SOLUTION: volume of a square based box is 256cm^3. 1.)Find h in terms of l. if the box has an opened top, find: 2.)the surface area, A in terms of l only 3.) dimensions of the box if the

Since the base (horizontal cross section) is square, 



The volume is given as 256, so








if the box has an opened top, find:
2.)the surface area, A in terms of L only

A rectangular solid has 6 sides, but without a top
it has only 5 sides, 4 of them have area LH and the
bottom is a square with area L² like this, but with 
the 4 rectangular sides (flaps) folded upward and taped 
to form a box:





And since 










3.) dimensions of the box if the surface area is to be a minimum



We find the derivative of A with respect to L



Rewrite it as







We set that equal to 0 to find candidates for x values at which
A might be a minimum or a maximum:



Multiply through by  to clear of fractions:











We need to determine if that is a maximum, minimum or neither.

Using 2nd derivative test:







Substituting 







That is positive so graph of A is concave upward
when L=8, so A has a minimum value when L=8

Since 



So the box has an 8cm² by 8cm² base with height 4cm.

4.) the minimum area.

Substitute in







So the minimum outside surface area is 192cm²

Edwin