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ADDITIONALMATHEMATICS PROJECT
2014
TASK : 2
TITTLE : POPCORN ANYONE?
NAME : NG WAI SAM
FORM : 5 UTARID
IC NUMBER : 970530 - 085193
TEACHER : MISS YUEN
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CONTENT
NO. Topic Page
1. Content 1 - 2
2. Introduction 3 - 4
3. Acknowledge 5
4. History 6
5. Objective and
Moral Values
7
6. Specification of Task
and Definition of
Problem
8
7. Section A 9 - 14
8. Section B 1522
8. Conclusion 23 - 24
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INTRODUCTION
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Title
POPCORN ANYONE?
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ACKNOWLEDGEMENT
First of all, I would like to express my special thanks to my additional
mathematics teacher, Miss Yuen who gave me the opportunity to do this project and
help me a lot throughout finishing this project. Without her guide, I may not done my
project nicely .
Secondly, I would like to thanks my parents and my family for providing
everything , such as money and energy 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.
I would like to thanks 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.
Besides 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 theco-operation given.
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HISTORY OF VOLUME
INVENTOR
Archimedes of Syracuse(Greek:;c.287BCc.212BC) was anancient
Greekmathematician,physicist,engineer,inventor,andastronomer.Although few
details of his life are known, he is regarded as one of the leadingscientists inclassical
antiquity.Among his advances inphysics are the foundation of hydrostatics,staticsand an explanation of the principle of thelever.He is credited with designing
innovativemachines,includingsiege engines and thescrew pump that bears his name.
Modern experiments have tested claims that Archimedes designed machines capable
of lifting attacking ships out of the water and setting ships on fire using an array of
mirrors.
Archimedes is generally considered to be the greatestmathematician of antiquity and
one of the greatest of all time. He used themethod of exhaustion to calculate the
area under the arc of aparabola with thesummation of an infinite series,and gave a
remarkably accurate approximation ofpi.He also defined thespiralbearing his name,
formulae for thevolumes ofsolids of revolution,and an ingenious system for
expressing very large numbers.
Archimedes died during theSiege of Syracuse when he was killed by aRoman soldier
despite orders that he should not be harmed.Cicero describes visiting the tomb of
Archimedes, which was surmounted by asphereinscribed within acylinder.
Archimedes had proven that the sphere has two thirds of the volume and surface area
of the cylinder (including the bases of the latter), and regarded this as the greatest of
his mathematical achievements.Unlike his inventions, the mathematical writings of Archimedes were little known in
antiquity. Mathematicians fromAlexandria read and quoted him, but the first
comprehensive compilation was not made until c.530 AD byIsidore of Miletus,
while commentaries on the works of Archimedes written byEutocius in the sixth
century AD opened them to wider readership for the first time. The relatively few
copies of Archimedes' written work that survived through theMiddle Ages were an
influential source of ideas for scientists during theRenaissance,while the discovery in
1906 of previously unknown works by Archimedes in theArchimedes Palimpsest has
provided new insights into how he obtained mathematical results.
http://en.wikipedia.org/wiki/Greek_languagehttp://en.wiktionary.org/wiki/%E1%BC%88%CF%81%CF%87%CE%B9%CE%BC%CE%AE%CE%B4%CE%B7%CF%82http://en.wiktionary.org/wiki/%E1%BC%88%CF%81%CF%87%CE%B9%CE%BC%CE%AE%CE%B4%CE%B7%CF%82http://en.wiktionary.org/wiki/%E1%BC%88%CF%81%CF%87%CE%B9%CE%BC%CE%AE%CE%B4%CE%B7%CF%82http://en.wikipedia.org/wiki/Ancient_Greecehttp://en.wikipedia.org/wiki/Ancient_Greecehttp://en.wikipedia.org/wiki/Greek_mathematicshttp://en.wikipedia.org/wiki/Physicisthttp://en.wikipedia.org/wiki/Engineerhttp://en.wikipedia.org/wiki/Inventorhttp://en.wikipedia.org/wiki/Astronomerhttp://en.wikipedia.org/wiki/Scientisthttp://en.wikipedia.org/wiki/Classical_antiquityhttp://en.wikipedia.org/wiki/Classical_antiquityhttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Fluid_staticshttp://en.wikipedia.org/wiki/Staticshttp://en.wikipedia.org/wiki/Leverhttp://en.wikipedia.org/wiki/Machinehttp://en.wikipedia.org/wiki/Siege_enginehttp://en.wikipedia.org/wiki/Archimedes%27_screwhttp://en.wikipedia.org/wiki/Mathematicianhttp://en.wikipedia.org/wiki/Method_of_exhaustionhttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Parabolahttp://en.wikipedia.org/wiki/Series_(mathematics)http://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Archimedes_spiralhttp://en.wikipedia.org/wiki/Volumehttp://en.wikipedia.org/wiki/Solid_of_revolutionhttp://en.wikipedia.org/wiki/Siege_of_Syracuse_(214%E2%80%93212_BC)http://en.wikipedia.org/wiki/Roman_Republichttp://en.wikipedia.org/wiki/Cicerohttp://en.wikipedia.org/wiki/Spherehttp://en.wikipedia.org/wiki/Inscribehttp://en.wikipedia.org/wiki/Cylinder_(geometry)http://en.wikipedia.org/wiki/Alexandriahttp://en.wikipedia.org/wiki/Isidore_of_Miletushttp://en.wikipedia.org/wiki/Eutocius_of_Ascalonhttp://en.wikipedia.org/wiki/Middle_Ageshttp://en.wikipedia.org/wiki/Renaissancehttp://en.wikipedia.org/wiki/Archimedes_Palimpsesthttp://en.wikipedia.org/wiki/Archimedes_Palimpsesthttp://en.wikipedia.org/wiki/Renaissancehttp://en.wikipedia.org/wiki/Middle_Ageshttp://en.wikipedia.org/wiki/Eutocius_of_Ascalonhttp://en.wikipedia.org/wiki/Isidore_of_Miletushttp://en.wikipedia.org/wiki/Alexandriahttp://en.wikipedia.org/wiki/Cylinder_(geometry)http://en.wikipedia.org/wiki/Inscribehttp://en.wikipedia.org/wiki/Spherehttp://en.wikipedia.org/wiki/Cicerohttp://en.wikipedia.org/wiki/Roman_Republichttp://en.wikipedia.org/wiki/Siege_of_Syracuse_(214%E2%80%93212_BC)http://en.wikipedia.org/wiki/Solid_of_revolutionhttp://en.wikipedia.org/wiki/Volumehttp://en.wikipedia.org/wiki/Archimedes_spiralhttp://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Series_(mathematics)http://en.wikipedia.org/wiki/Parabolahttp://en.wikipedia.org/wiki/Areahttp://en.wikipedia.org/wiki/Method_of_exhaustionhttp://en.wikipedia.org/wiki/Mathematicianhttp://en.wikipedia.org/wiki/Archimedes%27_screwhttp://en.wikipedia.org/wiki/Siege_enginehttp://en.wikipedia.org/wiki/Machinehttp://en.wikipedia.org/wiki/Leverhttp://en.wikipedia.org/wiki/Staticshttp://en.wikipedia.org/wiki/Fluid_staticshttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Classical_antiquityhttp://en.wikipedia.org/wiki/Classical_antiquityhttp://en.wikipedia.org/wiki/Scientisthttp://en.wikipedia.org/wiki/Astronomerhttp://en.wikipedia.org/wiki/Inventorhttp://en.wikipedia.org/wiki/Engineerhttp://en.wikipedia.org/wiki/Physicisthttp://en.wikipedia.org/wiki/Greek_mathematicshttp://en.wikipedia.org/wiki/Ancient_Greecehttp://en.wikipedia.org/wiki/Ancient_Greecehttp://en.wiktionary.org/wiki/%E1%BC%88%CF%81%CF%87%CE%B9%CE%BC%CE%AE%CE%B4%CE%B7%CF%82http://en.wikipedia.org/wiki/Greek_language8/11/2019 ADDITIONAL MATHEMATICS PROJECT.docx
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OBJECTIVES1. 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 .
7.Train students to appreciate the intrinsic values of mathematics and to become more
creative and innovative.
8. Realize the importance and the beauty of mathematics.
MORAL VALUESThe moral values that I learned from this would is to appreciate and understand how
mathematics intervenes our everyday life. Without mathematics to compute the gainand loss, there will be no economies, commerce, and businesses. Human would have
to make technological progress without mathematics to confirm computations of
theories and finding. There would be no navigational for lots of things. Even the
smallest feat of like telling the times involved mathematics.
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SECTION A
Specification Of Task
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For this project work there is two section which are section A and section B. For
section A I would like to the volume of two cylinders with different dimensions as
below. Then check whether the cylinders will hold the same amount of popcorn. Then
place the cylinder B on the paper plane with the cylinder A inside it. Use your cup to
pour popcorn into the cylinder A until is full. Carefully, lift cylinder A so that the
popcorn falls into cylinder B. Describe what happened.
Definition of Problem
Identifying the problem
Question asks to compare the volume of to cylinders creating 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 using the given dimensions according to the instructions given.
2. Measure the height and dimensions of the two labelled cylinder with a ruler.
3.Record a data obtained in a table
QUESTION 1
For this activity, you will be comparing the volume of 2 cylinders created using thesame sheet of paper. You will be determining which dimension can hold more
Take the white paper and roll it up along the longest side to form a
baseless cylinder.
Tape along the edges. Measure the dimensions with a ruler and
record your data below and on the cylinder.
Label it Cylinder A.
Take the colored paper and roll it up along the shorter side to form
a baseless cylinder
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.
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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. coloured paper, tape, popcorn plate, cup, ruler
1.Take the white paper and roll it up along the longest side to form a baseless cylinder
thatIs 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.8.5
11
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.
11
8.5
ANSWER 1
DIMENSION CYLINDER A CYLINDER B
HEIGHT 11.00 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 willhold more than the other? Which one? Why?
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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 4
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 sinceradius, 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 cylinderb) Calculate the volume of Cylinder A.
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c) Calculate the volume of Cylinder B.
d) Explain why the cylinders do or do not hold the same amount. Use the formula for
theformula for the volume of a cylinder to guide your explanation.
ANSWER 5
a) V =
b) V = =x 1.3 x 11= 58.4 inch
c) V == x 1.7 x 8.5
= 77.2 inch
d) The cylinders have different radius and heights, so the volumes are different
QUESTION 6
Which measurement impacts the volume more : the radius or the height? Workthrough the example below to help you answer the question .
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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?
CYLINDER RADIUS HEIGHT VOLUME
ORIGINAL 3 10 282.7
INCREASED RADIUS 4 10 502.7INCREASED 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.
0
100
200
300
400
500
600
Original Increased
Radius
Increased
Volume
Radius
Height
Volume
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SECTION B
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Specification Of Task
DefinitionIdentifying the problem:
The question requires us to identify the popcorn container that can the most value of
popcorn.
QUESTIONIf you were buying popcorn at the movie theater and wanted the most popcorn, whattype of container would you look for? Clue : You need more than one type
of containers. You are given 300 cm of thin sheet material. Explain the details.
ANSWER
First, calculate the maximum value that can be required from a cylinder
using 300 cm3
of thin sheet material.
Then calculate the maximum value that can be required from a cube using
300 cm3of thin sheet material.
Then calculate the maximum value that can be required from a cuboid 1
using 300 cm3 of thin sheet material.
Then calculate the maximum value that can be required from a cuboid 2usin 300 cm3 of thin sheet material.
Then calculate the maximum value that can be required from a hexagon
container using 300 cm3 of thin sheet material.
Then calculate the maximum value that can be required from a cone using
300 cm3of thin sheet material.
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1.Cylinder Containeropened top
Surface Area = 2r h += 300h = 300
2
Maximum Volume= dvdr
= 0
300 = 300 -r 4
2 2
= 300-r 4 r -1
2
= 300r3
2
=
-
= 0 h=
= 150 -
=
= 150 = 5.64 cm
3= 300
= 100r = 5.64
Volume= 563. 62 cm3
2.Cube Containeropened top
Surface Area = l2+4l2=300cm2
5l2=300cm2
l2=60 cm2
l = 7.75cm
volume = l3
=7.753
=465.48cm3
3.Cuboid Containeropened top
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that length is twice its width or others
Surface Area= 2l +4hl = 300cm
h =
Volume= 2l2h
= 2l
Maximum Volume =
= 0
2l2
=
=
= 150 l- l
= 1503l2= 0 h =
150 = 3l = 7.07 cm
l =50
l= 7.07cm
Volume = 2lh
= 2(7.07)(7.07)
= 706.79cm
4.Cuboid Containeropened top
Assume that length is equal to its width
Surface Area= l + 4hl= 300
h=
Volume= l h
= l2
Maximum Volume,
= 0
= 75-
= 0
75 =
300 = 3l2
100 = l2
l = 10
Volume = l2h
= 102 (5)
= 500 cm3
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5.Hexagon Containeropened top Assume that the length of the side = x
Area of the base= 6
ab
= 6
x(x)
=
(x2 )
Surface Area= 6hx+
(x2) =300
h =
Volume= base area height
=
=
Maximum Volume,
= 0
=
-
= 0
=
x = 4.39
h =
= 9.49
Volume=
= 475.17cm3
6.Cone Containeropened top
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From the diagram,
x = r + h
Surface Area = r x = 300cm
r x = 300
r ( r + h) = 90000
h =
Volume=
Volume=
Maximum Volume,
= 0
= 10000 -
= 0
10000 =
30000 = r = 7.42
h =
= 10.51cm
Volume =
= 605.95cm
Container Height Radius Length(cm) Width(cm) volume
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Cylinder 5.64 5.64 - - 563.69
Cube 7.75 - 7.75 7.75 465.48
Cuboid 1 7.07 - 7.07 14.14 706.79
Cuboid 2 5.00 - 10.00 10.00 500.00
Hexagon 9.49 - 4.39(side) - 475.15
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 orcuboid 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 due to its volume. 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.
Volume of container (cm3)
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0
100
200
300
400
500
600
700
800
cone hexagon cuboid 2 cuboid 1 cube cylinder
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CONCLUSION
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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 the height. I have
also realized that different dimensions of cylinder are used for different purposes.
Therefore, there are benefits and consequences depending on the purpose of its
dimensions.
If a person works as a popcorn seller, he need to find a container which has the
least volume to hold the least amount of popcorn. therefore, he will get the most profit
for his sale. The shape of the container holding the popcorn should be considered
before starting the business. A container, which hold the less volume, would raise theprofits of popcorn seller.
If a person is trying to produce a container to hold popcorns, he or she would
choose a container which is easy to produce in order to save the energy, money and
time. Thus it can increase the production rate. Eventually the particular producer
would be able to save more money thus make more profile if he were used a cylinder
container.