Chapter 3.3, Problem 82E

HOTEL OCCUPANCY RATES The occupancy rate of the allsuite Wonderland Hotel, located near an amusement park, is given by the function

$r\left(t\right)=\frac{10}{81}{t}^3-\frac{10}{3}{t}^2+\frac{200}{9}t+56.2$(0 ≤ t ≤ 12)

where t is measured in months, with t = 0 corresponding to the beginning of January. Management has estimated that the monthly revenue (in thousands of dollars per month) is approximated by the function

$R\left(r\right)=-\frac{3}{5000}{r}^3+\frac{9}{50}{r}^2$ (0 ≤ r ≤ 100)

where r is the occupancy rate.

1. a. Find an expression that gives the rate of change of Wonderland’s occupancy rate with respect to time.
2. b. Find an expression that gives the late of change of Wonderland's monthly revenue with respect to the occupancy rate.
3. c. What is the rate of change of Wonderland's monthly revenue with respect to time at the beginning of January? At the beginning of July?

Hint: Use the Chain Rule to find R'(r(0)) r'(0) and R'(r(6)) r'(6).