## holy CANolliBy: allie nichols, brooke dillon, and kazuki niwano

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

Volume Formula
Surface Area Formula

Explanation

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

Volume Equation

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.

Surface Area equation

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

Solved for "h"

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)

Rearrange Volume Equation Solved for "h"
Simplified Rearranged Volume Equation

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

Derivative of Rearranged Volume Equation

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.

Equation used to plug the radius in

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

Formula used to find percent optimized
Cream of Mushroom Soup

Original Dimensions:

Volume Optimized:

Percent Optimized

Sliced Ripe Olive Can

Original Dimensions:

Volume Optimized:

Percent Optimized

Hunts Tomato Paste

Original Dimensions

Volume Optimized:

Percent Optimized

Green Chiles Can

Original Dimensions:

Volume Optimized:

Percent Optimized