A cylinder shaped can needs to be constructed to hold 450 cubic centimeters of s

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 0.03 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 = pi r^2h Area of the sides: A = 2 pi rh Area of the top/bottom: A = pi r^2 To minimize the cost of the can: Radius of the can: Height of the can: Minimum cost: cents

Explanation / Answer

Area of the side = 2 r h

Cost for the side = 0.03(2 r h) = 0.06 r h

Area of the top and base = 2 r^2

Cost for the top and base = 0.05(2 r^2) = 0.1 r^2

Total cost C = 0.06 r h + 0.1 r^2

Volume = r^2 h = 450, so h = 450/( r^2)

C = 0.06 r (450/( r^2)) + 0.1 r^2 = (27/r) + 0.1 r^2

For minimum cost, dC/dr = 0

(-27/r^2) + 0.2 r = 0, from which we get r = 5.13 cm

h = 450/( r^2) = 5.44 cm

Minimum cost = (27/5.13) + 0.1 (5.13)^2 = 13.53 cents.

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