A box with a square base and open top must have a volume of 62,500 cm3. Find the

Question

A box with a square base and open top must have a volume of 62,500 cm3. Find the dimensions of the box that minimize the amount of material used. sides of base= height=

Explanation / Answer

Let x = length of side of the square base in cm and h = height of the box in cm => x^2h = 62500 Material used, y = x^2 + 4xh Plugging h = 62500/x^2 from the first eqn., y = x^2 + 4x*(62500/x^2) = x^2 + 250000/x For y to be minimum, dy/dx = 0 and d²y/dx² > 0 dy/dx = 0 => 2x - 250000/x^2 = 0 => x^3 = 125000 => x = 50 cm and h = 62500/(50)^2 = 25 cm d²y/dx² = 2 + 500000/x^3 > 0 => material used is minimum when x = 50 cm and h = 25 cm => Box has a square base of length = 50 cm and height = 25 cm.

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