Help with Optimization Problem

Question

1. A net enclosure for practicing golf shots is open at one end, as shown. Find the dimensions that will minimize the amount of netting needed and give a volume of 144m³. (Netting is required only on the sides, the top, and the far end.)

Answer 1

Volume, V = x²y = 144 --> y = 144/x²

Area, A = 3xy + x² = Area of 2 sides and top plus area of the far end

A = 3(144)/x + x² = 432/x + x²

To minimize, take the derivative and then set that to zero and solve for x.

A' = -432/x² + 2x = (-432 + 2x³)/x²

-432 + 2x³ = 0

x = (432/2)^1/3 = 6 m

y = 144/x² = 144/6² = 4 m