A box is constructed out of two different types of metal. The metal for the top

Question

A box is constructed out of two different types of metal. The metal for the top and bottom, which are both square, costs $1 per square foot and the metal for the sides costs $6 per square foot. Find the dimensions that minimize cost if the box has a volume of 25 cubic feet. Length of base x = Height of side z =

Explanation / Answer

Volume, x^2z = 25 Cost, C = 2x^2 * 1 + 4xz * 6 => C = 2x^2 + 24x * (25/x^2) ... [plugging z = 25/x^2 from the volume eqn.] => C = 2x^2 + 600/x For C to be minimum, dC/dx = 0 and d^2C/dx^2 > 0 dC/dx = 0 => 4x - 600/x^2 = 0 => x^3 = 150 => x ˜ 5.31 ft and z = 25/x^2 = 0.89 ft. Answer: Length of base x = 5.31 ft. Height of side z = 0.89 ft.

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