4)

A box with a square base and open top must have a volume of 340736 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 1:, the length of one side of the square base. [Hintz use the volume formula to express the height of the box in terms of 12.] Simplify your formula as much as possible. Awe Next, find the derivative, A'(a:). A'(m) = I Now, calculate when the derivative equals zero, that is, when A' (m) = 0. [Hintz multiply both sides by 332.] 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"(a:). m): Evaluate A"(m) at the m-value you gave above. :1 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'(:c) must indicate a local minimum for A(:c). (Your boss is happy now.) IA manufacturer has been selling 1750 television sets a week at $450 each. A market survey indicates that for each $21 rebate offered to a buyer, the number of sets sold will increase by 210 per week. a) Find the demand function p(a:), where a: is the number of the television sets sold per week. W) = b) How large rebate should the company offer to a buyer, in order to maximize its revenue? 5 c) If the weekly cost function is 131250 + 1501:, how should it set the size of the rebate to maximize its profit? 5:] For the given cost function C(x) = 48400 + 500x + x2find: a) The cost at the production level 1550 b) The average cost at the production level 1550 c) The marginal cost at the production level 1550 d) The production level that will minimize the average cost e) The minimal average cost