**Our Goal**

- Determine the amount of product each can is holding
- Based on each cans surface area, calculate the maximum product the can's structure has the ability to enclose

**Initial Equations**

**Explanation**

Initially we measured the height and radius of each can. With those dimensions, we calculated the Volume of each can using

Next, using the surface area equation (see below) we rearranged the equation to solve for "h" in attempt to replace height in the Volume equation with radius.

After rearranging the surface area equation to solve for "h" we got

We then plugged what we rearranged and solved for h into h of the volume equation, looking like this (see below). This simplifies to be (see below)

Next, we took the derivative of the simplified rearranged Volume equation because we are trying to find the critical points which are either the max or the mins and got

We set the equation above (Derivative of Rearranged Volume Equation) equal to zero because the critical points are where the slope of the function is zero or undefined

After plugging our surface area in for "A" and setting the equation equal to zero, we solved for "r" and got a positive and negative number. We knew the radius could not be negative because it would indicate a min when we are trying to find a max.

We plugged the radius into the equation (see below) which gave us our new Volume.

In order to calculate the percentage, it was optimized, we followed this formula (see below) and that gave us the percent it was optimized.