add math project 2011

Additional Mathematic Project

Acknowledge

This project could not have been written without Mdm. Lee, who encouraged and challenged me through my academic program. She never accepted less than my best efforts. Thank you.

What is done in this project are materials that I found in articles or in internet. I make no claim to be comprehensive. A special thanks to the authors. Without you, this project would have taken years off my life (which I don’t have many to spare).

I would like to acknowledge and extend my heartfelt gratitude to my family and friends words alone cannot express what I owe them for their encouragement and whose patient love enabled me to complete this project. A special thanks to Amni for help me doing some of the solving calculation.

And especially to God, who made all things possible.

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OBJECTIVEThe aims of carrying out this project work are:

to apply and 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 increases students’ 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

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Part

One


INTRODUCTION

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 others. Cakes are treasured not only because

of their wonderful taste but also in the art of cake baking and cake decorating

Baking a cake offers a tasty way to practice math skills, such as fractions and ratios, in a real-world

context. Many steps of baking a cake, such as counting ingredients and setting the oven timer, provide

basic math practice for young children. Older children and teenagers can use more sophisticated math to

solve baking dilemmas, such as how to make a cake recipe larger or smaller or how to determine what size

slices you should cut. Practicing math while baking not only improves your math skills, it helps you become

a more flexible and resourceful baker.

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http://www.ehow.com/how_11464_make-yellow-cake.html

http://www.ehow.com/recipes/


MATHEMATICS IN CAKE BAKING AND CAKE DECORATING

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

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When making a batch of cake batter, you end up with a certain volume, determined by the recipe.The baker must then choose the appropriate size and shape of pan to achieve the desired result. If the pan is too big, the cake becomes too short. If the pan is too small, the cake becomes too tall. This leads into the next situation.

The ratio of the surface area to the volume determines how much crust a baked good will have. The more surface area there is, compared to the volume, the faster the item will bake, and the less "inside" there will be. For a very large, thick item, it will take a long time for the heat to penetrate to the center. To avoid having a rock-hard outside in this case, the baker will have to lower the temperature a little bit and bake for a longer time.

We mix ingredients in round bowls because cubes would have corners where unmixed ingredients would accumulate, and we would have a hard time scraping them into the batter.

CALCULUS (DIFFERENTIATION)

To determine minimum or maximum amount of ingredients for cake-baking, to estimate min. or max. amount of cream needed for decorating, to estimate min. or max. Size of cake produced.

PROGRESSION

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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For example when we make a cake with many layers, we must fix the difference of diameter of the two layers. So we can say that it used arithmetic progression. When the diameter of the first layer of the cake is 8” and the diameter of second layer of the cake is 6”, then the diameter of the third layer should be 4”.

In this case, we use arithmetic progression where the difference of the diameter is constant that is 2. When the diameter decreases, the weight also decreases. That is the way how the cake is balance to prevent it from smooch. We can also use ratio, because when we prepare the ingredient for each layer of the cake, we need to decrease its ratio from lower layer to upper layer. When we cut the cake, we can use fraction to devide the cake according to the total people that will eat the cake.

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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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Part

Two


If a kilogram of cake has a volume of 38000cm3, and the height of the cake is to be 7.0 cm, the diameter of the baking tray to be used to fit the 5 kg cake ordered by your school 3800 is

Volume of 5kg cake = Base area of cake x Height of cake

3800 x 5 = (3.142)(d2

)² x 7

190007

(3.142) = (d2

)²

863.872 = (d2

)²

d2

= 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 cmin width and 45.0 cm in height.

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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)²h19000

(3.142)h = d ²4

24188.415h

= d²

d = 155.53

√h

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,

(i) state the range of heights that is NOT suitable for the cakes and explain your answers.

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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.

(ii) suggest the dimensions that you think most suitable for the cake. Give reasons for your answer.

Answer:

h = 8cm, d = 54.99cm, because it can fit into the oven, and the size is suitable for easy handling.

(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)(d2

)²h

19000/(3.142)h = d ²4

24188.415h

= d²

d = 155.53√ h

d = 155.53h−12

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log d = log 155.53h−12

log d = −12

log h + log 155.53

Log h 0 1 2 3 4Log d 2.19 1.69 1.19 0.69 0.19

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(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: based on the graph

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: based on the graph

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 = 54.99cmAmount 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)(54.99

2)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

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(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 = 19000

2length x width = 2375By 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

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2 – Triangle-shaped base

19000 = base area x heightbase area = 237512

x length x width = 2375

length x width = 4750By trial and improvement, 4750 = 95 x 50 (length = 95, width = 50)Slant length of triangle = √(95² + 25²)= 98.23Therefore, 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

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3 – Pentagon-shaped base

19000 = base area x heightbase area = 2375 = area of 5 similar isosceles triangles in a pentagontherefore:2375 = 5(length x width)475 = length x widthBy 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:

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Pentagon-shaped cake, since it requires only 3135 cm³ of cream to be used.

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Part Three


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 = 19000

(3.142)r ² → (3)

Sub. (3) into (2):

V = (3.142)r² + 2(3.142)r(19000

(3.142)r ²)

V = (3.142)r² + (38000r

)

V = (3.142)r² + 38000r-1

(dVdr

) = 2(3.142)r – (38000r ²

)

0 = 2(3.142)r – (38000r ²

) -->> minimum value, therefore dVdr

= 0

38000r ²

= 2(3.142)r

380002(3.142)

= r³

6047.104 = r³r = 18.22

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Sub. r = 18.22 into (3):

h = 19000

(3.142)(18.22)²h = 18.22therefore, 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 + 2h2

)² – (2h2

)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)

Sub. r = --h into (1):19000 = (3.142)(--h)²hh³ = 6047.104h = 18.22

Sub. h = 18.22 into (1):19000 = (3.142)r²(18.22)r² = 331.894r = 18.22therefore, 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.

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Further Exploration


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

Diagram 2.

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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 is10% 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 = 31cmradius of 2nd cake = 10% smaller than 1st cakeradius of 3rd cake = 10% smaller than 2nd cake

31, 27.9, 25.11, 22.599…

a = 31, r = 9

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V = (3.142)r²h

Radius of 1st cake = 31, volume of 1st cake = (3.142)(31)²(6) = 18116.772Radius of 2nd cake = 27.9, vol. of 2nd cake = 14674.585Radius of 3rd cake = 25.11, vol. of 3rd cake = 11886.414Radius 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

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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 bakery needs to bake. Verify your answer using other methods.

Answer:

Sn = (a (1−r n))

(1−r )

Sn = 57000, a = 18116.772 and r = 0.81

57000 =(18116.772(1– (0.81)n))

(1−0.81)

1 – 0.81n = 0.59779

0.40221 = 0.81n

og0.81 0.40221 = n

n = log 0.40221

log 0.81

n = 4.322

therefore, n ≈ 4

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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, Saturday and Sunday

From now on, I will do my best on every second that I will learn Additional Mathematics.

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add math project 2011

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