A box has a shape of a cuboid. Its height is y m. It has a square base of side x m. If the volume of the box is 64 m^3, find the value of x and y such that the total S.A of the box is a minimum.

Question

Are the equations

64=x^2y and S.A=2x^2+4xy correct for the question?

Answer 1

V = x²y = 64. So, y = 64 / x²

S = surface area = 4xy + 2x² = 4x(64/x²) + 2x² = 256x^-1 + 2x²

dS/dx = -256x^-2 + 4x = -256/x² + 4x = (-256 + 4x³) / x²

dS/dx = 0 when 4x³ = 256. So, x = 4.

When 0 < x < 4, dS/dx < 0. So, S is decreasing

When x > 4, dS/dx > 0. So, S is increasing

Relative and absolute minimum when x = 4 m and y = 4 m.