add math project work 2
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ADDITIONAL
MATHEMATICS
PROJECT WORK
2/2011
NAME:
IC. NUMBER:
CLASS:
TEACHER:
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Contents
No. Tittles Pg.
1 Acknowledgements
2 Objective
3 Introduction4 Part I
5 Part II
6 Part III
7 Further Exploration8 Reflection
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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.
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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
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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 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.
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PART I
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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
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PART II
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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
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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
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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.
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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
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Graph h against 1/d
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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
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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
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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
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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
Volume of cake without fresh cream
Vcream = 21 629.71 19 000
Vcream = 2 629.71 cm 5
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y Triangular-based shape
length = 57.27 cm, width = 57.27 cm, height = 13 cm
Volume of cake with fresh cream = x length x width x height
= x 57.27 x 57.27 x 13
= 21 319.04 cm6
Amount of fresh cream, Vcream = Volume of cake with fresh cream
Volume of cake without fresh cream
Vcream = 21 319.04 19 000
Vcream = 21 319.04 cm6
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y Pentagonal-based shape
length = 18.80 cm, width = 18.80 cm, height = 13 cm
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
Amount of fresh cream, Vcream = Volume of cake with fresh cream
Volume of cake without fresh cream
Vcream = 22 973.6 19 000Vcream = 3 973.6 cm
7
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PART III
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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.
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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.
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FURTHEREXPLORATION
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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
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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
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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.
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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 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
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