A standard piece of paper is 8.5 inches wide and 11 inches long. It can be formed into a cylinder two ways - one by joining the longer edges and one by joining the shorter edges, as shown here:

          

Answer the following 3 questions, showing and explaining your work clearly:

1. Find the volume of each of the cylinders.
   
   
2. Discuss your answer to question 1. If the volumes are the same, why do you think that is? If they are different, why is one of them larger than the other? In other words, which dimension of the cylinder has the biggest effect on the volume, the height or the diameter? Why do you think so?
   
   
3. Both cylinders started with an 8.5 by 11 inch piece of paper. Find the dimensions of another rectangular sheet of paper which has the same area as the 8.5 by 11 sheet, but which can be made into a cylinder with a greater volume than the cylinders in question 1. Show clearly that your rectangle has equal area and greater volume compared to the 8.5 by 11 inch paper.

Bonus: Suppose you have a rectangular sheet of paper with length 'x'. The paper has the same area as the 8.5 by 11 inch paper in this problem. Can you find a formula that will express the cylindrical volume of the paper in terms of 'x'? Can you use your formula to determine the largest possible cylindrical volume of a rectangular piece of paper with an area equal to that of the 8.5 by 11 inch sheet? Show and explain your work clearly.