Originally Posted by

**JeffM** I get a different result from romsek because I read the problem differently. Obviously

$32000 = V = s^2h \implies h = \dfrac{32000}{s^2}.$

But I read the OP as requiring the box to have a top and bottom, which changes the cost equation.

$cost\ of\ bottom = s^2 * 1 = s^2 \implies cost\ of\ top\ and\ bottom = 2s^2.$

$cost\ of\ a\ side = s * h * 0.25 = s * \dfrac{32000 * 0.25}{s^2} = \dfrac{8000}{s} \implies$

$cost\ of\ 4\ sides = \dfrac{32000}{s}.$

$total\ cost = C = 2s^2 + \dfrac{32000}{s} \implies \dfrac{dC}{ds} = 4s - \dfrac{32000}{s^2} \implies$

$\dfrac{dC}{ds} = 0 \implies 4s - \dfrac{32000}{s^2} = 0 \implies 4s^3 - 32000 = 0 \implies$

$s^3 = 8000 \implies s = 20 \implies h = \dfrac{32000}{20^2} = 80.$