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Problem 104

Research reports indicate that surveillance cameras at major intersections dramatically reduce the number of drivers who barrel through red lights. The cameras automatically photograph vehicles that drive into intersections after the light turns red. Vehicle owners are then mailed citations instructing them to pay a fine or sign an affidavit that they weren't driving at the time. The function \(N(t)=6.08 t^{3}-26.79 t^{2}+53.06 t+69.5 \quad(0 \leq t \leq 4)\) gives the number, \(N(t)\), of U.S. communities using surveillance cameras at intersections in year \(t\), with \(t=0\) corresponding to the beginning of 2003 . a. Show that \(N\) is increasing on \([0,4]\). b. When was the number of communities using surveillance cameras at intersections increasing least rapidly? What is the rate of increase?

Expert verified

a. The first derivative of the function \(N(t)\) is \(N'(t)=18.24t^2 - 53.58t + 53.06\). Since there are no critical points of \(N'(t)\) in the interval [0,4] and N'(2) > 0, \(N(t)\) is increasing on [0, 4].
b. The second derivative of the function \(N(t)\) is \(N''(t) = 36.48t - 53.58\). The critical point of \(N''(t)\) occurs at \(t \approx 1.47\). At this time, the rate of increase is \(N'(1.47) \approx 35.92\) communities per year. Thus, the number of communities using surveillance cameras at intersections was increasing least rapidly in the middle of 2004 at a rate of approximately 35.92 communities per year.

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