A box has a square base of side x and height h. Find the dimensions x, h for when the volume is 12 and the surface area is as small as possible.

Answer 1

Constraint that connects x and h V = x²h = 12

Equation of quantity minimized A = 2x² + 4xh

h = 12/x² sub in...

A = 2x² + 48/x

dA/dt = (4x -48/x²) = 0 for extremum

4x³ = 48 x = (12)^1/3 (Not surprising as cubes minimize SA for rectangular prisms.

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