Add Math Project 2011

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Additional Mathematic Project

Acknowledge This project could not have been written without Mdm. Lee, who encouraged 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 materials that I found in articles or in internet. internet. I make no claim to to be comprehensive. comprehensive. A special thanks thanks to to the authors. Without you, this this project would have taken years off my life (which I don¶t h ave many to spare).

I would like to acknowledge acknowledge and extend my heartfelt gratitude 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 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.












OBJECTIVE The aims of carrying out this t his 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

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to develop mathematical mathematical knowledge through problem solving in a way that increases students¶ interest and confidence co nfidence



to use the language o f mathematics to express mathematical ideas precisely

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to provide learning environment that stimulates and enhances effective learning



to develop positive attitude towards mathematics












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.












MATHEMATICS

IN CAK E BAK ING ING A ND CAK E

DECOR ATING

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 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 t he cake becomes too short. If the pan is too small, the cake beco mes too tall.














The ratio of the surface area to the volume determines how much crust a baked good will have. The more surface area there t here 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 t ime 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 little bit and bake for a longer time. We mix ingredients in round bowls bo wls because cubes would have corners cor ners where unmixed ingredients would accumulate, and we would have a hard time scraping them into the batter.

C ALCULUS

(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,












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.












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. ce lebration.












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)( )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 cmin 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:












d=

  

Height,h (cm)

Diameter,d(cm) Diameter,d( cm)

1.0 2.0

155.53 109.98

3.0

89.80

4.0 5.0 6.0 7.0 8.0 9.0 10.0

77.77 68.56 63.49 58.78 54.99 51.84 49.18

(b) Based on the values in your table, (i) state the range of heights that is answers.

NOT

suitable for the cakes and explain your














h = 8cm, d = 54.99cm, because it can fit into t he 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)( )²h  19000/(3.142)h =   = d² 


































(a) If Best Bakery received an order to bake a cake where the height of the cake is i s 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 s et 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).












(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-s haped base (cuboid)

19000 = base area x height














2

± Triangle-s haped base












3 ± Pentagon-s haped base












Part












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)












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)  




































(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?












(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












Reflection

Add Math Project 2011

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