ADDTIONAL MATHEMATICS
PROJECT WORK 2/2011
Title: The Art of Cake
Name : Siti Maisarah binti Zolkanine
Class : 5 Arif
I.C. No. :940409-01-5038
Teacher : Puan Zainon binti Hairon
CONTENT
| Title | Page |
| Preface | 1 |
| Appreciation | 2 |
| Objective | 3 |
| Introduction | 4–5 |
| Part I | 6 – 7 |
| Part II | 8 – 17 |
| Part III | 18 – 19 |
| Further Exploration | 20 – 21 |
| Conclusion | 22 |
| Reflection | 23 |
| Reference | 24 |
PREFACE
I have done many researches throughout the internet and discussing with a friend who have helped me a lot in completing this project. Through the completion of this project, I have learned many skills and techniques. This project really helps me to understand more about the uses of progression in our daily life. This project also helped expose the techniques of application of additional mathematics in real life situations.
While I was conducting this project work, I have gained consciousness in many other things in my life. Completing this project work in a team had gave me a chance to know each other and broaden my view on Mathematics. I’m now knows the correct way to apply mathematic knowledge in my daily life to solve a lot of problems.
Besides that, I also have learnt to accept other ideas from different people to make out all the possible results. Then, I know which is the best after those comparison made.
APPRECIATION
First of all, I want to express my utmost gratitude to Allah, for giving me the strength and health to do this project.
I would like to thank my parents for providing me everything, such as money, to buy anything that related to this project work and their advise which was to support me, which is the most needed to complete this project work. I also thanked to my grandmother, my aunt and my uncle who had gave me space and let me access the internet at their house.
Not forgotten my Additional Mathematics teacher, Puan Zainon for guiding me and my friends throughout this project. We had some difficulties in doing this project work, but she taught us patiently until we knew what to do. She tried and tried to teach us until we understand what we supposed to do with this project work. Thank you also to my tuition teacher, Puan Faridah for the extra information and guidance throughout this project.
Last but not least, my friends who were doing this project with me and sharing our ideas. They were very helpful that when we combined and discussed together, we had this project work done.
OBJECTIVE
The aims of carrying out this project work are :
§ to apply ad adapt a variety of problem-solving strategies to solve problems ;
§ to improve thinking skills ;
§ to promote effective mathematical communication ;
§ to develop mathematical knowledge through problem solving in a way that increase student’s interest and confidence ;
§ to use the language of mathematics to express mathematical ideas precisely ;
§ to provide learning environment that stimulates and enhances effective learning ;
§ to develop positive attitude towards mathematics ;
§ to learn the way to reply the formulas of mathematics in our daily life accurately.
INTRODUCTION
The history of cake dates back to ancient times. The first cakes were very different from what we eat today. They were more bread-like and sweetened with honey. Nuts and dried fruits were often added. According to the food historians, the ancient Egyptians were the first culture to show evidence of advanced baking skills. The Oxford English Dictionary traces the English word cake back to the 13th century. It is a derivation of 'kaka', an Old Norse word. Medieval European bakers often made fruitcakes and gingerbread. These foods could last for many months.
According to the food historians, the precursors of modern cakes (round ones with icing) were first baked in Europe sometime in the mid-17th century. This is due to primarily to advances in technology (more reliable ovens, manufacture/availability of food molds) and ingredient availability (refined sugar). At that time cake hoops--round molds for shaping cakes that were placed on flat baking trays--were popular. They could be made of metal, wood or paper. Some were adjustable. Cake pans were sometimes used. The first icing were usually a boiled composition of the finest available sugar, egg whites and [sometimes] flavorings. This icing was poured on the cake. The cake was then returned to the oven for a while. When removed the icing cooled quickly to form a hard, glossy [ice-like] covering. Many cakes made at this time still contained dried fruits (raisins, currants, citrons).
It was not until the middle of the 19th century that cake as we know it today (made with extra refined white flour and baking powder instead of yeast) arrived on the scene. A brief history of baking powder. The Cassell's New Universal Cookery Book [London, 1894] contains a recipe for layer cake, American (p. 1031). Butter-cream frostings (using butter, cream, confectioners [powdered] sugar and flavorings) began replacing traditional boiled icings in first few decades 20th century. In France, Antonin Careme [1784-1833] is considered THE premier historic chef of the modern pastry/cake world.
Height, h
| Diameter, d |
A basic round cake
Example of cakes’ shapes in 21st century.
| heart | triangle | pentagon |
| round | camera | Jabba the Hut |
| Rectangle | Umbrella | Strawberry |
| Fish | Flower | Car |
PART I
Question
Part I
Cakes come 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.
Answers:
1. 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.
Volume and Surface Area of a Rectangular Solid
Volume = L*W*H
Surface Area = 2(L*W + H*W + H*L)
Surface Area = 2(L*W + H*W + H*L)
Volume and Surface Area of a Sphere
Volume = (4/3)* Pi * r 3
Surface Area = 4 * Pi * r 2
Surface Area = 4 * Pi * r 2
Volume and Surface Area of a Right Circular Cylinder
Volume = Pi * r 2 * h
Surface Area = 2 * Pi * r * h
Surface Area = 2 * Pi * r * h
Volume and Surface Area of a Right Circular Cone
2. Calculus (differentiation)
To determine minimum or maximum amount of ingredients for cake-baking, to estimate minimum or maximum amount of cream needed for decorating, to estimate minimum or maximum size of cake produced.
Differentiation Formulae
- d/dx c = 0, c constant
- d/dx cf(x) = cf'(x), c constant
- d/dx [f(x) ± g(x)] = f'(x) ± g'(x)
- d/dx [f(x) * g(x)] = f(x) * g'(x) + g(x) * f'(x) (product rule)
- d/dx [f(x) / g(x)] = (g(x)f'(x) - f(x)g'(x))/([g(x)]2) (quotient rule)
- d/dx f[g(x)] = f'[g(x)] * g'(x)
OR
for u = g(x), d/dx f(u) = f'(u) * u' = f'(u) * g'(x)
OR
dy/dx = dy/du * du/dx (these are all chain rule)
3. 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.
Arithmetic Progression Formulae
Tn = a + (n-1)*d
d = T(n+1) - T(n)
T(m+n) - T(m-n) = 2T(m)
S(n) = (n/2)*[2a+ (n-1)d] d = T(n+1) - T(n)
T(m+n) - T(m-n) = 2T(m)
= (n/2)*(a + l)
T(n) = S(n+1) - S(n)
Geometric Progression Formulae
Tn = ar^(n-1)
r = T1/T2
Sn = =
S∞ =
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. (Diagram 11)
1) If a kilogram of cake has a volume of 3800 , 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 π = 3.142]
Answer:
Volume of 5kg cake = Base area of cake x Height of cake
3800 x 5 = (3.142)( )² x 7
(3.142) = ( )²
863.872 = ( )²
= 29.392
d = 58.784 cm
2) The cake will be baked in an oven with inner dimensions of 80.0 cm in length, 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 of heights, h cm, and the corresponding values of diameters of the baking tray to be used, d cm. Tabulate your answers.
Answer:
First, form the formula for d in terms of h by using the above formula for volume of cake, V = 19000, that is:
19000 = (3.142)(d/2)²h
=
= d²
d =
| Height,h (cm) | Diameter,d(cm) |
| 1.0 | 155.53 |
| 2.0 | 109.98 |
| 3.0 | 89.80 |
| 4.0 | 77.77 |
| 5.0 | 68.56 |
| 6.0 | 63.49 |
| 7.0 | 58.78 |
| 8.0 | 54.99 |
| 9.0 | 51.84 |
| 10.0 | 49.18 |
b) Based on the values in your table,
a) state the range of heights that is NOT suitable for the cakes and explain your answers.
Answer:
h< 7cm is NOT suitable, because the resulting diameter produced is too large to fit into the oven. Furthermore, the cake would be too short and too wide, making it less attractive.
| Too wide |
| Too short |
ii) suggest the dimensions that you think most suitable for the cake. Give reasons for your answer.
Answer:
| Hmm.. look nice and delicious!! |
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:
19000 = (3.142)( )²h
=
= d²
d =
d =
log d =
log d = log h + log 155.53
| Log h | 0 | 1 | 2 | 3 | 4 |
| Log d | 2.19 | 1.69 | 1.19 | 0.69 | 0.19 |
| Graph log d against log h |
(ii)
(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.
Answer:
h = 10.5cm, log h = 1.021, log d = 1.680, d = 47.86cm
(b) If Best Bakery used a 42 cm diameter round cake tray, use your graph to estimate the height of the cake obtained.
Answer:
d = 42cm, log d = 1.623, log h = 1.140, h = 13.80cm
3) Best Bakery has been requested to decorate the cake with fresh cream. The thickness of the cream 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 that you have suggested in 2(b)(ii).
Answer:
h = 8cm, d = 54.99cm
Amount of fresh cream = VOLUME of fresh cream needed (area x height)
Amount of fresh cream = Vol. of cream at the top surface + Vol. of cream at the side surface
Vol. of cream at the top surface
= Area of top surface x Height of cream
= (3.142)( )² x 1
= 2375 cm³
Vol. of cream at the side surface
= Area of side surface x Height of cream
= (Circumference of cake x Height of cake) x Height of cream
= 2(3.142)(54.99/2)(8) x 1
= 1382.23 cm³
Therefore, amount of fresh cream = 2375 + 1382.23 = 3757.23 cm³
(b) Suggest three other shapes for cake, that will have the same height and volume as those suggested in 2(b)(ii). Estimate the amount of fresh cream to be used on each of the cakes.
Answer:
1 – Rectangle-shaped base (cuboid)
19000 = base area x height
base area =
length x width = 2375
By trial and improvement, 2375 = 50 x 47.5 (length = 50, width = 47.5, height = 8)
Therefore, volume of cream
= 2(Area of left/right side surface)(Height of cream) + 2(Area of front/back side surface)(Height of cream) + Vol. of top surface
= 2(8 x 50)(1) + 2(8 x 47.5)(1) + 2375 = 3935 cm³
2 – Triangle-shaped base
19000 = base area x height
base area = 2375
x length x width = 2375
length x width = 4750
By trial and improvement, 4750 = 95 x 50 (length = 95, width = 50)
Slant length of triangle = √(95² + 25²)= 98.23
Therefore, amount of cream
= Area of rectangular front side surface(Height of cream) + 2(Area of slant rectangular left/right side surface)(Height of cream) + Vol. of top surface
= (50 x 8)(1) + 2(98.23 x 8)(1) + 2375 = 4346.68 cm³
3 – Pentagon-shaped base
19000 = base area x height
base area = 2375 = area of 5 similar isosceles triangles in a pentagon
therefore:
2375 = 5(length x width)
475 = length x width
By trial and improvement, 475 = 25 x 19 (length = 25, width = 19)
Therefore, amount of cream
= 5(area of one rectangular side surface)(height of cream) + vol. of top surface
= 5(8 x 19) + 2375 = 3135 cm³
(c) Based on the values that you have found which shape requires the least amount of fresh cream to be used?
Answer:
Pentagon-shaped cake, since it requires only 3135 cm³ of cream to be used.
Part III
Find the dimension of a 5 kg round cake that requires the minimum amount of fresh cream to decorate. Use at least two different methods including Calculus. State whether you would choose to bake a cake of such dimensions. Give reasons for your answers.
Answer:
Method 1: Differentiation
Use two equations for this method: the formula for volume of cake (as in Q2/a), and the formula for amount (volume) of cream to be used for the round cake (as in Q3/a).
19000 = (3.142)r²h → (1)
V = (3.142)r² + 2(3.142)rh → (2)
From (1): h = → (3)
Sub. (3) into (2):
V = (3.142)r² + 2(3.142)r( )
V = (3.142)r² + ( )
V = (3.142)r² + 38000r-1
( ) = 2(3.142)r – ( )
0 = 2(3.142)r – ( ) -->> minimum value, therefore = 0
= 2(3.142)r
= r³
6047.104 = r³
r = 18.22
Substitute r = 18.22 into (3):
h =
h = 18.22
therefore, h = 18.22cm, d = 2r = 2(18.22) = 36.44cm
Method 2: Quadratic Functions
Use the two same equations as in Method 1, but only the formula for amount of cream is the main equation used as the quadratic function.
Let f(r) = volume of cream, r = radius of round cake:
19000 = (3.142)r²h → (1)
f(r) = (3.142)r² + 2(3.142)hr → (2)
From (2):
f(r) = (3.142)(r² + 2hr) -->> factorize (3.142)
= (3.142)[ (r + )² – ( )² ] -->> completing square, with a = (3.142), b = 2h and c = 0
= (3.142)[ (r + h)² – h² ]
= (3.142)(r + h)² – (3.142)h²
(a = (3.142) (positive indicates min. value), min. value = f(r) = –(3.142)h², corresponding value of x = r = --h)
Substitute r = --h into (1):
19000 = (3.142)(--h)²h
h³ = 6047.104
h = 18.22
Substitute h = 18.22 into (1):
19000 = (3.142)r²(18.22)
r² = 331.894
r = 18.22
therefore, h = 18.22 cm, d = 2r = 2(18.22) = 36.44 cm
I would choose not to bake a cake with such dimensions because its dimensions are not suitable (the height is too high) and therefore less attractive. Furthermore, such cakes are difficult to handle easily.
FURTHER EXPLORATION
Best Bakery received an order to bake a multi-storey cake for Merdeka Day celebration, as shown 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 the second cake is 10% less than the 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 the fourth cakes. By comparing all these values, determine whether the volumes of the cakes form a number pattern? Explain and elaborate on the number patterns.
Answer:
height, h of each cake = 6cm
radius of largest cake = 31cm
radius of 2nd cake = 10% smaller than 1st cake
radius of 3rd cake = 10% smaller than 2nd cake
31, 27.9, 25.11, 22.599…
a = 31, r =
V = (3.142)r²h
Radius of 1st cake = 31, volume of 1st cake = (3.142)(31)²(6) = 18116.772
Radius of 2nd cake = 27.9, vol. of 2nd cake = 14674.585
Radius of 3rd cake = 25.11, vol. of 3rd cake = 11886.414
Radius of 4th cake = 22.599, vol. of 4th cake = 9627.995
18116.772, 14674.585, 11886.414, 9627.995, …
a = 18116.772, ratio, r = T2/T1 = T3 /T2 = … = 0.81
(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:
Sn =
Sn = 57000, a = 18116.772 and r = 0.81
57000 =
1 – 0.81n = 0.59779
0.40221 = 0.81n
og0.81 0.40221 = n
n =
n = 4.322
therefore, n ≈ 4
CONCLUSION
Cake is a type of sweet food, or a dessert during meal. Cake may be very important to a ceremony, or they may be merely decorative. Other terms for cake are the sign of delicious.
As I doing this project, I notice that geometry and progression can be so close in our life. There are many shape of cake outside there. Different shapes of the cake have different cost and attraction. From geometry, we can know area of the cake. From the area we can get volume of the cake. From the surface area of cake, we also can get the volume of cream to decorate the cake by using quadratic equation and differentiation. As the result, we can know the cost of a cake.
After we know concept of quadratic function and integration, we can apply it in our life. In order to meet the budget or saving money, we can capable to decide on which shape or design more favorable and reasonable.
REFLECTION
In the making of this project, I have spent countless hours doing this project. I realized that this subject is a compulsory to me. Without it, I can’t fulfill my big dreams and wishes….
I used to hate Additional Mathematics…
It always makes me wonder why this subject is so difficult…
I always tried to love every part of it…
It always an absolute obstacle for me…
Throughout day and night…
I sacrificed my precious time to have fun…
From..
Monday, Tuesday, Wednesday, Thursday, Friday
And even the weekend that I always looking forward to…
From now on, I will do my best on every second that I will learn Additional Mathematics..
REFERENCES
Add Maths Project Work 1/2010, Muhammad Syariff bin Aripin
Additional Mathematics Textbook Form 5 KBSM
SUCCESS Additional Mathematics Reference Book, Fajar Bakti Sdn. Bhd.
http://www. scribd.com
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