By: Harry Mizraji

The Penny Problem

The first challenge to complete is the Penny Problem. The radio station is giving the winner of this challenge a prize pack that includes tickets to see his or her favorite band in concert. To start off the challenge, the radio station has placed pennies in a cylindrical glass jar. Each penny is 0.75 inches in diameter and 0.061 inches thick. If the cylindrical glass jar containing the pennies has a diameter of 6 inches and a height of 11.5 inches, how many pennies can fit inside the jar? You must show all work to receive credit.

Volume of Cylindrical Glass Jar

V=3.14r²h

V=3.14(3²)(11.5)

V=3.14(9)(11.5)

V=3.14(103.5)

V=324.99 inches³

Volume Per Penny

V=3.14r²h

V=3.14(0.375)²(0.061)

V=3.14(0.140625)(0.061)

V=3.14(0.008578125)

V=0.0269353125

Divide the Volume of the Jar by the volume of the penny:

324.99 inches/0.0269353125=12065.57377≈ about 12,066 pennies can fit inside the jar!

The Tennis Trouble

Now it's time to move on to the second challenge called Tennis Trouble. The second challenge is to figure out how many tennis balls fit in a specially designed container. The radio station will give away a prize pack and a pair of front row tickets to the winner of this challenge! Each tennis ball is 2.63 inches in diameter. A sketch of the specially designed container is below. How many tennis balls can fit inside the container? How many more tennis balls could fit into the container if the container’s dimensions are doubled? You must show all work to receive credit.

Part 1:

Cylinder:

Height is 36 inches

Diameter is 13 inches

V=3.14r²h

V=3.14(6.5)²(36)

V=3.14(42.25)(36)

V=3.14(1521)

V=4,775.94

Cone:

Height is 8 inches

Diameter is 13 inches

V=1/3(3.14*r²)(h)

V=1/3(3.14*6.5²)(8)

V=1/3(3.14*42.25)(8)

V=1/3(132.665)(8)

V=1/3(1061.32)

V=353.7733333333333333

Tennis Ball:

Diameter is 2.63 Inches

V=4/3(3.14)(r³)

V=4/3(3.14)(1.315³)

V=4/3(3.14)(2.273930875)

V=4/3(7.1401429475)

V=9.520190597

Divide volume of entire whole figure by volume of tennis ball:

Volume of figure/volume of tennis ball=number of tennis balls that can fit in jar

5129.713333/9.520190597=538.8246464956777167

About 539 tennis balls can fit in the jar!

Tennis Trouble Part 2

How many more tennis balls could fit into the container if the container’s dimensions are doubled?

New Cylinder:

Height: 72

Diameter: 26

V=3.14r² h

V=3.14(13²)(72)

V=3.14(169)(72)

V=3.14(12,168)

V=38,207.52

New Cone:

Height:16

Diameter: 26

V=1/3(3.14*r² )(h)

V=1/3(3.14*13² )(16)

V=1/3(3.14*169)(16)

V=1/3(530.66)(16)

V=1/3(8,490.56)

V=2830.18667

Volume of whole new figure:

38,207.52+2830.18667=41037.70667

Tennis Ball:

Diameter is 2.63 Inches

V=4/3(3.14)(r³)

V=4/3(3.14)(1.315³)

V=4/3(3.14)(2.273930875)

V=4/3(7.1401429475)

V=9.520190597

Divide volume of whole figure by volume of tennis ball:41037.70667/9.520190597=4310.597173≈4311 tennis balls would fit in new figure.

Now, to find how many more tennis balls figure 2 can hold than figure 1, subtract 539 by 4311.

4311-539=3,772

If the containers dimensions were doubled, the new one would hold 3,772 more tennis balls than the original.

**Challenge #3- Giant Gum**

The Pharaoh Chewing Gum Company has decided to sponsor an additional prize in the radio station's contest. They are giving away backstage passes for the concert! Pharaoh Chewing Gum manufactures a new product they are trying to promote. The new product is a pyramid-shaped gum with a square base. In the spirit of the other challenges, the company has decided to place their pyramid-shaped gum inside a clear, glass, giant, bubblegum-shaped sphere. Each piece of gum has a base measurement of 1 inch and a height of 0.75 inches. The glass sphere container has a diameter of 17.25 inches. How many pieces of Pharaoh Chewing Gum can fit inside the glass sphere? You must show all work to receive credit.

Sphere:

Diameter:17.25 inches

V=4/3(3.14*r³)

V=4/3(3.14*8.625³)

V=4/3(3.14*641.6191406)

V=4/3(2014.684102)

V=2,686.245469

Gum:

Base=1 inch

Height=0.75 inch

V=1/3(B)(h)

V=1/3(1*1)(0.75)

V=1/3(1)(0.75)

V=1/3(0.75)

V=0.25

Divide the volume of the sphere by the volume of the gum:

2,686.245469/0.25=10744.981876≈10,745

10,745 Pieces of Gum can fit inside the sphere!

**Questions**

For the Penny Problem, how much empty space should exist inside the jar after being filled to capacity with pennies? Why doesn't this amount of space actually exist in the jar?

The space does not exactly have a correct way to be in the jar. Also, the pennies land in the jar at different angles so there is no way to tell the exact empty space in the jar. If it is filled to capacity, it is filled all the way, so the only empty space would be air.

Where does the formula for the volume of a cylinder derive from? Give an example, and provide evidence to support your claim.

The formula for the volume of a cylinder comes from the formula for finding area of a circle. For example, you use the formula for finding the area of a circle in the beginning of the volume formula when performing A=3.14(r²) is used for finding the base of a cylinder.

In the Tennis Challenge, a cone was used for calculations, and in Giant Gum, the formula for the volume of a pyramid was needed. Pick either the formula for the volume of a cone or the volume of a pyramid, and explain where the formula you chose derives from? Give an example, and provide evidence to support your claim.

The formula for volume of a cone comes from the formula for volume of a cylinder. They are both 1/3 times the base area times the height. Since the formula for volume of a cylinder requires the base area, they both use the area of a circle formula to find it.

In Tennis Trouble, the container used for the challenge is labeled "A" in the image below. If the container’s shape was modified to look like container "B," what effect would it have on the capacity (volume) of the container if the dimensions remained unchanged? What theory or principle helps to prove your point?

I believe that if container A and B were modified to look the same, they would have the same volume. Like in the lesson with the example of the pennies, I believe that the figures would have the same volume based on Cavalieri's principle. It states if the height and cross sections of the figures are congruent, then they have the same volume.

In Giant Gum, the gum is shaped like a pyramid. What shape do you think would best fit into the container? (Choose a shape other than a pyramid.) Explain why the shape you chose was better and back up your answer with proof, such as calculations and writing.

I think that the best shape other than a pyramid would be a cube to fit into the container. According to the lesson, 6 equal size pyramids form a cube. The volume of a pyramid is b3/6, therefore, for every 6 gums in that container, a cube should be formed. If you take the number of gums that fit into the container (10,745) and divide by 6, the number that comes out is about 1,791 pieces of gum, exhibiting that a cube is the best shape other than a pyramid to fit into the container.