A box with a square base and open top must have a volume of 237276 cm. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only a, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of x. ] Simplify your formula as much as possible. A(2) = Next, find the derivative, A' (x). A'(a) = Now, calculate when the derivative equals zero, that is, when A' (x) = 0. [Hint: multiply both sides by a2 . ] A'(x) = 0 when x = We next have to make sure that this value of x gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x). A " ( 20 ) = Evaluate A"(x) at the x-value you gave above