1. Rolled oats (dry oatmeal) come in cylindrical containers with a diameter of 5 inches and a height of 9 inches. These containers are shipped to grocery stores in boxes. Each shipping box contains six rolled oats containers. The shipping company is trying to figure out the dimensions of the box for shipping the rolled oats containers that will use the least amount of cardboard. They are only considering boxes that are rectangular prisms so that they are easy to stack.

a. What is the surface area of the box needed to ship these containers to the grocery store that uses the least amount of cardboard?

b. What is the volume of this box?

c. What percent is empty space within the box?

2. My sister's birthday is in a few weeks and I would like to buy her a new vase to keep fresh flowers in her house. She often forgets to water her flowers and needs a vase that holds a lot of water. In a catalog there are three vases available and I want to purchase the one that holds the most water. The first vase is a cylinder with diameter 10 cm and height 40 cm. The second vase is a cone with base diameter 16 cm and height 45 cm. The third vase is a sphere with diameter 18 cm.

a. Which vase should I purchase?

b. How much more water does the largest vase hold than the smallest vase?

c. Suppose the diameter of each vase decreases by 2 cm. Which vase would hold the most water?

3. The vase company designs a new vase that is shaped like a cylinder on bottom and a cone on top. The catalog states that the width is 12 cm and the total height is 42 cm. What would the height of the cylinder part have to be in order for the total volume to be 1224pi cm³?