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? About 0.01548485 inches cubed should be left in the container. This amount of volume doesn’t really exist as a whole though, because the shape of the pennies fit well in the container and thus leave very little- very spread out bits of volume that cannot be filled with more pennies.

2. Where does the formula for the volume of a cylinder derive from? Give an example and provide evidence to support your claim. The first step that is taken in finding the volume of a cylinder is to find the area of the base. The base is then multiplied by the height. This is because the volume can be described as how many bases thick the cylinder is. If we have a cylinder with a radius of 2 and a height of 5 inches, we known that the base is 12.56 inches squared, and that there is 5 inches worth of this base. We represent this by multiplying to find 62.8 inches cubed.

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. To find the volume of a cone, you start by multiplying the base by the height. As we know, this is the same formula as a cylinder, and because a cone is smaller than a cylinder with the same measurements, we know the formula can’t end there. If we dived by three the number we got by multiplying the base by the height, we will find the volume of a cone. If we have a cone with a radius of 2 and a height of 5 inches, we known that the base is 12.56

inches squared, and multiplying that by 5 will give us 62.8 inches cubed. This is the same volume as the cylinder in the previous question. Because the cone is smaller than the cylinder, we know we have to then dived this by 3 to get 20.93 inches cubed.

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? The volume would be the same, because of Cavalieri’s Principle. This principle states that if the measurements of two 3-D figures are the same, then their volumes are the same.

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. I believe that spheres would fit better inside the container, because it too is a sphere. If we were to use spheres with a diameter of 0.8, we would find that their volume is 0.267939968 which is very similar to the volume of the pyramid shaped pieces of gum. By dividing we can find that 10,025 pieces of the sphere shaped gum will fit in the container. Although less sphere shaped gum fits in the sphere then pyramid shaped, the sphere shaped leaves only 0.140574 cubic inches empty inside the sphere while the pyramid gum leaves 0.2387532 cubic inches empty.