substance used for leavening.

The most primitive peoples in the world began making cakes shortly after theydiscovered flour. In medieval England, the cakes that were described in writings werenot cakes in the conventional sense. They were described as flour-based sweet foodsas opposed to the description of breads, which were just flour-based foods withoutsweetening.

Bread and cake were somewhat interchangeable words with the term "cake" beingused for smaller breads. The earliest examples were found among the remains ofNeolithic villages where archaeologists discovered simple cakes made from crushedgrains, moistened, compacted and probably cooked on a hot stone. Today's versionof this early cake would be oatcakes, though now we think of them more as a biscuit

or cookie.

Cakes were called "plakous" by the Greeks, from the word for "flat." These cakeswere usually combinations of nuts and honey. They also had a cake called "satura,"which was a flat heavy cake.

During the Roman period, the name for cake (derived from the Greek term) became"placenta." They were also called "libum" by the Romans, and were primarily used asan offering to their gods. Placenta was more like a cheesecake, baked on a pastrybase, or sometimes inside a pastry case.

The terms "bread" and "cake" became interchangeable as years went by. The words

themselves are of Anglo Saxon origin, and it's probable that the term cake was usedfor the smaller breads. Cakes were usually baked for special occasions because theywere made with the finest and most expensive ingredients available to the cook. Thewealthier you were, the more likely you might consume cake on a more frequentbasis.

By the middle of the 18th century, yeast had fallen into disuse as a raising agent forcakes in favour of beaten eggs. Once as much air as possible had been beaten in, themixture would be poured into moulds, often very elaborate creations, but sometimesas simple as two tin hoops, set on parchment paper on a cookie sheet. It is fromthese cake hoops that our modern cake pans developed.

Cakes were considered a symbol of well being by early American cooks on the eastcoast, with each region of the country having their own favourites.

By the early 19th century, due to the Industrial Revolution, baking ingredients becamemore affordable and readily available because of mass production and the railroads.Modern leavening agents, such as baking soda and baking powder were invented.


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PART I

Cakes cone in a variety of forms and flavours and are among

favourite desserts served during special occasions such asbirthday parties, Hari Raya, weddings and etc. Cakes are


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treasured not only because of their wonderful taste but also inthe art of cake baking and cake decorating. Find out how,mathematics is used in cake baking and cake decorating andwrite about your findings.

Geometry

To determine suitable dimensions for the cake

To assist in designing and decorating cakes thatcomes in many attractive shapes and designs

To estimate volume of cake to be produced

Calculus (differentiation)

To determine minimum or maximum amount ofingredients for cake-baking

To estimate minimum or maximum amount of creamneeded for decorating

To estimate minimum or maximum size of cakeproduced,

Progressions

To determine total weight/volume of multi-storeycakes with proportional dimensions

To estimate total ingredients needed for cake-baking

To estimate total amount of cream for decoration


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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.


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1) If a kilogram of cake has a volume of 3800cm, and theheight of the cake is to be 7.0 cm, calculate the diameter ofthe baking tray to be used to fit the 5 kg cake ordered byyour school. [Use = 3.142]

Volume 1 kg cake = 3 800 cm

Volume 5 kg cake = 3 800(5)


Volume of cake, V = rh

19 000 = (3.142)(7)r

19 000 = 21.994r

r = 19 000/21.994

r = 863.872

r =

r = 29.39 cm

d = 2r

d = 2(29.392)

d = 58.78 cm

2) The cake will be baked in an oven with inner dimensions of80.0 cm in length, 60.0 cm in width and 45.0 cm in height.


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a) If the volume of cake remains the same, explore by usingdifferent values of heights, h cm, and the correspondingvalues of diameters of the baking tray to be used, d cm.

Tabulate your answers.


19 000 = 3.142hr

r = 19 000/3.142h

r =

d = 2r

d = 2( )

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

1.0 155.532.0 109.973.0 89.80

4.0 77.765.0 66.556.0 63.507.0 58.788.0 54.999.0 51.8410.0 49.1811.0 46.8912.0 44.90

13.0 43.1414.0 41.5715.0 40.1616.0 38.8017.0 37.7218.0 36.6619.0 35.6820.0 34.77

b) Based on the values in your table,


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i. State the range of heights that is NOT suitable for thecake and explain your answers.

h>7 is not suitable for the cakes, because theheight is too short for a cake as its diameter ofthe cake also will be too large to fit into the oven.

ii. Suggest the dimensions that you think most suitable forthe cake. Give reasons for your answer.

h = 12cm, d = 44.90cm, because the dimensionsof the cake is suitable for a cake as it not tooshort or too high and it also can fit into the oven.Moreover, the cake will be easy to handle and thecake also will take a short time to be completelybaked.

c)


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i. Form an equation to represent the linear relationbetween h and d. Hence, plot a suitable graph based onthe equation that you have formed. [You may draw yourgraph with the aid of computer software.]

d = 2r

r = d/2


V = (d/2)h

V = (d/4)h

19 000 = (d/4)h

76 000 = dh

dh = 76 000/

h = (76 000/) (1/d)

Y = m X

h 4 8 12 16 20

d 77.76 54.99 44.90 38.80 34.77

1/d 1.65 x10-4

3.31 x10-4

4.96 x10-4

6.64 x10-4

8.27 x10-4


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Graph h against 1/d

a) If Best Bakery received an order to bake a cakewhere the height


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of the cake is 10.5 cm, use your graph todetermine the diameter

of the round cake pan required.

h = 10.5 cm

1/d = 4.35x104

d = 4 350

d =

d = 65.95 cm

b) If Best Bakery used a 42 cm diameter round caketray, use your . graph toestimate the height of the cake obtained

d = 42 cm

d = 1 764

1/d = 5.67x 104

h = 13.90 cm


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3. Best Bakery has been requested to decorate the cake withfresh cream. The thickness of the cream is normally set to auniform layer of about 1 cm.

a) Estimate the amount of fresh cream required to decorate thecake using the dimensions that you have suggested in 2(b)(ii).

22.45 cm

12 cm

Cake without fresh cream:

h = 12 cm

d = 44.9 cm

r = 44.9/2

r = 22.45 cm

Volume of cake, V = 19 000 cm


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

Square-based shape

length = 40.79 cm, width = 40.79 cm, height = 13 cm

Volume of cake with fresh cream = length x width xheight

= 40.79 x 40.79 x 13

= 21 629.71 cm

Amount of fresh cream, Vcream = Volume of cake withfresh cream

Volume of cakewithout fresh cream

Vcream = 21 629.71 19 000

Vcream = 2 629.71 cm


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Triangular-based shape

length = 57.27 cm, width = 57.27 cm, height =13 cm

Volume of cake with fresh cream = x length xwidth x height

= x 57.27 x 57.27 x 13


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= 21 319.04 cm

Amount of fresh cream, Vcream = Volume of cake

with fresh cream Volume of

cake without fresh cream

Vcream = 21 319.04 19000

Vcream = 21 319.04cm

Pentagonal-based shape

length = 18.80 cm, width = 18.80 cm, height =13 cm


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Volume of cake with fresh cream = 5 x length xwidth x height

= 5 x 18.80 x 18.80 x 13

= 22 973.6 cm

Amount of fresh cream, Vcream = Volume of cakewith fresh cream

Volume ofcake without fresh cream

Vcream = 22 973.6 19 000

Vcream = 3 973.6 cm

PART III


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Find the dimension of a 5 kg round cake that requires theminimum amount of fresh cream to decorate. Use at least twodifferent methods including Calculus.

State whether you would choose to bake a cake of suchdimensions. Give reasons for your answers.

METHOD 1: Quadratic function

Vcream, f(r) = Surface area of the cake= r + 2rh

Formula: f(x) = a(x + (b/2a)) + 4ac-b/4a

a = , b = 2h, c = 0

f(r) = (r+(2h/2)) + 4(0)-(2h)/4= (r + h) - 4h/4= (r + h) - h

Minimum value (-h, -rh)

V = rh

19 000 = (3.142)(-h)h19 000 = (3.142)h


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h = 19 000/3.142

h =

h = 18.22 cm

V = rh19 000 = (3.142)(18.22)r19 000 = 57.247r

r = 19 000/57.247

r =

r = 18.22 cm

d = 2rd = 2(18.22)d = 36.44 cm

So, the dimension is h = 18.22 cm, r = 18.22cm, d =36.44cm.

METHOD 2: Differentiation

Vcream = Surface area of the cake

= r + 2rh

V = rh19 000 = rh

h = 19 000/r

Vcream = r + 2r(19 000/ r)= r + 2(19 000/r)= r + 38 000/r-1

dV/dr = 2(3.142)r 38 000/r= 6.284r 38 000/r

Minimum value, dV/dr = 0

6.284r 38 000/r = 06.284r = 38 000/r

6.284r = 38 000r = 38 000/6.284

r =

r = 18.22 cm


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d = 2rd = 2(18.22)d = 36.44 cm

h = 19 000/(3.142)(18.22)

h = 18.22 cm

So, the dimensions is h = 18.22 cm, r = 18.22cm, d =36.44cm.

Therefore, I would not choose to bake a cake suchthat dimensions because it is not suitable for a cake

as the height too high. Moreover, it also will bedifficult to handle.

FURTHER


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EXPLORATION

Best Bakery received an order to bake a multi-storey cake forMerdeka Day celebration, as shown in Diagram 2.


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The height of each cake is 6.0 cm and the radius of the largestcake is 31.0 cm. The radius of the second cake is 10% less thanthe radius of the first cake, the radius of 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 thefourth cakes. By comparing all these values, determinewhether the volumes of the cakes form a number pattern?Explain and elaborate on the number patterns.

a = 31.00 cm

r = 90/100


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r = 0.9

Radius of cake, Tn = arn-1


Cake Radius (cm) Volume (cm)

1st T1 = a= 31.00

V = (3.142)(31.00)(6)

= 18 116.77

2nd

T2 = ar2-1

= 31.00(0.9)= 27.90

V = (3.142)(27.90)(6)

= 14 674.593rd T3 = ar3-1

= 31.00(0.9)= 25.11

V = (3.142)(25.11)(6)

= 11 886.41

4th T4 = ar4-1

= 31.00(0.9)

= 22.60

V = (3.142)(22.60)(6)

= 9 628.85

The volume of the cakes of 18 116.77, 14 674.59, 11886.41, 9 628.85, form a number pattern, whichis a geometric progression with a common ratio of0.81 .

14 674.59/18 116.77 = 0.81

11 886.41/14 674.59 = 0.81

9 628.85/11 886.41 = 0.81


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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 bakeryneeds to bake. Verify your answer using other methods.


Volume 15 kg cake = 3 800(15)


a = 18 116.77 cm

r = 0.81

Sn 57 000

a(1- rn)/1- r 57 000

18 116.77(1-0.81n)/(1-0.81) 57 000

18 116.77(1-0.81n)/0.19 57 000

18 116.77(1-0.81n) 10 830

1-0.81n 0.5978

-0.81n -0.4022

0.81n 0.4022

n log 0.81 log 0.4022

n log 0.4022/log 0.81

n 4.3223

Therefore, the maximum number of cakes is 4.


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Reflection

I spent so many hours doing this project.But, it was worth it.

It never occurs to me that Mathematics isused in our daily life like baking and decorating

a cake. From now, I will more appreciate theimportance of mathematics, even though I findit difficult to understand.

Besides that, this project had thought meso many moral values. I learned to be more

disciplined student by using my time wisely inorder to complete this project on time.

The essence of mathematics is not to make simple thingscomplicated, but to make complicated things simple. ~S.

Gudder

additional mathematics project work 2/2011

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