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"