Let the length of the square base be x inches .
And the height of the box be y inch .
Volume = x*x*y
4000 = x *x *y
[tex] y = \frac{4000}{x^2} [/tex]
S = x*x + 2xy + 2 x y
[tex] S = x^2 + 4 xy [/tex]
[tex] S = x^2+ 4x* \frac{4000}{x^2} [/tex]
[tex] S = x^2 + \frac{16000}{x} [/tex]
[tex] S ' = 2x - \frac{16000}{x^2} [/tex]
S ' =0
[tex] 2x - \frac{16000}{x^2}=0 [/tex]
[tex] 2x^3 - 16000 = 0 [/tex]
[tex] x^3 = 8000
[/tex]
x = 20
[tex] y = \frac{4000}{20^2} = \frac{4000}{400} = 10 [/tex]
