1. ## Chain Rule

The occupancy rate of the suite wonderland hotel located near an amusement park is given by the function

r(t)=10/81t^3 -10/3t^2+ 200/9t+60 (0<t<12)

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

R(r)=-3/5000r^3 + 9/50r^2 (0<r<100)

where r is the occupancy rate

a) Find an expression that gives the rate of change of Wonderland's occupancy rate with respect to time
b) Find an expression that gives the rate of change of Wonderland's montly revenue with respect to the occupancy rate
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 June?

HINT: use the chain rule to find R'(r(0))r'(0) and R'(r(6))r'(6)

2. Originally Posted by lemontea
The occupancy rate of the suite wonderland hotel located near an amusement park is given by the function

r(t)=10/81t^3 -10/3t^2+ 200/9t+60 (0<t<12)

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

R(r)=-3/5000r^3 + 9/50r^2 (0<r<100)

where r is the occupancy rate

a) Find an expression that gives the rate of change of Wonderland's occupancy rate with respect to time
This is asking for $\displaystyle \frac{dr}{dt}$ which you should be able to do.

b) Find an expression that gives the rate of change of Wonderland's montly revenue with respect to the occupancy rate
This is asking for $\displaystyle \frac{dR}{dr}$ which again should be simple.

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 June?

HINT: use the chain rule to find R'(r(0))r'(0) and R'(r(6))r'(6)

Final part is uses the chain rule:

$\displaystyle \frac{dR}{dt}=\frac{dR}{dr}\times \frac{dr}{dt}$

You have both of the terms on the right of this equation from the first two parts of the question, so you can rewrite this as a function of $\displaystyle t$ only, then plug in the $\displaystyle t$'s corresponding to Jan and June.

RonL