0 Question 4 v A box with a square base and open top must have a volume of 296352 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 9:, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of .10.] Simplify your formula as much as possible. A(m) =l ' Next, find the derivative, A'(m). A'(x) =l l Now, calculate when the derivative equals zero, that is, when A'(.7;) = 0. [Hintz multiply both sides by 3:2 -] A'(x) = 0 when x = We next have to make sure that this value of 3 gives a minimum value for the surface area. Let's use the second derivative test. Find A"($). A"(a:) =l l Evaluate A"(m) at the alt-value you gave above. NOTE: Since your last answer is positive, this means that the graph of A(2:) is concave up around that value, so the zero of A'(m) must indicate a local minimum for A(:c). (Your boss is happy now.) Check Answer . Question 2 A manufacturer has been selling 1700 television sets a week at $510 each. A market survey indicates that for each $24 rebate offered to a buyer, the number of sets sold will increase by 240 per week. a) Find the demand function p($), where a: is the number of the television sets sold per week. pm =l J b) How large rebate should the company offer to a buyer, in order to maximize its revenue? Si J c) If the weekly cost function is 144500 -i 170x, how should it set the size of the rebate to maximize its profit? $l_l Check