Purpose

The purpose of this project is to find out how *volume optimized* cans are in a grocery store.

Define Optimization

"An act, process, or methodology of making something as fully perfect, functional, or effective as possible" -Merriam-Webster

In Other Words..

We want to know how much "stuff" can a cylinder container actually hold while maintaining the same amount of material (or surface area).

Our 4 Cans

Formulas

Surface Area = A = (2*π*r*h)+(2*π*r*r)

Volume = V = π*r*r*h

Great Value: Canola Oil

**Radius**: 2.75 centimeters

**Height**: 23.30 centimeters

**Surface Area**: 450.11 centimeters squared

**Volume**: 553.57 centimeters cubed

Coca-Cola Can

**Radius**: 2.60 centimeters

**Height**: 12.40 centimeters

**Surface Area**: 263.34 centimeters squared

**Volume**: 245.04 centimeters cubed

Bush's Best: Grillin' Beans

**Radius**: 4.25 centimeters

**Height**: 11.50 centimeters

**Surface Area**: 420.58 centimeters squared

**Volume**: 652.57 centimeters cubed

Morton's: Season-All

**Radius**: 3.10 centimeters

**Height**: 16.00 centimeters

**Surface Area**: 372.03 centimeters squared

**Volume**: 483.05 centimeters cubed

Our variables

**Surface Area (A)**: Since we are not adding or taking away material, the surface area will remain *constant *throughout the entire process.

**Radius (r)**: Finding the optimized radius of the container is the first step to the process.

**Height (h)**: In order to find the optimized height, we first had to find the optimized radius. Once we solved for the radius, that number was plugged into the surface area formula.

**Volume (v)**: Once the optimized height and optimized radius are found, we plug those numbers into the original volume formula. The answer will give you the maximum amount of space the container can contain.

**Percent of Optimization**: This will be the final product of the project. The percentage of optimization can be found by dividing the initial volume by the optimized volume and multiplying that number by one hundred. This percentage lets us know how optimized our containers are.

Lets Get Started!

#### Step 1: Find the Optimized Radius

Change the Surface Area Equation:

Plug the Height Formula Into the Volume Formula:

Find the Derivative of the "New" Formula:

Set the Slope of the Volume Equal to Zero and Solve for the Radius:

#### Step 2: Find the Optimized Height

Plug the Optimized Radius into the Original Height Formula:

#### Step 3: Plug Optimized Radius and Height into the Volume Formula

Our Answers

#### Great Value: Canola Oil

Maximum Radius: 4.89 centimeters

Maximum Height: 9.77 centimeter

Maximum Volume: 733.18 centimeters cubed

This can is 75.50% optimized

#### Coca-Cola Can

Maximum Radius: 3.61 centimeters

Maximum Height: 7.21 centimeter

Maximum Volume: 294.51 centimeters cubed

This can is 89.42% optimized

#### Bush's Best: Grillin' Beans

Maximum Radius: 4.72 centimeters

Maximum Height: 9.45 centimeters

Maximum Volume: 662.22 centimeters cubed

This container is 98.54% optimized.

#### Morton's: Season All

Maximum Radius: 4.44 centimeters

Maximum Height: 8.89 centimeters

Maximum Volume: 550.92 centimeters cubed

This container is 87.68% optimized.

Bonus

#### Great Value: Canola Oil

#### Coca-Cola Can

#### Bush's Best: Grillin' Beans

#### Morton's: Season All

Thank You For Watching Our 'Can'tastic Project!

Credits:

Created with images by skeeze - "beverage cans tops aluminum"