A cylinder shaped can needs to be constructed to hold 450 cubic centimeters of soup. The material for the sides of the can costs 0.04 cents per square centimeter.

Question

A cylinder shaped can needs to be constructed to hold 450 cubic centimeters of soup. The material for the sides of the can costs 0.04 cents per square centimeter. The material for the top and bottom of the can need to be thicker, and costs 0.05 cents per square centimeter. Find the dimensions for the can that will minimize production cost.

Helpful information:

h : height of can, r : radius of can

Volume of a cylinder: V=πr^2h

Area of the sides:  A=2πrh

answers:

To minimize the cost of the can:

Radius of the can:     

Height of the can:     

Minimum cost: cents.

Answer 1

You want to minimize cost given the volume constraint

C = (.04)2πrh = 2*(.05)πr² and 450 = πr²h (This implies h = 450/πr²

Substitute the constraint for h into the C equation

Take the derivative of C(r) and equate to zero

Find the roots, which are the critical points: one will make sense for r_Min

Use constraint to solve for h_Min

Plug in r_Min and h_Min to solve for minimum cost.