admath project

8/2/2019 Admath Project

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NAME : KHAVILESHWARHY A/P SATHASIVAN I/C NO : 941222-14-5382 CLASS : 5 UNGGUL TEACHER : PN.YUSNITA BT. MUHAMAD YUSOFF


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CONTENTS

Acknowledgement

Objective

Introduction

Part I

Part II

Part III

Further Exploration

Reflection

Conclusion


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ACKNOWLEDGEMENT

During the implementation of this Additional Mathematicsproject, many people have helped me to carry out the projecteffectively. Therefore, I would like to take this opportunity tosay thank you to everyone who has played a part in this project.Firstly, I would like to thank Cik Yusnita, my AdditionalMathematics teacher for providing guidance to me while doing

this project. She was also there to give useful and importantinformation for me to complete this project.Next, I would like to thank my friend and my parents forhelping and guiding me during the project.Lastly, I would like to thank Pn.Shamsiah bt. Ahmad, our

excellent principal of SMK Cochrane for allowing us to takepart in this project.


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Objective

The aims of carrying out this project work are : To apply and adapt a variety of problem-solving strategiesto solve problems.

To improve thinking skills

To promote effective mathematical communication

To develop mathematical knowledge through problemsolving in a way that increase students interest andconfidence.

To develop positive attitude towards mathematics.


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introductionDuring form 5, student who takes additional mathematics as an elective isrequired to carry out the project. This year, the Curriculum DevelopmentDivision, Ministry of Education has prepared task for us. We have tocomplete the task based on the methods. Through this project, we are able:

To wider our knowledge

Apply and adapt a variety of problem. Solving strategies to solveroutine and non-routine problems.

Experience a classroom environment which is challenging andmeaningful, thus, improving thinking skills.


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Experience a classroom environment where knowledge and skills areapplied in showing real life problems.

Experience a classroom environment where expressing onesmathematical thinking is highly encouraged.

Experience a classroom environment that stimulate and enhanceeffective learning.

Acquire effective learning mathematical communications throughoral and writing and to use the language of mathematics to expressidea correctly and precisely.


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Part ICakes 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 aretreasured not

only because of their wonderful taste but also inthe art of cake baking and cake decorating.

Find out howmathematics is used in cake baking and cake decorating andwrite about your

findings.


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Answer:

Geometry To determine suitable dimensions for the cake, to assist in designing

and decorating cakes that comes in many attractive shapes anddesigns, to estimate volume of cake to be produced

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.

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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Best Bakery shop received an order from your school to bake a 5 kg of round cake as shownin

Diagram 1 for the Teachers Daycelebration.

h cm

dcm

1) If a kilogram of cake has a volume of 3800 , and the height of the cake is to be7.0cm, calculate the diameter of the baking tray to be used to fit the 5 kg cake ordered

by your school.

[Use = 3.142]

Answer:

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

3800 x 5 = (3.142)( )x 7

(3.142) =( )

863.872 =( )

= 29.392

d = 58.784 cm


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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 height s, hcm, 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 forvolume of cake, V = 19000, that is:

19000 = (3.142)(d/2)h

=

= d

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


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(b) Based on the values in your table,

(i)state the range of heights that isNOTsuitable for the cakes andexplain your

answers.

Answer:

h< 7cm is NOT suitable, because the resulting diameter produced is toolarge to fit into the oven. Furthermore, the cake would be too short andtoo 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)( )h

19000/(3.142)h =


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= d

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


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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:

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 yourgraph 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


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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)( ) x 1= 2375 cmVol. 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 cm

Therefore, amount of fresh cream = 2375 + 1382.23 = 3757.23 cm

(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


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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/backside surface)(Height of cream) + Vol. of top surface

= 2(8 x 50)(1) + 2(8 x 47.5)(1) + 2375 = 3935 cm

2 Triangle-shaped base

19000 = base area x height

base area = 2375

x length x width = 2375length x width = 4750

By trial and improvement, 4750 = 95 x 50(length = 95, width = 50)

Slant length of triangle = (95 + 25)= 98.23


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Therefore, amount of cream:

= Area of rectangular front side surface(Height of cream) + 2(Area of slantrectangular left/right side surface)(Height of cream) + Vol. of top surface

= (50 x 8)(1) + 2(98.23 x 8)(1) + 2375 = 4346.68 cm

3 Pentagon-shaped base

19000 = base area x heightbase 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 cm


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(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 cm of cream to be used.


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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

todecorate. Use at least two different methods including Calculus.State whether you would

choose to bake a cake of such dimensions. Give reasons for youranswers.

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 (asin Q3/a).

19000 = (3.142)rh (1)

V = (3.142)r + 2(3.142)rh (2)

From (1): h = (3)

Sub. (3) into (2):

V = (3.142)r + 2(3.142)r( )

V = (3.142)r + ( )

V = (3.142)r + 38000r-1

( ) = 2(3.142)r ( )

0 = 2(3.142)r ( ) -->> minimum value, therefore = 0

= 2(3.142)r

= r


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6047.104 = r

r = 18.22

Sub. 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 ofcream is the main equation used as the quadratic function.

Let f(r) = volume of cream,

r = radius of round cake:

19000 = (3.142)rh (1)

f(r) = (3.142)r + 2(3.142)hr (2)

From (2):

f(r) = (3.142)(r + 2hr) -->> factorize (3.142)

= (3.142)[ (r + ) ( ) ] -->> completing square,

a = (3.142), b = 2h c = 0

= (3.142)[ (r + h) h ]


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= (3.142)(r + h) (3.142)h

(a = (3.142) (positive indicates min. value), min. value = f(r) = (3.142)h,corresponding value of x = r = --h)

Sub. r = --h into (1):

19000 = (3.142)(--h)h

h = 6047.104

h = 18.22

Sub. h = 18.22 into (1):

19000 = (3.142)r(18.22)

r = 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 lessattractive. Furthermore, such cakes are difficult to handle easily.


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FURTHER EXPLORATIONBest Bakery received an order to bake a multi-storey cake for Merdeka Day celebration, asshown 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

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. Bycomparingall 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)rh

Radius of 1st cake = 31, volume of 1st cake = (3.142)(31)(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


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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, calculatethe maximumnumber of cakes that the bakery needs to bake. Verify your answer usingother methods.

Answer:

Sn =

Sn = 57000, a = 18116.772 and r = 0.81

57000 = 1 0.81n = 0.59779

0.40221 = 0.81n

log0.81 0.40221 = n

n =n = 4.322

Therefore: n 4


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REFLECTIONIn 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 cant 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


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I LOVE YOU

ADDITIONAL

MATHEMATICS

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

Additional Mathematics.


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CONCLUSIONIn the process of solving the problems and calculationsgiven, we need to use the suitable formulas from anyappropriate topics of additional mathematics. In this project, thedifferentiation method, integration method, completing ofsquare method had been used to find the value needed.

admath project

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