20
Additional Mathematics Project Work 2 Written by: ALVIN SOO CHUN KIT I/C Num : Angka Giliran: School : Date :
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Alvin Soo Chun Kit
Additional Mathematics Project Work 2
2011
TABLE OF CONTENTS
Num. 1
Part I Part II ~ Question 1 ~ Question 2 (a) ~ Question 2 (b)
2 ~ Question 2 (c) ~ Question 3 (a) ~ Question 3 (b) ~ Question 3 (c) 3
Part III
4
Further Exploration
Question
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Alvin Soo Chun Kit
Additional Mathematics Project Work 2
2011
PART I History of cake baking and decorating Although clear examples of the difference between cake and bread are easy to t o find, the precise classification has always a lways been elusive. For example, banana bread may be properly considered either a quick bread brea d 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 a sweet 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 were also essentially bread: the most obvious differences between a "cake" and "bread" were the round, flat shape of the cakes, cakes, and the cooking method, which turned cakes over once while cooking, while bread was left upright throughout 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 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, persons, places and things. In many areas of the world, decorated cakes are often a focal point of a special celebration such a s a birthday, graduation, bridal shower, shower, wedding, or anniversary. Mathematics are often used to bake and decorate cakes, especially in the following actions: y y y y y
Measurement of Ingredients Calculation of Price and Estimated Cost Estimation of Dimensions Calculation of Baking Times Modification of Recipe according to scale
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Alvin Soo Chun Kit
Additional Mathematics Project Work 2
2011
PART II 1)
1 kg = 3800 cm h = 7 cm
3
5 kg = 3800 x 5 3 = 19000 cm V = r 2 h 2 19000 = 3.142 x r x 7 r 2 = 19000 3.142 x 7 2 r = 863.872 r = 29.392 cm .
d = 2r d = 58.783 cm 2) Maximum dimensions of cake: d = 60.0 cm h = 45.0 cm a) h/cm 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
d/cm 155.5262519 109.9736674 89.79312339 77.76312594 69.5534543 63.49332645 58.78339783 54.98683368 51.84208396 49.18171919 46.89292932 44.89656169 43.13522122 41.56613923 40.15670556
h/cm 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
d/cm 38.88156297 37.72065671 36.65788912 35.68016921 34.77672715 33.93861056 33.15830831 32.42946528 31.74666323 31.10525037 30.50120743 29.93104113 29.39169891 28.88049994 28.39507881
h/cm 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
d/cm 27.93333944 27.49341684 27.07364537 26.67253215 26.2887347 25.92104198 25.56835831 25.2296896 24.90413158 24.59085959 24.28911983 23.99822167 23.71753106 23.44646466 23.18448477
b) i) h < 7 cm , h > 45 cm This is because any heights lower 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 cm will cause the cake being too tall to fit into the baking oven
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Alvin Soo Chun Kit
c) i)
Additional Mathematics Project Work 2 2
V = r h V = 19000 cm
3
d
r = /2 d
2
19000 = 3.142 x ( /2) x h d2 = 19000 2 4 3.142 x (d /4) .
2
d =
76000 3.142 x h
.
d = 155.53 x h-1/2 1
log10 d = - /2 log10 h + log 10 155.53
log10 h
log10 d
1
1.691814
2
1.191814
3
0.691814
4
0.191814
c) ii) a) When h = 10.5 cm, log 10 h = 1.0212 According to the graph, log 10 d = 1.7 when log 10 h = 1.0212 Therefore, d = 50.12 cm b) When d = 42 cm, log10 d = 1.6232 According to the graph, log 10 h = 1.2 when log 10 d = 1.6232 Therefore, h = 15.85 cm
2011
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Alvin Soo Chun Kit
Additional Mathematics Project Work 2
2011
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Alvin Soo Chun Kit
3) a)
Additional Mathematics Project Work 2
2011
h = 29 cm r = 14.44 cm 14.44 cm
29 cm
Diagram 1: Cake without Cream
1 cm 15.44 cm 1 cm
30 cm
Diagram 2: Cake with Cream
To calculate volume of crea m used, the cream is symbolised as the larger cylinder and the cake is symbolised as the s maller cylinder. 2
Vcream = 3.142 x 15.44 x 30 ± 19000 = 22471 ± 19000 3 = 3471 cm
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Alvin Soo Chun Kit
Additional Mathematics Project Work 2
3) b) i) Square shaped cake
Estimated volume of cream used = 30 x 27.6 x 27.6 - 19000 = 22852.8 ± 19000 3 = 3852.8 cm b) ii) Triangle shaped cake
Estimated volume of cream used = ½ x 39.7 x 39.7 x 30 ± 19000 = 23641.4 ± 19000 = 4641.4 cm3 b) iii) Trapezium shaped cake
2011
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Alvin Soo Chun Kit
Additional Mathematics Project Work 2
2011
PART III Method 1: By comparing values of height against volume of cream used
h/cm
volume of cream 3 used/cm
h/cm
volume of cream 3 used/cm
h/cm
volume of cream 3 used/cm
1
19983.61
18
3303.66
35
3629.54
2
10546.04
19
3304.98
36
3657.46
3
7474.42
20
3310.62
37
3685.67
4
5987.37
21
3319.86
38
3714.13
5
5130.07
22
3332.12
39
3742.81
6
4585.13
23
3346.94
40
3771.67
7
4217.00
24
3363.92
41
3800.67
8
3958.20
25
3382.74
42
3829.79
9
3771.41
26
3403.14
43
3859.01
10
3634.38
27
3424.89
44
3888.30
11
3533.03
28
3447.80
45
3917.65
12
3458.02
29
3471.71
46
3947.04
13
3402.96
30
3496.47
47
3976.46
14
3363.28
31
3521.98
48
4005.88
15
3335.70
32
3548.12
49
4035.31
16
3317.73
33
3574.81
50
4064.72
17
3307.53
34
3601.97
3
According to the table above, the minimum volume of cream used is 3303.66 cm when h = 18cm. When h = 18cm, r = 18.3 cm
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Alvin Soo Chun Kit
Additional Mathematics Project Work 2
2011
Method 2: Using differentiation Assuming that the surface area of the cake is proportionate to the amount of fresh cream needed to decorate the cake.* Formula for surface area 2 = r + 2rh h = 19000 / 3.142r 2 Surface area in contact with cream 2 2 = r + 2r(19000 / 3.142r ) = r 2 + (38000/r) The values, when plotted into a graph will from a minimum value that can be obtained through differentiation. dy = 0 dx 2
dy = 2r ± (38000/r ) dx 2 0 = 2r ± (38000/r ) 0 = 6.284r 3 ± 38000 3 38000 = 6.284r 3 6047.104 = r 18.22 = r When r = 18.22 cm, h = 18.22 cm The dimensions of the cake that requires the minimum amount of fresh cream to decorate is approximately 18.2 cm in height and 18.2 cm in radius. I would bake a cake of such dimensions because the cake would not be too large for the cutting or eating of said cake, and it would not be too big to bake in a conventional oven.
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Alvin Soo Chun Kit
Additional Mathematics Project Work 2
2011
FURTHER EXPLORATION
a) Volume of cake 1 2 = r h = 3.142 x 31 x 31 x 6 3 = 18116.772 cm
Volume of cake 2 2 = r h 2 = 3.142 x (0.9 x 31) x 6 2 = 3.142 x (27.9) x 6 = 14676.585 cm 3
Volume of cake 3 2 = r h = 3.142 x (0.9 x 0.9 x 31) 2 x 6 2 = 3.142 x (25.11) x 6 3 = 11886.414 11886.414 cm cm
Volume of cake 4 2 = r h = 3.142 x (0.9 x 0.9 x 0.9 x 31) 2 x 6 2 = 3.142 x (22.599) x 6 3 = 9627.995 9627.995 cm cm
The values 118116.772, 14676.585, 11886.414, 9627.995 form a number pattern. The pattern formed is a geometrical progression. progression. This is proven by the fact that t hat there is a common ratio between subsequent numbers, r = 0.81. 14676.585 = 0.81 18116.772
11886.414 = 0.81 14676.585
9627.995 = 0.81 11886.414 .
b) Sn = a(1-r n) = 18116.772 ( 1-0.8 1-0. 8 n) 1-r 1-0.8 15 kg = 57000 cm
3
n
57000 > 18116.772(1-0.8 18116.772(1- 0.8 ) 0.2 n
11400 > 18116.772(1-0.8 18116.772(1 -0.8 )
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Alvin Soo Chun Kit
Additional Mathematics Project Work 2
Verification of answer If n = 4 Total volume of 4 cakes 3 3 3 3 = 18116.772 cm + 14676.585 cm + 11886.414 cm + 9627.995 cm 3 = 54307.766 cm Total mass of cakes = 14.29 kg If n = 5 Total volume of 5 cakes = 18116.772 cm 3 + 14676.585 cm3 + 11886.414 cm 3 + 9627.995 cm 3 + 7798.676 cm 3 3 = 62106.442 cm Total mass of cakes = 16.34 kg Total mass of cakes must not exceed 15 kg. Therefore, maximum number of cakes needed to be made = 4
2011
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Alvin Soo Chun Kit
Additional Mathematics Project Work 2
2011
Reflection In the process of conducting this project, I have lear nt that perseverance pays pays off, especially when you obtain a just reward for all your hard work. For me, succeeding in completing this project work has been reward enough. enough. I have also learnt t hat mathematics is used everywhere in daily life, from the most simple things like baking and decorating a cake, to designing and building monuments. monuments. Besides that, I have learned many moral values that I practice. This project work had taught me to be more confident when doing something especially the homework given by the teacher. I also learned to be a more disciplined student who is punctual and independent.