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ADDITIONAL MATHEMATICS PROJECT WORK

2/2012Project Popcorn

Name: Calan Eskandar Bin Shamsudin

Class: 5C

Set: 2

Index Number: WL501D032

IC No: 950926-10-5551

Teacher’s Name: Madam Sarah Tan

School: Wesley Methodist School Kuala Lumpur

Contents

TOPIC PAGE

1) Contents 2

2) Introduction 3

3) Questions 4

4) Section A 5-12

5) Section B 13-23

6) Conclusions 24

2

Introduction

OBJECTIVES

1) Apply and adapt a variety of problem-solving strategies to solve routine

and non-routine problems.

2) Acquire effective mathematical communication through oral and writing,

and to use the language of mathematics to express mathematical ideas

correctly and precisely.

3) Increase interest and confidence as well as enhance acquisition of

mathematical knowledge and skills that are useful for career and future

undertakings.

4) Realize that mathematics is an important and powerful tool in solving

real-life problems and hence develop positive attitude towards

mathematics.

5) Train students not only to be independent learners but also collaborate,

to cooperate, and to share knowledge in an engaging and healthy

environment.

6) Use technology especially the ICT appropriately and effectively. Train

students to appreciate the intrinsic values of mathematics and to become

more creative and innovative.

7) Realize the importance and the beauty of mathematics.

MORAL VALUES

The moral value of this project would be the appreciation and understanding

how mathematics intervenes our everyday life. Without mathematics to compute

profit and loss, there would be no economies, commerce and businesses.

Humans would have to make technological progress without mathematics to

confirm computations of theories and findings. There would be no navigational

systems for cars, ships, trains, planes and rockets. Even the smallest feat of like

telling the time involves mathematics!

3

Question 1Question:

 Take the white paper and roll it up along the longest side to form a baseless cylinder that is tall and narrow. Do not overlap the sides. Tape along the edge. Measure the dimensions with a ruler. Record data below and on the cylinder. Label the cylinder, cylinder A.

Take the colour paper and roll it up along the shorter side to form a baseless cylinder

that is short and stout. Do not overlap the sides. Tape along the edge. Measure the

height and diameter with a ruler. Record data below and on the cylinder. Label it

Cylinder B.

11”

8.5”

4

Identify the Problem:Question asks to compare the volume of two cylinders created using the same sheet of

paper, determine the dimensions to hold more popcorn and find a pattern for the

dimensions for containers.

Strategy:

1) Create two baseless cylinders using the given dimensions according to the

instructions given

2) Measure the height and diameter of the two-labeled cylinders with a ruler

3) Record the data obtained in a table

5

Dimension Cylinder A Cylinder B

Height 11.00 in 8.50 in

Diameter 2.706 3.5014

Radius 1.3528 1.7507

Question 2Question:

Do you think the two cylinders will hold the same amount? Do you think one will

hold more than the other? Which one? Why?

Identify Problem:

Question asks which cylinder will hold more popcorn than the other and to

explain.

Strategy:

1. Using mathematical knowledge about the effect of radius and height of cylinder

on its volume, determine the most suitable answer.

2. The answer provided is explained.

Working:

The two cylinders hold different amount. Cylinder B will hold more than cylinder A.

This is because the radius of cylinder B is longer than cylinder A. Therefore, the

volume of Cylinder B is larger than Cylinder A. Although the height of Cylinder B is

shorter than Cylinder A, the height does not affect the volume as much as the radius.

Conclusion:

The two cylinders will hold different amounts of popcorn. Cylinder B will hold

more popcorn than Cylinder A.

6

Question 3

Question:

Place Cylinder B on the paper plate with Cylinder A inside it. Use your cup to pour

popcorn into Cylinder A until it is full. Carefully, lift Cylinder A so that the popcorn

falls into Cylinder B. Describe what happened. Is Cylinder B full, not full or

overflowing?

Identify Problem:

Question asks for the condition of Cylinder B after filling it with the same amount of

popcorn used to fill up Cylinder A.

Strategy:

1. Following the instructions of the question, the scenario is carried out.

2. The condition of Cylinder B is observed and recorded.

Working:

When the popcorn from Cylinder A falls into Cylinder B, the popcorn will not fill up

Cylinder B completely. Therefore, Cylinder B is not full. There is still room in the

cylinder for more popcorn.

Conclusion:

Cylinder B can hold more popcorn compared to Cylinder A.

7

Question 4Question:

a) Was your prediction correct? How do you know?

b) If your prediction was incorrect, describe what actually happened.

Identify Problem:

Find out whether the prediction on the size of Cylinder A and B is correct.

Strategy:

1. Calculate the volume of Cylinder A and B using the formula πr2 .

2. Compare the volume a of Cylinder A and B.

Working:

Volume of Cylinder A: rπ ²h = 3.142 x (1.3528) ²(11)= 63.27 in.3

Volume of Cylinder B: rπ ²h

=3.142 x(1.7507) ²(8.5)

= 81.88³ in.3

a) My prediction was correct. Based on the observation, Cylinder B holds more

popcorn than Cylinder A as its volume is much bigger than Cylinder A.

b) Cylinder B will have a smaller volume than Cylinder A. The popcorn will fall out

from the cylinder.

Conclusion:

The prediction is correct. Cylinder B has a bigger volume than Cylinder A.

8

h = 11.00 in.

Question 5Question:

a) State the formula for finding the volume of a cylinder.

b) Calculate the volume of Cylinder A?

c) Calculate the volume of Cylinder B?

d) Explain why the cylinders do or do not hold the same amount. Use the formula for

the volume of a cylinder to guide your explanation.

Identify Problem:

To determine the volume of Cylinder A and Cylinder B.

Strategy:

1. Calculate the volume of Cylinder A and Cylinder B by multiplying the base area

and height of the cylinders.

2. By substituting π = 3.142 into the formula.

Working:

a) The formula for finding the volume of a cylinder is v= πr2h.

b) Volume of Cylinder A r = 1.38 in.

v= πr2h

π= 3.142, r=1.38, h=11.00

v= π (1.38)2(11.00)

v= 63.92 in.3

Volume of Cylinder B r = 1.89 in.

v= πr2h

π= 3.142, r=1.75, h=8.50

v= π (1.75)2(8.50) h = 8.50 in.

v= 81.88 in.3

9

d) The cylinders do not hold the same amount because the radius of their circular base

is different from each other. Based on the formula for the volume of a cylinder, the

radius is squared. In Cylinder A, the square of its smaller radius brings about a

smaller value than Cylinder B. Therefore, Cylinder B has a bigger volume than the

volume of Cylinder A.

Conclusion:

The volume of Cylinder A is 63.92 in.3, the volume of Cylinder B is 81. in.3

10

Question 6Which measurement impacts the volume more: the radius or the height? Work

through the example below to help you answer the question.

Assume that you have a cylinder with a radius of 3 inches and a height of 10 inches.

Increase the radius by 1 inch and determine the new volume. Then using the original

radius, increase the height by 1 inch and determine the new volume.

Which increased dimension had a larger impact on the volume of the cylinder? Why

do you think this is true?

Identify Problem:

To find out radius or the height impacts the volume more.

Strategy:

1. Find out the new volume using by multiplying the base area and the height of the

cylinders.

2. By using the formula πr2h, the volume of cylinder can be found.

3. By substituting π = 3.142 into the formula.

Working:

When r = 3, h = 10

V = π (3)2(10.00)

= 282.74 in3

When r = 4, h = 10

V = π (4)2(10.00)

=502.65 in3

When r = 3, h = 11

V = π (3)2(11.00)

=311.02 in3

11

CYLINDER RADIUS , r HEIGHT , h VOLUME , V

ORIGINAL 3 10 282.74 in3

INCREASED

RADIUS

4 10 502.65 in3

INCREASED

HEIGHT

3 11 311.02 in3

By examining the calculations done, it clearly show that when the radius of a cylinder

increases, the volume of cylinder increases more dramatically than when the height is

increased. This is because from the formula πr2h, the radius, r is squared. Therefore,

radius, r had a larger impact on the volume of the cylinder.

Conclusion:

Radius, r has a larger impact on the volume of the cylinder than height, h.

Original Increased radius Increased height0

100

200

300

400

500

600

RadiusHeightVolume

12

Section B

13

QUESTION

If you were buying popcorn at the movie theater and wanted the most popcorn, what type of container would you look for?

Clue : You need more than one type of containers.

You are given 300 cm² of thin sheet material. Explain the details.

ANSWER

Identifying the problem:

The question requires us to identify the popcorn container that can carry the most

popcorn.

Strategy:

1. First, calculate the maximum volume that can be acquired from a cube using

300 cm2 of thin sheet material.

2. Then calculate the maximum volume that can be acquired from a cylinder using

300 cm2 of thin sheet material.

3. Then calculate the maximum volume that can be acquired from a cuboid using

300 cm2 of thin sheet material.

4. Then calculate the maximum volume that can be acquired from a cone using

300 cm2 of thin sheet material.

5. Compare all the volume and determine the shape with the highest volume

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ANSWER

I ) Cylinder Container – Opened top

Surface area=2 πrh+π r2=300 cm2

h=300−π r2

2 πr

Volume=π r2h

¿ π r2(300−π r2

2 πr )

Maximum volume=dVdr

=0

π r2(300−π r2

2 πr )=(300 π r2−π r4

2 πr ) ¿( 300 π r2−π r4

2πr )n r−1

¿( 300 πr−π r3

2 ) dV

dr=300

2−3 π2

2=0

¿150−3 π2

2

3 π2

2=150 h=( 300−π r2

2πr )

15

πr2=100 ¿( 300−π (5.64 )2

2 π (5.64 ) ) r=5.64cm h=5.64 cm

Maximum volume ,d2 Vdr2 =−3 π

2r 2

¿−4.71(5.64 )2

¿−149.82 (< 0, maximum)

Volume=π (5.64 )2 (5.64 )

¿563.62cm3

16

II- Cube Container – Opened top

Surface Area = l² + 4l² = 300cm²

5 l² = 300 l² = 60 l = 7.75cm

Volume = l³ = (7.75) ³ = 465.48 cm³

17

III) Cuboid Container – Opened top

Assume that length is twice its width or others

Surface area=l2+4 hl=300 cm2

h=300−l2

4 lVolume=2l2 h

¿ l2( 300−l2

4 l )

Maximum volume ,dVdr

=0

dVdr

=75−3 l2

4

0=75−3 l2

4

3l2

4=75

l2=100

18

l=10 cm

h=300−(10)2

4(10)

h=5 cm

d2Vd r2 =−2( 3

4 )l¿−1.5 (10 )

¿−15 (< 0, maximum)

Volume=l2h

¿(10)2(5)

¿500 cm3

19

IV) Cone Container – Opened top

¿ the diagram, x2=r2+h2

Surface area=πrx=300 cm2

π2 r2 x2=3002

π2 r2(r¿¿2+h2)=90000 ¿

h2=90000−π2r 4

π2 r2

Volume=13

π r2h

Volume=19

π2 r4 h2

¿19

π2 r4( 90000−π2r 4

π2 r2 )¿10 000 r2−1

9π2 r6

Maximum volume ,dVdr

=0

2 vdVdr

=20 000 r−23

π2 r5

¿20 000 r−2

3π2 r5

2 v

0=20 000 r−23

π2 r5

r5

r=20 000

23

π 2

r 4=3039.63r=7.425cm

20

h2=90000−π2r 4

π2 r2

h=300−π ¿¿h=5.436 cm

d2Vd r2 =20 000−5¿

¿20 000−99 995.8¿−79 995.8(< 0 , maximum)

Volume=13

π (7.425 )2(5.436)

¿313.83cm3

21

CONTAINER

HEIGHT(cm) RADIUS LENGTH (cm)

WIDTH(cm) VOLUME

(cm3)Cube 7.75 __ 7.75 7.75 465.48

Cylinder 5.64 5.64 __ __ 563.69

Cuboid 10 __ 10 5 500.00

Cone 5.436 7.425 __ __ 313.83

From the activity earlier, I found that increasing the radius increased the volume more than increasing the height. This is because the radius is squared to find the volume, which increases its impact on the volume. From the calculations, it has been found that the cuboid can hold the most amount popcorn. Which is then followed by the cone, cylinder and cube. This means that cube is the shape that can be filled with the least amount of popcorn. At the movie theater, no cuboid shapes can be found. This is to avoid too much popcorn being sold in one container.

I) If you were the popcorn seller, what type of container would you look for?

- If I were the popcorn seller, I would look for cube shaped container. This is because the cube shaped container can hold the least amount of popcorn due to its volume. Hence, less popcorn would be sold to customers and my profit would be maximized.

II) You are the producer of the containers, what type of container would you choose to have the most profit?

-If I were the producer of containers, I would choose the cylinder shape. This is mainly because it is the easiest to make with a simple structure. Despite this, it has a very high volume.

22

Cube Cylinder Cuboid Cone0

100

200

300

400

500

600

700

800

Volume of container

Volume of container

23

Conclusion

Based on the assessment that I have done, I have realized that the volume of a

cylinder is based on mainly the radius of its circular base rather than its height. I have also

realized that different dimensions of cylinders are used for different purposes. Therefore,

there are benefits and consequences depending on the purpose of its dimensions.

If a person work as a popcorns seller, he would need to find a container, which has the

least volume to hold the least amount of popcorn. Thus, he will make more money whenever

a customer buys from him. The shape of the container holding the popcorn should be

considered before starting business. A container, which holds less volume, would raise the

profits of the popcorn seller.

If a person is trying to produce containers to hold popcorns, he or she would choose a

container which is easy to manufacture in order to save manpower as well as workforce. Not

only that, he would choose a container which does not take long to shape in order to save

production time, thus increasing its production rate. Eventually, that particular producer

would be able to save more money thus make more profit if he were to use a cylinder shaped

container.

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