Question 30833
Let a side of the square base = x
Let the height of the box = y 
The volume, V =
{{{V = 32*10^3}}}
{{{32*10^3 = x^2 * y}}}
The dimensions of the materials = 4*x*y for the sides
and x^2 for the base
M = materials
{{{M = 4*x*y + x^2}}}
from above
{{{y = 32*10^3 / x^2}}}
{{{M = 4 * x * (32*10^3/ x^2) + x^2}}}
{{{M = 4 * (32*10^3/ x) + x^2}}}
find M'= MP the dirivative of M
{{{MP = 32*10^3* (-4*x^(-2)) + 2 * x}}}
set MP = 0  to find a minimum
{{{0 = 32*10^3* (-4*x^(-2)) + 2 * x}}}
{{{(32 * 10^3 * 4) / x^2 = 2* x}}}
multiply both sides by x^2
{{{x^3 = 4 * 32 * 10^3}}}
{{{x^3 = 64 * 10^3}}}
{{{x = 40}}}
The base is 40X40 cm
{{{y = 32*10^3 / x^2}}}
{{{y = 32*10^3 / 40^2}}}
{{{y = 32 * 10^3 / 16 * 10^2}}}
{{{y = 20}}}
The height is 20 cm