A box with a square-base is to be constructed with no top. The bottom face will cost 4 times as much as the sides. The volume of the box will be 0.5 m3. What dimensions (length, width and height) will minimize the cost to construct this box?

Answer 1

If the base has side length x and the box has height h, you know that
x^2 h = 1/2
So, h = 1/(2x^2)
The cost of the box will be
c = 4x^2 + 4(xh) = 4x^2 + 4x/(2x^2) = 4x^2 + 2/x
the minimum cost occurs when dc/dx = 0
c' = 8x - 2/x^2 = 2(4x^3-1)/x^2
c'=0 when x = 1/∛4
So the box with least cost is 1/∛4 x 1/∛4 x ∛2
or approximately 0.63 x 0.63 x 1.26