**Module 8 Quiz**

Solve: Part A

cone height =8 inches

Diameter = 13 inches

V = 1/3 ( 3.14*r)^2 (h)

V = 1/3 (3.14 * 6.5) (8)

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

V = 1/3 (132.665)(8)

V =1/3 (1,061.32)

V = 353.7733333

Solve : Part B

New cylinder height = 72

Diameter = 26

V = 3.14r^2 h

V = 3.14 (13)^2 (72)

V = 3.14 (169) (72)

V = 3.14 (12,168)

V = 38,207.52

**Conclusion**

challenge # 1 - Penny problem

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

Solve

Glass sphere: diameter 17.25

Gum : base 1 inch and height 0.75 inches

Answering the Questions

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?

Challenge # 2 Tennis trouble

The second challenge is to figure out how many tennis balls fit in a specially designed container. 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

Solve

Each Penny: 0.75 in diameter

0.065 inches thick.

Cylindrical Glass jar: 6 inches in diameter 11.5 inches in height.

Challenge #3: Giant Gum

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

Volume of cylindrical glass

V = 3.14r^2 h

V = 3.14 (3)^2 (11.5)

V = 3.14 (9) (11.5)

V = 3.14 (103.5)

V = 324.99 inches.

Volume of Pennies

V = 3.14r ^2 h

V = 3.14 (0.375)^2 (0.061)

V = 0.4415625 ( 0.061)

V = 0.026935312

Then you divide the cylindrical glass jar's volume by th pennies volume

324.99/0.026935312 =12065.57399 or 12065.57

Cylinder Height = 36 inches

Diameter= 13 inches.

V = 3.14r^2 h

V = 3.14 (6.5)^2 (36)

V = 3.14 (42.25) (36)

V = 3.14 ( 1,521)

V = 4,775.94

Tennis Ball Diameter 2.63

V = 4/3 (3.14 r)^2

V = 4/3 (3.14)(1.315)

V = 4/3 (3.14) 2.273930875)

V = 4/3 (7.140142948)

V = 9.520190597

To find the total volume of the figure you would have to add the cone volume with the cylinders volume.

353.7733333 + 4775.94 = 5129.713333

To find how many tennis balls can fit in the speacial container you would have to divide the total volume of the figure by the tennis balls volume. (5129.713333 /9.520190 597) = 538.82464665 or 539 tennis balls can fit.

New cone height = 16

Diameter = 26

V = 1/3 (3.14r^2)( h)

V = 1/3 (3.14 )(13)^2 (16)

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

V = 1/3 (530.66) (16)

V = 1/3 (8,490.56)

V =2,830.186667

Tennis Ball Diameter 2.63

V = 4/3 (3.14 r)^2

V = 4/3 (3.14)(1.315)

V = 4/3 (3.14) 2.273930875)

V = 4/3 (7.140142948)

V = 9.520190597

To find the total volume of the new figure you would have to add the new cone volume with the new cylinder volume 2830.186667 + 38207.52 = 41037.70667

you can find how many tennis balls can fit into the new container by dividing the total volume of the new figure by the volume of the tennis balls.

41037.70667 / 9.520190597=4310.5971726 or 4311. To find out how many more tennis balls

fit into the container when the dimension is doubled you would subtract how many

balls fitted for the original problem from the new problem ( 4311- 539 )= 3772 more balls can fit

Sphere

V = 4/3 (3.14r^2)

V = 4/3 (3.14)( 8.625)^2

V = 4/3 ( 3.14 )( 641.6191406)

V = 4/3 ( 2014.684102)

V = 2,686.245469

Gum

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

To find how many pieces of Pharaoh Chewing Gum can fit inside the glass sphere you would have to divide the volume of sphere ny the volume of the gum ( 2686.245469/0.25)=10744.981876 or 10745 gum pieces

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, you can 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 so this means that the empyt space does nto exist because pennies will fall in a jar and land in any position, because there is not a proper way to put pennies in a 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 for the volume of the cylinder derives from the formula of the area of a circle. Which is A = 3.14 (r)^2. and this is also similar to the formula to find the volume of the cylinder. And in order to find the volume of the cylinder, you first have to find the base of the cylinder using the formula of the area of a circle.

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 formula for the volume of a cone comes from the formula of the volume of a cylinder. both these formulas equal to 1/3 of the base area and times the height of the figure. to find the area of the base of the cone you will need to use the formula of the area of a circle. The formula of a to find the volume of a cone is V = 1/3 (area of the base)(Height) And in order to find the volume of the cone you have to know the area of the base. Once you find this you can multiply it by the height.

4. 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?

In figure A you have a cylinder a cylinder height and altitude can be drawn by drawing a straight line in the center perpendicular to the outer walls, and a line from the center to the wall. In Figure B this a oblique cylinder it does not have a height or a altitude. I do not think the volume of the tennis balls would be affected if they were placed in container B because both container look similar. I think the principle that would best support this is Cavalieri's Principle. Was when he stacked the pennies differently in two opposite stacks and proved they both had the same volume because their radius, diameter, and height. He also proved that if the cross section, and the height were congruent in a 3-D figure then the two figures have the same volume.

5. 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.

The shape that 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.