Question
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?
Answers
oobleck
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
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