Pharaoh Chewing Gum manufactures a new product to promote in the shape of a pyramid gum with a square base

Decide to place the pyramid shaped gum inside a clear glass giant bubble gum shaped sphere

Each piece of hum has a base of 1 inch and a height of 0.75 inches

Glass sphere container has a diameter of 17.25 inches

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 bubble-gum 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.

Answering the Questions:

Challenge #1 - 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.

Challenge #2 - 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.

Contest Directions

A local radio station, Really Rad Radio is having a contest. There are three challenges within the contest and each must be completed to enter. Several prizes will be given to the winners of the contest by the radio station.

**08.03 Module Eight Quiz**

The Penny Problem

Each penny is 0.75 inches in diameter, 0.061 inches thick

Cylindrical glass jar has diameter of 6 inches, height of 11.5 inches

Radius is half of the diameter

*How many pennies can fit inside the jar?*

Tennis Trouble = Part A

Each tennis ball is 2.63 inches in diameter

*Figure out how many tennis balls fit in a specially designed container*

1.) 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?

Volume of Cylindrical Glass Jar:

Volume of Penny

*Divide the volume of cylindrical glass jar by the volume of the penny*

Approximately 12,066 pennies can fit inside the jar.

Cylinder:

Height = 36 inches

Diameter= 13 inches

Cone:

Height = 8 inches

Diameter = 13 inches

Radius = half of diameter

Add the volumes of both the cone and the cylinder to get the volume of the whole figure

Tennis Ball:

Diameter : 2.63 inches

Divide the volume the whole figure by the volume of the tennis ball

About 539 Tennis Balls can fit in the designed container

Tennis Trouble = Part B

* How many tennis balls can fit into the container if the container's dimensions are doubled?*

Original Cylinder:

Height = 36

Diameter= 13

Original Cone:

Height =8

Diameter = 13

New Cylinder :

Height = 72

Diameter = 26

New Cone:

Height= 16

Diameter = 26

Add the volumes of both the new cylinder and the new cone to get the volume of the whole figure

Tennis ball measures are still the same

*Divide the volume of whole figure by the volume of tennis ball *

*Subtract the result from part A from the result of this part to find how many more tennis balls can this new container hold than the original container*

If the container's dimensions were doubled, It would hold 3,772 more tennis balls than the original container.

*How many pieces of pharaoh chewing gum can fit inside the glass sphere?*

Sphere:

Diameter = 17.25 inches

radius = 1/2 diameter

Gum:

Base = 1 inch

Height = 0.75 inch

*Divide the volume of the sphere by the volume of the gum*

10,745 Pieces of gum can fit inside the glass sphere

The empty space that exists inside the jar being filled with pennies can be found subtracting the volume of the pennies from the volume of the jar. The volume of each penny is 0.026935312, and since approximately 12,066 pennies can fit in the jar, i multiplied the number of pennies by the volume of 1 penny to find the volume of all the pennies together which is 325.0014746. The volume of the jar is 324.99. Basically there is not exact empty space left, since the empty space left is -0.0114746.This space doesn't exist in the jar because the pennies are all together and don't exactly have a correct way to be in the jar. Like when are putting pennies in a jar, they land in different positions so basically there is space all around the jar.

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

The formula of the volume of a cylinder comes from the formula of the area of a circle. When finding the area of a circle, the formula is A= "pi" times the radius squared.In the formula of a cylinder, to find the volume, you must first find the area of the base and multiply it by "h" the height. So either way, in order to find the volume, you must first find the base of the cylinder using the formula of the area of a circle. For example, to find the volume of a stack of CD's the area of the base must be found first (A= 3.14 *radius squared) then the result must be multiplied by the height of the stack of CD's.

Volume of a stack of CD's

Volume = (area of base) * (height)

or

Volume = (3.14 r -squared) (height)

3.) 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 volume of a cone comes from the volume of a cylinder. Both formulas are calculated the same way and are equaled to 1/3 of the base area times the height of the figure. The formula of the volume of a cone is V = 1/3 (area of the base) (height). The area of the base of a cone is found using the formula of area of a circle, since the base is the shape of a circle. The area formula of a circle is 3.14 r2 (radius squared).In order to find the volume of a cylinder, the area of the base must be found and then the result of that can be multiplied by the height. Both formulas help each other .

4.) In Tennis Trouble, the container used for the challenged is labeled "A". 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?

The figure labeled A is a right cylinder, since the height and altitude can be drawn to connect the centers of the circular bases. The figure labeled B is an oblique cylinder, it has no height or altitude so it cannot be drawn to connect the centers of the circular bases.

If the shape of container "A" was modified to look like container "B" , i think the volume would not be affected. The volume would not be affected because both figures appear to be the same, in figure A, it shows the height and diameter while figure b displays and irregular shaped cylinder with not measurements at all. Based on the lesson, the principle that would support this is Cavalieri's Principle. In the lesson, two stacks of pennies were shown, both of the same height, same radius, same diameter, same amount of pennies, thus both had the same volume even though a stack was not properly set up. Based on Cavalieri's Principle, it shows that if the area of the cross sections of two 3-D figures are congruent and the height of the figures is also congruent, then it can be concluded that the volumes of the two figures are congruent. If figure A was changed to look like figure B, the volume would remain the same because the dimensions would not be changed. No matter how the figure is set up, it can be tilted like figure B, yes of course the shape would look different but as long as the dimensions are the same, the volume remains the same.

5.) In Giant gum, the gum was 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.

The shape i think would best fit into the container shaped like a sphere would be a cube. Since the gum was shaped like a pyramid, its volume come from the volume of a cube. Based on the lesson, 6 equal size pyramids can form a cube. The volume of a cube is base x base x base or (B3) , so the volume of a pyramid is 1/6 (b3) = b3 /6. IF in the Giant Gum challenge container about 10,745 pyramid shaped gum could fit, then for each 6 gums a cube is formed. To find how many cubes are made with 10,745 pyramid shaped gums, we would have to divide the amount of gum by 6, since 6 equally size pyramids make a cube. AFter doing calculations, over 1,791 cubes are formed with the 10,745 pyramid shaped gums. Based on this, the shape that would best fit into the container other than the shape of a pyramid would be a cube.

Thank You for Watching :D