Open box, so there is no top
SA = 2*l*h + 2*w*h + l*w
V = l*w*h = 32cm^3
h = 32/(w*l)
SA = 2*l*32/(w*l) + 2*w*32/(w*l) + l*w
SA = 64/w + 64/l + l*w
Find minimum SA: take partial derivatives to get critical point(s)
SAw = -64/w^2 + l
SAl = -64/l^2 + w
Both the partials have to be 0, so...
0 = -64/w^2 + l and 0 = -64/l^2 + w
64/w^2 = l
0 = -64/(64/w^2)^2 + w (plug into second equation)
0 = -w^4/64 + w
0 = w(1-w^3/64)
1 = w^3/64 or 0 = w (impossible answer)
64 = w^3
4 = w
Plug w back into 64/w^2 = l
64/4^2 = l
4 = l
Plug w and l back into h = 32/(w*l)
h = 32/(4*4)
h = 2
The Surface Area:2*l*h + 2*w*h + l*w
SA = 2*4*2 + 2*4*2 + 4*4
SA = 48
So the answer is length = 4cm, width = 4cm, and height = 2cm, with Surface Area of 48cm^2
