0 Question 1 " B 0/1 pt '0 2 8 99 (9 Details 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 :13, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of ax] Simplify your formula as much as possible. 11(3) ='\ ' Next, find the derivative, A'(:n). A'(;n) ='\ ' Now, calculate when the derivative equals zero, that is, when A'(m) = 0. [Hint: multiply both sides by $2.] A'($) = 0 when :3 = 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"(3:). A"(:1:) 2" Evaluate A"(:1:) 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'(:c) must indicate a local minimum for A(:1:). (Your boss is happy now.) Question Help: E] Video Submit Question 0 Question 11 " B 0/1 pt '0 2 8 99 (D Details The function P(:I:) = 43:2 + 1850m 6500 gives the profit when 3: units of a certain product are sold. Find a) the profit when 80 units are sold ' ' dollars b) the average profit per unit when 80 units are sold ' ' dollars per unit c) the rate that profit is changing when exactly 80 units are sold ' ' dollars per unit d) the rate that profit changes on average when the number of units sold rises from 80 to 160. ' ' dollars per unit e) The number of units sold when profit stops increasing and starts decreasing. (Round to the nearest whole number if necessary.) ' ' units Question Help: E] Video Submit Question 0 Question 5 " B 0/1 pt '0 2 8 99 (D Details 2 , 15625 \ a) The cost at the production level 1000' ' For the given cost function C(33) = 128V; + find b) The average cost at the production level 1000' ' c) The marginal cost at the production level 1000' ' d) The production level that will minimize the average cost. e) The minimal average cost. ' ' Give answers to at least 3 decimal places. Question Help: E] Video Submit Question 0 Question 8 7 B 0/1 pt '0 2 8 99 G) Details A manufacturer has been selling 1250 television sets a week at $390 each. A market survey indicates that for each $23 rebate offered to a buyer, the number of sets sold will increase by 230 per week. a) Find the demand function p(:1:), where a; is the number of the television sets sold per week. plan) = l \ b) How large rebate should the company offer to a buyer, in order to maximize its revenue? six J c) If the weekly cost function is 81250 + 1303:, how should it set the size of the rebate to maximize its profit? Si J Question Help: E] Video Submit Question 0 Question 13 " B 0/1 pt 0 2 8 99 (9 Details Suppose a company's revenue function is given by R(q) = q3 + 380(12 and its cost function is given by C(q) = 240 + 10g, where q is hundreds of units sold/ produced, while R(q) and C(q) are in total dollars of revenue and cost, respectively. A) Find a simplified expression for the mginal profit function. (Be sure to use the proper variable in your answer.) mm) = [ B) How many items (in hundreds) need to be sold to maximize profits? Answer: hundred units must be sold. (Round to two decimal places.) Question Help: E] Video Submit