The document discusses the application of mathematics in popcorn packaging. It begins with an introduction to popcorn and its history. It then presents the objectives and sections of the project. Section A involves an experiment comparing the volume of cylinders made from paper. Calculations show that a cylinder with a larger radius holds more popcorn. Section B discusses finding the container shape from given materials that can hold the most popcorn volume. A cuboid is determined to have the greatest volume. In conclusion, different container shapes are compared and a cuboid is found to hold the most popcorn.

1. NAME: NURUL IZZAH BINTI SULAIMAN WADI
CLASS: 5 BESTARI
ADD MATH TEACHER: PN YEO SAI KIU
ADDITIONAL MATHEMATICS
APPLICATION OF MATHEMATICS IN POPCORN PACKAGING

2. 1
ADDITIONAL
MATHEMATICS
APPLICATION OF MATHEMATICS
IN POPCORN PACKAGING
ACKNOWLEDGEMENT
First 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 projectand 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 projectand their
advises, which is the most needed to do this project.I
am grateful for their constant supportand help. Not
forgottenIzzah , 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 forhelping and spending
their time to answer my questions for this project.
Without respondents,I might not be able to complete
this projectbecause their co-operationin answering the
questions,I have the conclusionfor 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.

3. 2
CONTENTS
NO TOPICS PAGE
1 INTRODUCTION
2 ACKNOWLEDGEMENT
3 HISTORY
4 OBJECTIVE
5 SECTIONA
6 SECTIONB
7 CONCLUSION

4. 3
INTRODUCTION TO
POPCORN
Popcorn is a type of corn that expands from the kernel and puffs up when heated.
Popcorn is able to pop like amaranth grain, sorghum, quinoa, and millet. When heated,
pressure builds within the kernel, and a small explosion (or "pop") is the end result. Some
strains of corn are now cultivated specifically as popping corns.
There are various techniques for popping corn. Along with prepackaged popcorn, which is
generally intended to be prepared in a microwave oven, there are small home appliances
for popping corn. These methods require the use of minimally processed popping corn.
A larger-scale, commercial popcorn machine was invented by Charles Cretors in the late
19th century.
Unpopped popcorn is considered nonperishable and will last indefinitely if stored in ideal
conditions.
Depending on how it is prepared and cooked, some consider popcorn to be a health food,
while others caution against it for a variety of reasons. Popcorn can also have non-food
applications, ranging from holiday decorations to packaging materials.

5. 4
HISTORY OF POPCORN
Popcornwas first discovered thousands of years ago by Native Americans. It is
one of the oldestforms of corn: evidence of popcornfrom 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 Popcornhas been found in early 1900s to be a purple
color. The English who came to Americain the 16th and 17th centuries learned
about popcornfrom the Native Americans.
During the Great Depression,popcornwas comparatively cheap at 5– 10 cents a
bag and became popular. Thus, while other businesses failed,the popcorn
business thrived and became asource of income for many struggling farmers.
During World War II,sugar rations diminished candy production, causing
Americans to eat three times as much popcornthan they had before.
At least six localities (all in the Midwestern United States)claim to be the
"PopcornCapital 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 popcornproductionis specificallyplanted
for this purpose;most is grown in Nebraska and Indiana, with increasing area in
Texas.
As the result of an elementary schoolproject, popcornbecame the officialstate
snack food of Illinois ,U.S.A.

6. 5
OBJECTIVE
Apply and adapt a variety of problem-solving strategies to solve routine and non-
routine problems.
Acquire effective mathematical communicationthrough oral and writing, and to
use the language of mathematics to express mathematical ideas correctlyand
precisely.
Increase interest and confidenceas 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 independentlearners but also collaborate, to
cooperate,and to share knowledge in an engaging and healthy environment.
Use technology especiallythe 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.

7. 6
SECTION A
QUESTION 1
For this activity, you will be comparing the volume of 2 cylinders created using the same sheetof paper.
You will be determining which dimension can hold more popcorn. To do this, youwill 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”.
2. Take the colored paper and roll it up along the shorter side to form a baseless cylinderthat is
short and stout. Do not overlap the sides. Tape along the edge. Measure the heightand diameter with a
ruler and record you data below and on the cylinder. Label it “Cylinder B”.

8. 7
DIMENSION CYLINDER A CYLINDER B
HEIGHT 11 8.5
DIAMETER 2.6 3.4
RADIUS 1.3 1.7
QUESTION 2
Do you think the two cylinders will hold the same amount? Do you think one will hold more than
the other? Which one? Why?
ANSWER:
The 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 3
Place Cylinder B on the paper plate with Cylinder A inside it. Use your cup to pour popcorninto
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:
Cylinder B is not full. There is still space in the cylinder for more popcorn.

9. 8
QUESTION 4
a) Was your prediction correct? How do you know?
b) If your prediction is incorrect, describe what actually happened
ANSWER:
a) Yes, the prediction is correct. It is based on the formula, volume of cylinder equals to .
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 5
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 formula
for the volume of a cylinder to guide your explanation.
ANSWER:
a) 𝑣 = 𝜋𝑟2ℎ
b) 𝑣 = 𝜋𝑟2ℎ
= 𝜋 × 1.3 × 1.3 × 11
= 58.4 𝑖𝑛𝑐ℎ3
c) 𝑣 = 𝜋𝑟2ℎ
= 𝜋 × 1.7 × 1.7 × 8.5
= 77.2 incℎ3
d) The cylinders have different radius and heights, so the volumes are different.

10. 9
DATA AND OBSERVATION
 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 B
HEIGHT, incℎ2 11 8.5
DIAMETER, incℎ2 2.6 3.4
RADIUS, incℎ2 1.3 1.7
VOLUME, incℎ3 58.4 77.2
QUESTION 6
Which 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.
ANSWER:
CYLINDER RADIUS HEIGHT VOLUME
ORIGINAL 3 10 282.7
INCREASEDRADIUS 4 10 502.7
INCREASEDHEIGHT 3 11 311
Which increased dimension had a larger impact on the volume of the cylinder? Why do you think
this is true?
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.

11. 10
SECTION B
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:
1- CYLINDER CONTAINER – OPENED TOP
𝑆𝑈𝑅𝐹𝐴𝐶𝐸 𝐴𝑅𝐸𝐴 = 2𝜋rℎ + 𝜋𝑟2 = 300
ℎ =
300− 𝜋𝑟2
2𝜋r
𝑣𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟2ℎ
= 𝜋𝑟2(
300− 𝜋𝑟2
2𝜋r
)
Maximum volume=
𝑑𝑣
𝑑𝑟
= 0
𝜋𝑟2 (
300− 𝜋𝑟2
2𝜋r
) = (
300𝜋𝑟2−𝜋𝑟4
2𝜋r
)
=(
300𝜋𝑟2−𝜋𝑟4
2
)𝜋𝑟−1
=
300𝜋r−𝜋𝑟3
2
𝑑𝑣
𝑑𝑟
=
300
2
−
3𝜋𝑟2
2
= 0 h=
300− 𝜋𝑟2
2𝜋r
= 150 − 3𝜋𝑟2 =
300− 𝜋(5.64)2
2𝜋(2.64)
3𝜋𝑟2
2
= 150 h= 5.64cm
3𝜋𝑟2 = 300
𝜋𝑟2 = 100
r = 5.64 cm
VOLUME = 563.62 cm

12. 11
2- CUBE CONTAINER – OPENED TOP
Surface Area = ² + 4² = 300cm²
5 ² = 300
² = 60
 = 7.75cm
Volume = ³
= (7.75)³
= 465.48 cm³
3- CUBOID CONTAINER – OPENED TOP
 Assume that length is twice its width or others
Surface Area = 2² +4h = 300cm²
h =
300−2∫
2
4
Volume = 2² h
2² (
300−2∫
2
4
)
Maximum volume =
𝑑𝑣
𝑑𝑟
= 0 ∫
2
= 50
2² (
300−2∫
2
4
) = (
600 ∫
2
−4∫
2
4
)  = 7.07 cm
= (
600 ∫
2
−4∫
4
4
)∫ −1 h =
300−2(7.07)2
4(7.07)
= 150 - ∫
3
h = 7.07 cm
𝑑𝑣
𝑑𝑟
= 150 − 3∫
2
=0 VOLUME = 2∫ ℎ
2
150 = 3∫
2
= 2 (7.07)2 (7.07)
= 706.79 c𝑚3

13. 12
4- 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− 2 𝑟4
2 𝑟2
Volume =
1
3
 𝑟2ℎ
volum𝑒2 =
1
9
2 𝑟4ℎ2
=
1
9
2 𝑟4(
90000−2 𝑟4
2 𝑟2
)
Maximum volume =
𝑑𝑣
𝑑𝑟
= 0
𝑑𝑣
𝑑𝑙
= 10000 −
2 𝑟4
3
= 0
300√3
4
=
2 𝑟4
3
r = 7.42
h = 10.51
VOLUME =
1
3
 (7.42)2 (10.51)
= 605.95 c𝑚3

14. 13
CONCLUSION
CONTAINER HEIGHT RADIUS LENGTH WIDTH VOLUME
CYLINDER 5.64 5.64 - - 563.69
CUBE 7.75 - 7.75 7.75 465.48
CUBOID 7.07 - 7.07 14.14 706.79
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 cuboid can be filled in with the most amount
popcorn. These means that cube is the container that can be filled with the least amount of
popcorn. Randomly, surveying at the movie theatre , no cube or cuboid shapes can be found.
Therefore, in this case, the cuboid was the most preferable container that can have the most
popcorns.
 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.
 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 shaped
container. It is because this shape is the easiest production and it takes less effort and
also no time consuming to produce.