Add Math Project Work 2

8/6/2019 Add Math Project Work 2

1/28

ADDITIONAL

MATHEMATICS

PROJECT WORK

2/2011

NAME:

IC. NUMBER:

CLASS:

TEACHER:


2/28

Contents

No. Tittles Pg.

1 Acknowledgements

2 Objective

3 Introduction4 Part I

5 Part II

6 Part III

7 Further Exploration8 Reflection


3/28

Acknowledgements

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 financial support on making of

this project.


4/28

Objectives

y Appreciate the importance of mathematics in everydaylives.

y Improve problem-solving skills and thinking skills.

y Develop positive attitude and personalities such asconfidence.

y Develop mathematical knowledge


5/28

Introduction


6/28

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 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 the amateur cook can

easily become and expert baker. There are five basic types of cake, depending on the substance used for

leavening.

The most primitive peoples in the world began making cakes shortly after they discovered flour. In

medieval England, the cakes that were described in writings were not cakes in the conventional sense.

They were described as flour-based sweet foods as opposed to the description of breads, which were just

flour-based foods without sweetening.

Bread and cake were somewhat interchangeable words with the term "cake" being used for smaller

breads. The earliest examples were found among the remains of Neolithic villages where archaeologistsdiscovered simple cakes made from crushed grains, moistened, compacted and probably cooked on a hot

stone. Today's version of 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 cakes were 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 as an offering to their gods. Placenta

was more like a cheesecake, baked on a pastry base, 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 used for the smaller breads. Cakes wereusually baked for special occasions because they were made with the finest and most expensive

ingredients available to the cook. The wealthier you were, the more likely you might consume cake on a

more frequent basis.

By the middle of the 18th century, yeast had fallen into disuse as a raising agent for cakes in favour of

beaten eggs. Once as much air as possible had been beaten in, the mixture would be poured into moulds,

often very elaborate creations, but sometimes as simple as two tin hoops, set on parchment paper on a

cookie sheet. It is from these cake hoops that our modern cake pans developed.

Cakes were considered a symbol of well being by early American cooks on the east coast, with each region

of the country having their own favourites.

By the early 19th century, due to the Industrial Revolution, baking ingredients became more affordableand readily available because of mass production and the railroads. Modern leavening agents, such as

baking soda and baking powder were invented.


7/28

PART I


8/28

Cakes cone 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 not only 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.

Geometry

To determine suitable dimensions for the cake To assist in designing and decorating cakes that comes in many

attractive shapes and designs

To estimate volume of cake to be produced

Calculus (differentiation)

To determine minimum or maximum amount of ingredients for cake -baking

To estimate minimum or maximum amount of cream needed fordecorating

To estimate minimum or maximum size of cake produced,

Progressions To determine total weight/volume of multi-storey cakes with

proportional dimensions

To estimate total ingredients needed for cake-baking To estimate total amount of cream for decoration


9/28

PART II


10/28

Bes

B e y s ece ve e y sc

e 5

c e s s D 1

e e c e s D y ce e

1 ! c e s v e 3800c " e e e c e s

e 7

0 c

"c

c

e

e

e

e

e

y

e

se

e

5 c e e e y y sc [Use = 3 142]

Volu #$

1 %& ' (

%

$

= 3 800'

#

Volu#

$

5%

& ' (

%

$

= 3 800)5

0

Volu#

$

5%

& ' (

%

$

= 19 000'

#

Volu #$

of' (

%

$

1 V = rh19000= (3

2142

0

(70

r

19000=21 2 994r

r =19000/212994

r =863 2 872

r =

r =29 2 39 cm

d=2r

d=2(29 2 3920

d=58278 cm


11/28

2) The cake will be baked in an oven with inner dimensions of 80.0 cm inlength, 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 ofheights, h cm, and the corresponding values of diameters of the baking trayto be used, d cm. Tabulate your answers.

Volume of cake, V = rh19 000 = 3.142hr

r = 19 000/3.142h

r =

d = 2r

d = 2( )

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

1.0 155.53

2.0 109.97

3.0 89.80

4.0 77.76

5.0 66.55

6.0 63.50

7.0 58.78

8.0 54.99

9.0 51.84

10.0 49.18

11.0 46.89

12.0 44.90

13.0 43.14

14.0 41.57

15.0 40.16

16.0 38.80

17.0 37.72

18.0 36.66

19.0 35.68

20.0 34.77


12/28

b) Based on the values in your table,i. State the range of heights that is NOT suitable for the cake and

explain your answers.

h>7 is not suitable for the cakes, because the height is too short for

a cake as its diameter of the cake also will be too large to fit into the

oven.

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

h = 12cm, d = 44.90cm, because the dimensions of the cake is

suitable for a cake as it not too short or too high and it also can fit

into the oven. Moreover, the cake will be easy to handle and the

cake also will take a short time to be completely baked.


13/28

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.]

d = 2r

r = d/2

Volume of cake, V = rhV = (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 x 10

-4

3.31 x 10

-4

4.96 x 10

-4

6.64 x 10

-4

8.27 x 10

-4


14/28

Graph h against 1/d


15/28

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.

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 cake tray, use your

. graph to estimate the height of the cake obtained

d = 42 cm

d = 1 764

1/d = 5.67x 104

h = 13.90 cm


16/28

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 cake usingthe 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 cm3


17/28

23.45 cm

13 cm

Cake with fresh cream:

h = 13 cm

r = 23.45 cm

Volume of cake, V = rhV = (3.142)(23.45)(13)

V = 22 461.32cm4

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

Volume of cake without fresh cream

Vcream = 22 461.32 19 000

Vcream = 3461.32 cm4


18/28

b) Suggest three other shapes for cake, that will have the same height andvolume as those suggested in 2(b)(ii). Estimate the amount of fresh cream

to be used on each of the cakes.

y Square-based shape

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

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

= 40.79 x 40.79 x 13

= 21 629.71 cm5

Amount of fresh cream, Vcream

= Volume of cake with fresh cream


Vcream = 21 629.71 19 000

Vcream = 2 629.71 cm 5


19/28

y Triangular-based shape


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

= x 57.27 x 57.27 x 13

= 21 319.04 cm6



Vcream = 21 319.04 19 000

Vcream = 21 319.04 cm6


20/28

y Pentagonal-based shape


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

= 5 x 18.80 x 18.80 x 13

= 22 973.6 cm7



Vcream = 22 973.6 19 000Vcream = 3 973.6 cm

7


21/28

PART III


22/28

Find the dimension of a 5 kg round cake that requires the minimum amount of

fresh cream to decorate.8

se at least two different methods including Calculus.

State whether you would choose to bake a cake of such dimensions. 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)h

19 000 = (3.142)h9

h9

= 19 000/3.142

h = h = 18.22 cm

V = rh

19 000 = (3.142)(18.22)r

19 000 = 57.247r

r = 19 000/57.247

r = r = 18.22 cm

d = 2r

d = 2(18.22)d = 36.44 cm

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


23/28

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 = 0

6.284r = 38 000/r

6.284r@

= 38 000

r@

= 38 000/6.284

r = r = 18.22 cm

d = 2r

d = 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 such that dimensionsbecause it is not suitable for a cake as the height too high. Moreover, it

also will be difficult to handle.


24/28

FURTHEREXPLORATION


25/28

Bes A BB C e D y D ece E ve F B G H D F e D A H I B C e B P Q R A E -s A H D ey c B C e S H D T e D F e C B DB y

ce R e I D B A E H G U B s s V H W G E G DE B X D B P 2 Y

` V e V e E X V A H S e B c V c B C e E s 6 Y 0 c P B G F A V e D B F E Q s H S A V e R B D X es A c B C e E s 31 Y 0 c P Y

` V e D B F E Q s H S A V e sec H G F c B C e E s 10% R ess A V B G A V e D B F E Q s H S A V e S E D s A c B C e U A V e

D B F E Q s H S A V e A V E D F c B C e E s 10% R ess A V B G A V e D B F E Q s H S A V e sec H G F c B C e B G F s H

H G Y

B a F

E G F A V e v

H R Q P e

H S A V e

S E Ds

A U A V e sec

H G F U A V e

A V E D F B G F A V e

S H Q D A V c

B Ces

YBy

c H P b B D E G X B R R A V ese v B R Q es U F e A e D P E G e W V e A V e D A V e v H R Q P es H S A V e c B C es

S H D P B G Q P I e D b B A A e D G ? c d b R B E G B G F e R B I H D B A e H G A V e G Q P I e D b B A A e D G s Y


26/28

a = 31.00 cm

r = 90/100

r = 0.9

Radius of cake, Tn = arn-1

Volume of cake, V = rh

Cake Radius (cm) Volume (cme

)

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)

= 14 674.59

3rd

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)e

= 22.60

V = (3.142)(22.60)(6)

= 9 628.85

The volume of the cakes of 18 116.77, 14 674.59, 11 886.41, 9 628.85,

form a number pattern, which is a geometric progression with a

common ratio of 0.81 .

14 674.59/18 116.77 = 0.81

11 886.41/14 674.59 = 0.81

9 628.85/11 886.41 = 0.81


27/28

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.

Volume 1 kg cake = 3 800 cmf

Volume 15 kg cake = 3 800(15)

Volume 15 kg cake = 57 000 cmf

a = 18 116.77 cmf

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.


28/28

Reflection

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

worth it.

It never occurs to me that Mathematics is used in our

daily life like baking and decorating a cake. From now, I

will more appreciate the importance of mathematics, even

though I find it difficult to understand.

Besides that, this project had thought me so many

moral values. I learned to be more disciplined student by

using my time wisely in order to complete this project on

time.

The essence of mathematics is not to make simple things complicated, but to

make complicated things simple. ~S. Gudder

Add Math Project Work 2

Documents