The occupancy rate of the all-suite Wonderland Hotel, located near an amusement

Question

The occupancy rate of the all-suite Wonderland Hotel, located near an amusement park, is given by the function

r(t) =

(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/month) is approximated by the function

R(r) = -3/500(r^3) +90/50(r^2)    (0 ? r ? 100)

where r is the occupancy rate.

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)?

Explanation / Answer

r(t)= (10/81)t^3 - (10/3)t^2 +(200/9)t +60
Take the derivative by power rule and you get
r'(t) = (10/27)t^2 - (20/3)t + 200/9

R(r) =(-3/500)r^3 +(9/50)r^2
Same idea here
R'(r) = (-9/500)r^2 + (9/25)r
Assuming the parenthesis are written correctly, I'm going to rewrite it to make the problem look easier to see

R'(r(0))r'(0) can be rewritten as R'(r(0)) * r'(0)
Remember R'(r) = (-9/500)r^2 + (9/25)r, so R'(r(0)) is the same as (-9/500)*(r(0))^2 + (9/25)*(r(0))

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

so R'(r(0)) is (-9/500)*60^2 + (9/25)*60 = 20.952. Now, what's r'(0)? Recall that r'(t) = (10/27)t^2 - (20/3)t + 200/9 so simply plug in 0 and you will get 200/9

R'(r(0)) * r'(0) = 20.952 * 200/9 = 465.6

For R'(r(6))r'(6), it is the same idea but instead of plugging in 0, plug in 6

r(6) = (10/81)*6^3 - (10/3)6^2 + (200/9)*6 + 60 = 100

so R'(r(6)) is (-9/500)*100^2 + (9/25)*100 = 34.2
r'(6) = (10/27)6^2 - (20/3)6 + 200/9 = - 40/9

R'(r(6)) * r'(6) = 34.2 * -40/9 = -152

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