Science:Math Exam Resources/Courses/MATH102/December 2016/Question B 07 (b)

MATH102 December 2016

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Question B 07 (b)

Each day 100,000 people use Vancouver’s public transit system. Translink estimates that 2000 of those people ride on transit without paying (called free riders). The number of free riders caught by Translink officers in any particular day is given by $R(N)=2000(1-e^{-kN})$ where $k=1/500$ and $N$ is the number of Translink officers on duty that day. Each officer on duty costs Translink 1000 dollars per day. (b) How many officers should be hired in order to maximize the net revenue for a fixed fine $F$? Check that your answer is in fact a maximum.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Find the critical point of $I(N)$ on the domain $\mathbb {N} \cup \{0\}$.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. To maximize the net revenue $I(N)$ on the domain $\mathbb {N} \cup \{0\}$, we first find its derivative. Recall that $I(N)=2000(1-e^{-kN})F-1000N$ from part (a). Then, we have $I'(N)=(2000(1-e^{-kN})F-1000N)'=2000F(1-e^{-kN})'-1000=2000Fke^{-kN}-1000$. Since $2000Fke^{-kN}-1000=0$ at $N=500\ln \left({\frac {F}{250}}\right)=N_{0}$, we have $I'(N)>0$ for $N<N_{0}$ and $I'(N)<0$ for $N>N_{0}$. Therefore, $I(N)$ achieves its maximum at $N=N_{0}$, provided that $N_{0}\geq 0\iff F\geq 250$. Answer: $\color {blue}N=500\ln \left({\frac {F}{250}}\right),F\geq 250$