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. So first I want to fine what the volume of the jar is so I am going to use or
Now we know how much space we have inside the jar.
Now we have to find the volume of the pennies.
Now you will divide 325 by .33 and we find that there is a total of 985 pennies in the jar.
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.
In order to find the volume of this container we are going to have to use two formulas. The first formula will be so we can find the volume of the cylinder.
Now we have to find the volume of the cone
And after doing that I find that
Now all we have to do is add the two together and we will know the volume of
Now we have to find the volume of the tennis balls. In order to find the volume of a tennis ball we have to find the area of it.
After doing that I am going to divide 87.9 from 5.43 to find how many tennis balls will fit into the container.
So a total of 16 tennis balls would fit into the container. If the containers dimensions were doubled then the number of tennis balls would have probably doubled.
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. Now to find the volume of the sphere we are going to use
So the volume of the sphere is 51.8 inches
Finally we have to find the volume of the pyramid gum.
Now we go ahead and divide them by each other and find that a total of 207 pieces of gum fit into the sphere.
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? There should only be enough room for there to be about 3 or 4 more penny’s but you would not be able to put them into the jar because the pennies are going everywhere and they jar is not made to form around the pennies.
2. Where does the formula for the volume of a cylinder derive from? Give an example and provide evidence to support your claim. It derives from the formula of area of a circle the only difference between them is for the cylinder you have to what the height is in order to use the formula.
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.
This equation is almost like the formula for the area of a circle crossed with the area of a triangle. The differences between them are that for a circle you don’t have to find the height and you don’t have a fraction and for a triangle you don’t use pie and don’t use the radius and the fraction is not 1/3 it is ½.
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?
Nothing because they amount of space has not changed just the way it looks has and the theory that supports this is cavallies principle. 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 would thing that balls would best fit into the container because it is shaped as a ball. Also if they were shaped as a ball there wouldn’t have been that much space left in the sphere.