additional mathematics project.docx

8/11/2019 ADDITIONAL MATHEMATICS PROJECT.docx

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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_language


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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.

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