If a rectangular box with a square base and an open top is to have a volume of 4 cubic feet , find he dimensions that require the least amount of material .

Answer 1

Let x is side of box and y is height.

Then we have minimise function S = x² + 4xy if V = x²y = 4; y = 4/x²_;

S = x² + 16/x; S' = 2x - 16/x² = 0; 2x³ = 16; x³ = 8; x = 2 feet.

Now S'' = 2 + 32/x³ > 0 if x = 2. So, at x = 2 we have minimum.

x = 2 feet, y = 4/2² = 1 feet