Determine the dimensions of the 800 cc can that will be cheapest to produce.

Question

The material for the top and bottom of aluminum cans sells for .02 cents per square centimeter, and the material for the thinner cylindrical portion of aluminum cans sells for .01 cents per square centimeter.

1. The area of a circle with radius r is pi r^2. Using information above, we determine that the cost for the top and bottom portion of each can will be 2 x .02 x pi r^2 = .04pi r^2. The area of the cylindrical portion of the can is 2pi x r x h where h is the height of the can. Use this information to find a formula for the total cost, C, (that is the total cost for both the cylindrical portion and the top and bottom) for the material in one can.

Answer 1

Hi Krista,

The total cost of the can is C = .04π*r² + .02π*r*h

But he volume V, must be 800cc's so V=π*r² *h this is called a constraint because it "constrains" the relationship between r and h.

To solve use the constraint function to determine r as a function of h and V (V is a constant=800) and then

substitute this relationship in the equation for the cost C. This will make the cost only a function of h. To minimize the cost C differentiate it with respect to h and set it equal to zero. This will determine the min h you then substitute this in the volume equation to determine the corresponding r that minimizes the cost.

Let me know if you have any questions.

Jim