A company is offering a coupon for money off at a popular chain restaurant if people respond to an on-line survey in the month of June. The rate at which people answer surveys is modeled by the function A defined by A(t) = 6000/t2 - 8t + 74
The rate at which coupons are mailed out to people who answered the survey is modeled by the function M defined by
M(t) = 8000/t2 - 25t + 270
Both A(t) and M(t) are measured in people per day and time t is measured in days. These functions are valid for the month of June 0 ≤ t ≤ 270 the days at which the survey is valid. At time t = 0, one hundred surveys have already been taken but no coupons have been sent out.
a) How many people answered surveys by June 15 (t = 15)?
b) The company will send a $10 coupon for all people who return surveys by June 15 and $5 for all surveys returned after June 15. What is the dollar value in coupons that the company will send for the month of June?
c) Let F(t) = A(0) - M(0) + [A(x) - M(x)] dx. Find the value of F(15) and explain its meaning in the context of the problem.
d) Find the value of F'(15) and explain its meaning in the context of the problem.
e) At what time t for 0 ≤ t ≤ 30 does the model predict that the number of people who have answered the survey but their coupon hasn’t been mailed out is a maximum?