Question

A cardboard box of 32 in^3 volume with a square base and open top is to be constructed. What is the length of base that will minimize the surface area?

Answer 1

Volume = Area_Base * height
32 = x^2 * h --> h = 32/x^2

Surface area = perimeter_Base*h + Area_Base
SA = 4x(32/x^2) + x^2
SA = 128/x + x^2

to minimize, set derivative equal to zero
SA' = -128/x^2 + 2x = 0
2x = 128/x^2
2x^3 = 128
x^3 = 64
x = 4

Surface area minimized when length of base is 4 in.