sample addmath project

AdditionalMathematics

Project Work 2Name:I/C Number :Angka Giliran:School :Date :

TABLE OF CONTENTS

Additional Mathematics Project Work 2/2011

Num. Question Page


1 Part I

2 Part II~ Question 1~ Question 2 (a)~ Question 2 (b)~ Question 2 (c)

3 Part III~ Question 3 (a)~ Question 3 (b)~ Question 3 (c)

4 Further Exploration

INTRODUCTION OF ADDITIONAL MATHEMATICS PROJECT WORK

2/2011


The aims of carrying out this project work are to enable students to :

a) Apply mathematics to everyday situations and appreciate the importance and the beauty of mathematics in everyday lives ;

b) Improve problem-solving skills, thinking skills , reasoning and mathematical communication ;

c) Develop positive attitude and personalities and instrinsic mathematical values such as accuracy , confidence and systematic reasoning ;

d) Stimulate learning environment that enhances effective learning inquiry-base and teamwork ;

e) Develop mathematical knowledge in a way which increase students’ interest and confidence.

Introduction.There are a lot of things around us related to circles or parts of a circle. A circle is

a simple shape of Euclidean geometry consisting of those points in a plane which is the same distance from a given point called the centre. The common distance of the points of a circle from its center is called its radius.

Circles are simple closed curves which divide the plane into two regions, an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure (known as the perimeter) or to the whole figure including its interior. However, in strict technical usage, "circle" refers to the


perimeter while the interior of the circle is called a disk. The circumference of a circle is the perimeter of the circle (especially when referring to its length).

A circle is a special ellipse in which the two foci are coincident. Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone.

The circle has been known since before the beginning of recorded history. It is the basis for the wheel, which, with related inventions such as gears, makes much of modern civilization possible. In mathematics, the study of the circle has helped inspire the development of geometry and calculus. Circles had been used in daily lives to help people in their living.

Definition.

Pi, π has the value of 3.14159265. In Euclidean plane geometry, π is defined as the ratio of a circle's circumference to its diameter.


π=Cd

The ratio Cd

is constant, regardless of a circle's size. For example, if a circle has twice the

diameter of another circle it will also have twice the circumference, C, preserving the

ratioCd

.

Alternatively π can be also defined as the ratio of a circle's area (A) to the area of a square whose side is equal to the radius.

π= Ar

History.Pi or π is a mathematical constant whose value is the ratio of any circle's

circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. It is approximately equal to 3.14159 in the usual decimal notation. π is one of the most important mathematical and physical constants: many formulae from mathematics, science, and engineering involve π.


π is an irrational number, which means that its value cannot be expressed exactly as a fractionm/n, wherem andn are integers. Consequently, its decimal representation never ends or repeats. It is also a transcendental number, which means that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can be equal to its value; proving this was a late achievement in mathematical history and a significant result of 19th century German mathematics. Throughout the history of mathematics, there has been much effort to determine π more accurately and to understand its nature; fascination with the number has even carried over into non-mathematical culture.

The Greek letter π, often spelled outpi in text, was adopted for the number from the Greek word forperimeter "περίμετρος", first by William Jones in 1707, and popularized by Leonhard Euler in 1737. The constant is occasionally also referred to as the circular constant, Archimedes' constant (not to be confused with an Archimedes number), or Ludolph's number (from a German mathematician whose efforts to calculate more of its digits became famous).

The name of the Greek letter π is pi, and this spelling is commonly used in typographical contexts when the Greek letter is not available, or its usage could be problematic. It is not normally capitalised (Π) even at the beginning of a sentence. When referring to this constant, the symbol π is always pronounced like "pie" in English, which is the conventional English pronunciation of the Greek letter. In Greek, the name of this letter is pronounced /pi/.

The constant is named "π" because "π" is the first letter of the Greek words περιφέρεια (periphery) and περίμετρος (perimeter), probably referring to its use in the formula to find the circumference, or perimeter, of a circle. π is Unicode character U+03C0 ("Greek small letter pi").

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.


Find out how mathematics is used in cake baking and cake decorating and write about your findings.

Constructing the structure of a cake

These cakes are made by using different sizes of circular pans, then stacking the baked cake sections on top of each other.

You are to plan for a cake that will serve between 200 and 250 people.

The wedding cake must feed between 200 and 250 people. You have 4 different sizes of pans of you can use. ( All pans have the same height

)

r = 10 cm r = 15 cm r = 20 cm r = 25 cm

Each layer of cake must remain a cylinder You can stack layers . Each layer can then be separated and cut individually.

Each layer of cake will be cut into sectors that have a top area of exactly 50 cm2

You may have some left-over cake from a layer

Example of 50 cm2

top area of sector

One sector feeds one person.

Your final ingredients list must be proportional to the ingredients list provided for

you.

By using the theory of arithmetic and geometric progressions in Chapter 1 Form 5, the concept can be used to

Decide on how many layers of each size of cake you will need for your cake.


Show how you can cut the layers of the cake into equivalent sectors having a top

area of 50 cm2 each, in order to feed between 200 and 250 people. Complete the ingredients list by identifying the quantities needed for each

ingredient in the cake.

Work and calculations to determine the ingredients of the cake

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.

Calculate the proportions of different ingredients. For example, a frosting recipe that calls for 2 cups cream cheese, 2 cups confectioners' sugar and 1/2 cup butter has a cream cheese, sugar and butter ratio of 4:4:1. Identifying ratios can also help you make recipes larger or smaller.

Use as few measuring cups as possible. For example, instead of using a 3/4 cup, use a 1/4 cup three times. This requires you to work with fractions.

Determine what time it will be when the oven timer goes off. For example, if your cake has to bake for 30 minutes and you set the timer at 3:40, the timer will go off at 4:10.

Calculate the surface area of the part of the cake that needs frosting. For example, a sheet cake in a pan only needs the top frosted, while a sheet cake on a tray needs the top and four sides frosted. A round layer cake requires frosting on the top, on each layer and on the sides.

Determine how large each slice should be if you want to serve a certain amount of people. For example, an 18 by 13 inch sheet cake designed to serve 25 people should be cut into slices that measure approximately 3 by 3 inches.

Add up the cost of your ingredients to find the cost of your cake. Estimate the cost of partially used ingredients, such as flour, by determining the fraction of the container used and multiplying that by the cost of the entire container

Initial draft of the cake


Answer:

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

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.

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 3800cm3, 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)(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.

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.

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

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


(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 = 54.99cmAmount 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

(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

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

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:

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: DifferentiationUse 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

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 FunctionsUse 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 = 6047.104


h = 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.

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 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

10

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

(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 ≈ 4REFLECTION

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 progressions in our daily life. This project also helped expose the techniques of application of additional mathematics in real life situations. While conducting this project, a lot of information that I found. I have learnt how to bake a wedding tiered cake stands with good quality and proper height.


Apart from that, this project encourages the student to work together and share their knowledge. It is also encourage student to gather information from the internet, improve thinking skills and promote effective mathematical communication.

Last but not least, I proposed this project should be continue because it brings a lot of moral values to the student and also test the students understanding in Additional Mathematics.


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