Cans and Rational Equations
A) Determine a formula for the volume of a can in terms of h, r, and pi. Explain how you can use the basic shapes present in a cylinder to construct your formula.
- V=hπr², the pi and r² is the formula for surface area of a circle. The depth the multiples the number of circles that could go into the cylinder depth. This then finds the total area inside the cylinder.
B) Draw a net diagram for the coffee can. Determine a formula for the cost of the can based upon your net diagram. Write a verbal model of the formula and then write a mathematical model.
- Surface Area of the cylinder = .04 times (two pi times radius times height) plus .06 times (two pi times radius squared)
- SA = .04(2πrh) + .06(2πr²)
C) Find an equation for the cost of the coffee can that includes all the requirements mentioned in the scenario at the top of the page. List all the specific requirements and then write your equation. Simplify your result.
- (0.04)(2πr)(43/r) + (0.06)(2πr²)
D) What is the minimum cost of the coffee can? What are the dimensions of this coffee can?
- 12π(2.4)^3 + 10.8
- C = $6.67
- We found the minimum dimensions by using the volume of the can and then the size requirements of the can. The minimum cost was found by using the same equation as in part C but instead found the cost instead of the size. The solution make sense due to the fact that each side has a cost of either .04 or .06.
Using a selected cylinder of our choice determine the height and radius of the can. Calculate the cost of the can for .6 cents on the top and bottom and .4 cents for the sides. Using these specifications calculate the minimum cost of the can. Explain why there may be a difference between your results and the actual product.
- C = (.04)(2πrh) + (.06)(2πr²)
- C = $2.89
- Our solution may be different than the original solution due the the different sizes of the can. Even though the cost for the separate parts are the same the size specifications of the can are different.