Question 1155310
Hi,
Taking the measurements of the length and breadth of the base and top of the box = x
Taking it to be a closed box (Closed top and bottom)
Volume = x^2 h (h being height)
Volume = 64 ins^3
x^2 h = 64
Therefore h = 64/x^2
Surface area = 2x^2 (top and bottom)+ 4xh (4 sides)
Surface Area = 2x^2 + 4xh
(Removing the h)  by multiplying 4xh by 64/x^2 (h)
4xh * 64/x^2   ( * means times)
= 256/x
Surface Area = 2x^2 + 256/x
S.A.(x) = 2x^2 + 256x^-1
S.A.'(x)= 4x - 256x^-2
S.A.' (x) = 4x - 256/x^2
S.A.' (x) = 0
   4x - 256/x^2 = 0
     - 256/x^2 = - 4x  (Multiply both sides by -1)
       256/x^2 = 4x (Cross multiply)
         256 = 4x^3
          4x^3 = 256
           x^3 = 64
            x = cube root of 64
            x = 4
Nature Table  shows x = 4 to be a minimum value
Therefore Length = Breadth = 4 ins
Height = 64/x^2 = 64/16 = 4 ins.
Hope this helps :-)