ADDITIONAL MATHEMATICS

PROJECT WORK

TASK 2 / 2012

NAME: NAME:

CLASS: CLASS:

IC. NUMBER: IC. NUMBER:

TEACHER: TEACHER:

CONTENTSNO TOPICS PAGE1. Title of projects2. Introduction3. Acknowledgement4. History5. Objective6. Section A7. Section B8. Conclusion9. Reflection10. Rubrics

INTRODUCTION

TITLEAPPLICATION

OF MATHEMATICS

IN POPCORN PACKAGING

ACKNOWLEDGEMENTFirst of all, I would like to express my special thanks of gratitude to my additional

mathematics teacher, Puan Rubiah who gave me the opportunity to do this project and help me

a lot throughout finishing this project. Without her guide, I may not finish my project and do it

properly.

Secondly, I would like to thanks my parents and my family for providing everything,

such as money to buy anything that are related to this project and their advises, which is the

most needed to do this project. I am grateful for their constant support and help.Not forgotten to

my friends who have contributed lots of idea in finding the topic that would be interesting to do

and gave their comments on my research. I really appreciate their kindness and help.

Beside that, I want to thanks to the respondents for helping and spending their time to

answer my questions for this project. Without respondents, I might not be able to complete this

project because their co-operation in answering the questions, I have the conclusion for this

project.

Last but not least, I would like to express my thankfulness to those who are involved

either directly or indirectly in completing this project. Thank you for all the co-operation given.

HISTORY OF POPCORN

Popcorn was first discovered thousands of years ago by Native Americans. It is one of the oldest forms of corn: evidence of popcorn from 3600 B.C. was found in New Mexico and even earlier evidence dating to perhaps as early as 4700 BC was found in Peru. Some Popcorn has been found in early 1900s to be a purple color.

The English who came to America in the 16th and 17th centuries learned about popcorn from the Native Americans.

During the Great Depression, popcorn was comparatively cheap at 5–10 cents a bag and became popular. Thus, while other businesses failed, the popcorn business thrived and became a source of income for many struggling farmers. During World War II, sugar rations

diminished candy production, causing Americans to eat three times as much popcorn than they had before.

At least six localities (all in the Midwestern United States) claim to be the "Popcorn Capital of the World": Ridgway, Illinois; Valparaiso, Indiana; Van Buren, Indiana; Schaller, Iowa; Marion, Ohio; and North Loup, Nebraska. According to the USDA, most of the corn used for popcorn production is specifically planted for this purpose; most is grown inNebraska and Indiana, with increasing area in Texas.

As the result of an elementary school project, popcorn became the official state snack food of Illinois, U.S.A.

OBJECTIVESApply and adapt a variety of problem-solving strategies ti solve routine and non-routine

problems.

Acquire effective mathematical communication through oral and writing, and to use the

language of mathematics to express mathematical ideas correctly and precisely.

Increase interest and confidence as well as enhance acquisition of mathematical

knowledge and skills that are useful for career and future undertakings.

Realize that mathematics is an important and powerful tool in solving real-life problems

and hence develop positive attitude towards mathematics.

Train students not only to be independent learners but also collaborate, to cooperate, and

to share knowledge in an engaging and healthy environment.

Use technology especially the ICT appropriately and effectively.

Train students to appreciate the intrinsic values of mathematics and to become more

creative and innovative.

Realize the importance and the beauty of mathematics.

SECTION A

QUESTION 1For this activity, you will be comparing the volume of 2 cylinders created using the same sheet of paper. You will be determining which dimension can hold more popcorn. To do this, you will have to find a pattern for the dimensions for the containers.

Materials :

8.5 x 11 in. white paper, 8.5 x 11 in. colored paper, tape, popcorn plate, cup, ruler

1. 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 edges. Measure the

dimensions with a ruler and record your data below and on the cylinder. Label it

“Cylinder A”.

diagram

2. Take the colored 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 and record you data below and on the cylinder. Label it

“Cylinder B”.

diagram

ANSWER 1DIMENSION CYLINDER A CYLINDER B

HEIGHT 11.0 8.5DIAMETER 2.6 3.4

RADIUS 1.3 1.7

QUESTION 2Do you think the two cylinders will hold the same amount?

Do you think one will hold more than the other? Which one? Why?

ANSWER 2The two cylinders will hold the different amount. Cylinder B will hold more than Cylinder A. This is because the radius of Cylinder B is longer and this make the volume is bigger than Cylinder A. Although the height of Cylinder B is shorter than Cylinder A, but this does not affect much compare the affect of different in radius.

QUESTION 3Place Cylinder B on the paper plate with Cylinder A inside it. Use your cup to pour popcorn into Cylinder A until is full. Carefully, lift Cylinder A so that the popcorn falls into Cylinder B. Describe what happened. Is Cylinder B full, not full or over flowing?

ANSWER 3Cylinder B is not full. There is still space in the cylinder for more popcorn.

QUESTION 4a) Was your prediction correct? How do you know?b) If your prediction is incorrect, describe what actually happened?

ANSWER 4a) Yes, the prediction is correct. It is based on the formula, volume of cylinder equals to

π r2h. According to the formula, radius,r has more effect than height,h since radius,r is squared. Thus, the Cylinder B with greater radius,r have the greater volume,V than Cylinder A.

b) Cylinder B has a greater volume than Cylinder A.

QUESTION 5a) State the formula for finding the volume of a cylinderb) 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 formula for the volume of a cylinder to guide your explanation.

ANSWER 5a) V = π r2h

b) V =π r2h

=π x 1.3² x 11 =58.4 inch³

c) V = π r2h = π x 1.7² x 8.5 = 77.2 inch³

d) The cylinders have different radius and heights, so the volumes are different.

Data and Observations:The cylinder with have the greater radius and diameter will have the greater volume

The radius of Cylinder B is greater than Cylinder A.

The volume of Cylinder B is greater than Cylinder A.

So, Cylinder B holds more popcorn than Cylinder B.

DIMENSION CYLINDER A CYLINDER BHEIGHT, inch² 11.0 8.5

DIAMETER, inch² 2.6 3.4RADIUS, inch² 1.3 1.7

VOLUME, inch³ 58.4 77.2

Height,h Diameter,d Radius,r Volume,V

Cylinder A 11 2.6 1.3 58.4

Cylinder B 8.5 3.4 1.7 77.2

5

15

25

35

45

55

65

75

85

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.

CYLINDER RADIUS HEIGHT VOLUMEORIGINAL 3 10

INCREASED RADIUS

INCREASED HEIGHT

Which increased the dimension had a larger impact on the volume of the cylinder? Why do you think this is true?

ANSWER 6CYLINDER RADIUS HEIGHT VOLUMEORIGINAL 3 10 282.7

INCREASED RADIUS

4 10 502.7

INCREASED HEIGHT

3 11 311

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.

SECTION B

QUESTIONIf 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

1. Cylinder Container – opened top

Surface Area = 2π r h + π r2 = 300

h = 300−π r2

2 π r diagram

Volume = π r2h

= π r2(300−π r2

2 π r )

Maximum Volume =dVdr = 0

Volume = 563.62

2. Cube Container – opened top

Surface Area = l² + 4l² = 300cm²

5 l² = 300

l² = 60

l = 7.75cm diagram

Volume = l³

= (7.75)³

= 465.48 cm³

3. Cuboid Container – opened top

Assume that length is twice its width or others

Surface Area = 2l² +4hl = 300cm²

h = 300−2 l ²

4 l diagram

Volume = 2l² h

= 2l² ( 300−2 l ²4 l )

Maximum Volume,dVdr = 0

2l² ( 300−2 l ²4 l ) = ( 600 l ²−4 l4

4 l ) = ( 600 l ²−4 l4

4 l ) l−1

= 150l - l³

dVdr

= 150 – 3l² = 0

150 = 3l² h = 300−2(7.07) ²

4 (7.07)

l² = 50 h = 7.07cm

l = 7.07cm

Volume = 2l²h

= 2(7.07)²(7.07)

= 706.79cm³

4. Cuboid Container – opened top

Assume that length is equal to its width

Surface Area = l² + 4hl = 300

h = 300−l ²

4 l diagram

Volume = l² h

= l² ( 300−l ²4 l )

Maximum Volume,dVdr = 0

dVdr

= 75 – 3l ²4

= 0

75 = 3l ²4

l = 10

h = 5

Volume = 500

5. Hexagon Container – opened top Assume that the length of the side = x

Area of the base = 6[ 12

ab sin θ]= 6[ 1

2x . x . sin θ]

= 3√32

x ²

Surface Area = 6hx + ( 3√32

x ²) = 300 diagram

h = 300−(3 √3

2x ²)

6 x

Volume = base area × height

= 3√32

x ²h

= 3√32

x ² (300−( 3√32

x ²)6 x

)Maximum Volume,

dVdr = 0

dVdl = 300√3

4 - 27 x2

4

300√34

= 27 x2

4

x = 4.39

h = 9.49

Volume = 475.17

6. Cone Container – opened top

From the diagram, x² = r² + h²

Surface Area = π r x = 300cm²

π² r² x² = 300²

π² r² ( r² + h²) = 90000

h² = 90000−π2 r4

π 2r ²

Volume = 13π r² h

Volume² = 19 π² r4 h² diagram

= 19 π² r4 ( 90000−π 2r4

π2r ² )

Maximum Volume, dVdr = 0

dVdl = 10000 –π 2r4

3= 0

300√34

= π 2r4

3

r = 7.42

h = 10.51

Volume = 13π (7.42)² (10.51)

= 605.95cm³

CONCLUSIONContainer height radius length width volumeCylinder 5.64 5.64 - - 563.69

Cube 7.75 - 7.75 7.75 465.48Cuboid 1 7.07 - 7.07 14.14 706.79Cuboid 2 5.00 - 10.00 10.00 500.00Hexagon 9.49 - 4.39(side) - 475.17

Cone 10.51 7.42 - - 605.95

Shape of containers that give the most popcorn reflect the maximum volume. 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 cuboid1 can be filled in with the most amount popcorn. It followed by cone, cuboid2, and hexagon. These means that cube is the container that can be filled with the least amount of popcorn. Randomly, surveying at the movie theater, no cube or cuboid shapes can be found. Therefore, in this case, the cuboid1 was the most preferable container that can have the most popcorns.

i. You are the popcorn seller, what type of container would you look for?If I was the popcorn seller, I will look for cube shape container. It is because the least popcorns will be in. So, I will get the most profit for my sale. Furthermore, it is cute and simple shape.

ii. You are the producer of the containers, what type of container would you choose to have the most profit?If I was the producer of the popcorns containers, I will look for cylinder shape container. It is because this shape is the easiest production and it takes less effort and also no time consuming to produce .

Cylinder

Cube

Cuboid1

Cuboid2

Hexagon

Cone

563.69

465.48

706.79

500

475.17

605.95

Volume of container,V (cm³)Volume of container

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 can’t fulfill my big dreams and wishes.

I used to hate Additional Mathematics…

It always make me wonder why this subject is so difficult…

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…

From now on, I will do my best on every second that I will learn Additional Mathematics full of effort!