Question 1. Score: 25/25 A box with a square base and open top must have a volume of 32000 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 x, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of ac.] Simplify your formula as much as possible. A(a) = Next, find the derivative, A' (x). A'(a) = Now, calculate when the derivative equals zero, that is, when A' (@) = 0. [Hint: multiply both sides by ac 2. ] A' (x) = 0 when x = We next have to make sure that this value of a gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x). A"( a) = Evaluate A"() at the x-value you gave above.A piece of cardboard measuring 13 inches by 12 inches is formed into an open-top box by cutting squares with side length a: from each corner and folding up the sides. Find a formula for the volume of the box in terms ofm W: Find the value for :2 that will maximize the volume of the box A cylinder shaped can needs to be constructed to hold 400 cubic centimeters of soup. The material for the sides of the can costs 0.02 cents per square centimeter. The material for the top and bottom of the can need to be thicker, and costs 0.08 cents per square centimeter. Find the dimensions for the can that will minimize production cost. Helpful information: h : height of can, r : radius of can Volume of a cylinder: V = mr2h Area of the sides: A = 2rrh Area of the top/bottom: A = mr2 To minimize the cost of the can: Radius of the can: Height of the can: Minimum cost: cents