A box with a square base and open top must have a volume of 500000 (37713. 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 33, the length of one side of the square base. [Hintz use the volume formula to express the height of the box in terms of m] Simplify your formula as much as possible. Ala?) : [: Next, find the derivative, A'.(a:) W) = I: Now, calculate when the derivative equals zero, that is, when A(m) = 0. [Hintz multiply both sides by $2.] A(.:1:) : Owhen :1: : We next have to make sure that this value of :I: gives a minimum value for the surface area. Let's use the second derivative test. Find A".(a:) .1.) = S Evaluate A"(3:) at the m-value you gave above. NOTE: Since your last answer is positive, this means that the graph of A(:c) is concave up around that value, so the zero of A(a:) must indicate a local minimum for 14(3)). (Your boss is happy now.)