Chapter 17.6, Problem 70E

Time In Exercise 39 of Section 17.3, we saw that the time (in hours) that a branch of Amalgamated Entities needs to spend to meet the quota set by the main office can be approximated by

T(x, y) = x⁴ + 16y⁴ – 32xy + 40,

where x represents how many thousands of dollars the factory spends on quality control and y represents how many thousands of dollars they spend on consulting. Find the average time if the amount spent on quality control varies from $0 to $4000 and the amount spent on consulting varies from $0 to $2(XX). (Hint: Refer to Exercises 61–64.)

39.    Time The time (in hours) that a branch of Amalgamated Entities needs to spend to meet the quota set by the main office can be approximated by

T(x, y) = x⁴ + 16y – 32xy + 40,

where x represents how many thousands of dollars the factory spends on quality control and y represents how many thousands of dollars they spend on consulting. Find the amount of money they should spend on quality control and on consulting to minimize the time spent, and find the minimum number of hours.

(b)    Use the solver capability of a spreadsheet to find a function of the form y = ab′ that fits the data above. (Hints: Using the ideas from part (a), find values for a and b that minimize the function

f(a, b) = [ln(0.275) – 0 ln b – ln a]²

+[ln(1.2) – 4 ln b – ln a]²

+[ln(3.1) – 8 ln b – ln a]²

+[ln(7.5) – 12 ln b – ln a]²

+[ln(9.5) – 14 ln b – ln a]²

+[ln(42) – 15 ln b – ln a]²

+[ln(291) – 20 ln b – ln a]²

+[ln(820) – 22 ln b – ln a]²

+[ln(1900) – 24 ln b – ln a]².)

(c)    Compare your answer to this problem with the one found with a graphing calculator in the chapter on Nonlinear Functions, Section 10.4 Exercise 54.