A box with a square base and open top must have a volume of 256000 cm3. 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 an, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of 33.] Simplify your formula as much as possible. Aw) = Next, find the derivative, A'(:1:). Aw Now, calculate when the derivative equals zero, that is, when A'(a':) = 0. [Hintz multiply both sides by m2 -] A'(m) = Owhen :1: = We next have to make sure that this value of m gives a minimum value for the surface area. Let's use the second derivative test. Find A".(ac) mm = Evaluate A"(m) at the m-value you gave above. NOTE: Since your last answer is positive, this means that the graph of A(:1:) is concave up around that value, so the zero of A'(w) must indicate a local minimum for A(w). (Your boss is happy now.)