Addmaths Project 2014

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ADDITIONAL

MATHEMATICS

PROJECT WORKTASK 2 / 2012

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INTRODUCTION


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TITLE

APPLICATIONOF

MATHEMATICSIN POPCORN

PACKAGING


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HISTORY OFPOPCORN

Popcorn was first discovered thousands of years ago by Native Americans. It is one of theoldest forms of corn: evidence of popcorn from 3600 B.C. was found in New Mexico and evenearlier 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 asource 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 theyhad 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 usedfor popcorn production is specifically planted for this purpose; most is grownin Nebraska and Indiana, with increasing area in Texas.

As the result of an elementary school project, popcorn became the official state snackfood of Illinois, U.S.A.
http://en.wikipedia.org/wiki/Indigenous_peoples_of_the_Americashttp://en.wikipedia.org/wiki/Great_Depressionhttp://en.wikipedia.org/wiki/World_War_IIhttp://en.wikipedia.org/wiki/Rationing#United_Stateshttp://en.wikipedia.org/wiki/Rationing#United_Stateshttp://en.wikipedia.org/wiki/Candyhttp://en.wikipedia.org/wiki/Midwestern_United_Stateshttp://en.wikipedia.org/wiki/Ridgway,_Illinoishttp://en.wikipedia.org/wiki/Valparaiso,_Indianahttp://en.wikipedia.org/wiki/Van_Buren,_Indianahttp://en.wikipedia.org/wiki/Schaller,_Iowahttp://en.wikipedia.org/wiki/Schaller,_Iowahttp://en.wikipedia.org/wiki/Marion,_Ohiohttp://en.wikipedia.org/wiki/North_Loup,_Nebraskahttp://en.wikipedia.org/wiki/United_States_Department_of_Agriculturehttp://en.wikipedia.org/wiki/Cornhttp://en.wikipedia.org/wiki/Nebraskahttp://en.wikipedia.org/wiki/Indianahttp://en.wikipedia.org/wiki/Texashttp://en.wikipedia.org/wiki/Elementary_schoolhttp://en.wikipedia.org/wiki/Illinoishttp://en.wikipedia.org/wiki/Illinoishttp://en.wikipedia.org/wiki/Elementary_schoolhttp://en.wikipedia.org/wiki/Texashttp://en.wikipedia.org/wiki/Indianahttp://en.wikipedia.org/wiki/Nebraskahttp://en.wikipedia.org/wiki/Cornhttp://en.wikipedia.org/wiki/United_States_Department_of_Agriculturehttp://en.wikipedia.org/wiki/North_Loup,_Nebraskahttp://en.wikipedia.org/wiki/Marion,_Ohiohttp://en.wikipedia.org/wiki/Schaller,_Iowahttp://en.wikipedia.org/wiki/Schaller,_Iowahttp://en.wikipedia.org/wiki/Van_Buren,_Indianahttp://en.wikipedia.org/wiki/Valparaiso,_Indianahttp://en.wikipedia.org/wiki/Ridgway,_Illinoishttp://en.wikipedia.org/wiki/Midwestern_United_Stateshttp://en.wikipedia.org/wiki/Candyhttp://en.wikipedia.org/wiki/Rationing#United_Stateshttp://en.wikipedia.org/wiki/World_War_IIhttp://en.wikipedia.org/wiki/Great_Depressionhttp://en.wikipedia.org/wiki/Indigenous_peoples_of_the_Americas


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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.


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SECTION A


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QUESTION 1For 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.

diagram

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 height

and diameter with a ruler and record you data below and on the cylinder. Label it

Cylinder B.

diagram


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ANSWER 1DIMENSION CYLINDER A CYLINDER B

HEIGHT 11.0 8.5DIAMETER 2.6 3.4

RADIUS 1.3 1.7


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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 thanCylinder A. Although the height of Cylinder B is shorter than Cylinder A, but this does notaffect much compare the affect of different in radius.


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


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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

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

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


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QUESTION 5a) 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 theformula for the volume of a cylinder to guide your explanation.

ANSWER 5a) V =

b) V = = x 1.3 x 11= 58.4 inch

c) V = h = x 1.7 x 8.5= 77.2 inch

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


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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, inch 11.0 8.5

DIAMETER, inch 2.6 3.4RADIUS, inch 1.3 1.7

VOLUME, inch 58.4 77.2


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ANSWER

1. Cylinder Container opened top

Surface Area = 2 r h + = 300

h = diagram

Volume = h

=

Maximum Volume = = 0


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Volume = 563.62

2. Cube Container opened topSurface Area = l + 4l = 300cm

5 l = 300

l = 60

l = 7.75cm diagram

Volume = l

= (7.75)

= 465.48 cm


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3. Cuboid Container opened top

Assume that length is twice its width or others

Surface Area = 2l +4hl = 300cm

h = diagram

Volume = 2l h

= 2l

Maximum Volume, = 0

2l =

=

= 150l - l

= 150 3l = 0


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150 = 3l h =

l = 50 h = 7.07cm

l = 7.07cm

Volume = 2lh

= 2(7.07)(7.07)

= 706.79cm

4. Cuboid Container opened top Assume that length is equal to its width

Surface Area = l + 4hl = 300

h = diagram

Volume = l h

= l

Maximum Volume, = 0

= 75 = 0


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75 =

l = 10

h = 5

Volume = 500

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

Area of the base = 6

= 6

=

Surface Area = 6hx + = 300 diagram

h =


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Volume = base area height

= h

=

Maximum Volume, = 0

= -

=

x = 4.39

h = 9.49

Volume = 475.17

6. Cone Container opened topFrom the diagram, x = r + h

Surface Area = r x = 300cm

r x = 300

r ( r + h) = 90000

h =


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Volume = r h

Volume = r 4 h diagram

= r 4

Maximum Volume, = 0

= 10000 = 0

=

r = 7.42

h = 10.51

Volume = (7.42) (10.51)

= 605.95cm

CONCLUSIONContainer height radius length width volumeCylinder 5.64 5.64 - - 563.69Cube 7.75 - 7.75 7.75 465.48

Cuboid 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


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Shape of containers that give the most popcorn reflect the maximum volume. From theactivity earlier, I found that increasing the radius increased the volume more than increasing theheight. This is because the radius is squared to find the volume, which increases its impact onthe 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 thecontainer that can be filled with the least amount of popcorn. Randomly, surveying at the movietheater, no cube or cuboid shapes can be found. Therefore, in this case, the cuboid1 was themost 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 cuteand simple shape.

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


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REFLECTIONIn the making of this project, I have spent countless hours doing this project. I realized that thissubject is a compulsory to me. Without it, I cant fulfill my big dreams and wishes.


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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 ofeffort!

Addmaths Project 2014

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