## The Classic Coke Can Problem

Let's see what more we can grasp about using the graphing calculator while moving up one dimension from the perimeter vs. area problem.

Imagine how many 12 ounce cans of soda, beer, etc. are produced each year in the United States! Is the "standard" shape for such cans the most efficient packaging possible?
Let's think about it by considering the can as a right cylinder. This may be easier done in metric units, so the volume of a standard soda can is 355 cubic centimeters.
1. Express the surface area, A, as a function of its radius, r, measured in centimeters. Use the standard formulas for surface area and volume of a cylinder.
[Hint: First, use the volume formula to find the height, h, in terms of r; substitute that in for h in the surface area formula.]

2. What units are being used for radius and surface area? What does this suggest about max and min values for an appropriate window?

3. Why isn't the graph a nice line, parabola, or some other readily recognized form?

4. How many ways are there to use the graphing calculator to answer the question?

5. The radius of a "standard" 12 ounce can used in the United States is 3.25 cm. Is this the most efficient shape?

6. Are soda cans in the United States the same shape as cans used in other countries? How could we find out more about this?

7. While we are thinking about it, is the standard 6-pack the most efficient pacaging for multiple cans? Is this a fuction of the relative height and radius of the can?