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)= (10/81)t^3 - (10/3)t^2 +(200/9)t +60 (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) =(-3/50000)r^3 +(9/50)r^2 (0 ? r ? 100)
where r is the occupancy rate.
(a) Find an expression r'(t) that gives the rate of change of Wonderland's occupancy rate with respect to time t.

(b) Find an expression R'(r) that gives the rate of change of Wonderland's monthly revenue with respect to the occupancy rate r

(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).

Explanation / Answer

a. 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 b. R(r) =(-3/50000)r^3 +(9/50)r^2 Same idea here R'(r) = (-9/50000)r^2 + (9/25)r c. 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) = (-9/50000)r^2 + (9/25)r, so R'(r(0)) is the same as (-9/50000)*(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 (-9/50000)*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) = 60 * 200/9 = 12000/9 or approximately 1333.33 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 - (10/3)6^2 + (200/9)*6 + 60 = 2000/27 so R'(r(6)) is (-9/50000)*(2000/27)^2 + (9/25)*(2000/27) = 2080/81 r'(6) is (10/27)6^2 - (20/3)6 + 200/9 = -40/9 R'(r(6)) * r'(6) = (2080/81)*(-40/9) = -83200/729 or approximately -114.13 I may have made a slight arithmetic error somewhere but the process should be right. Hope this helps!

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