Will the number of cars on GT5 prologue increase?

At what speed can the greatest number of cars travel safely on the road?

• A traffic expert wants to estimate the maximum number of cars that can safely travel on a particular road at a given speed. He assumes that each car is 18 feet long, travels at speed , and follows the car in front of it at a safe distance for that speed. He finds that the number of cars that can pass a given spot per minute is modeled by the function A traffic expert wants to estimate the maximum number of cars that can safely travel on a particular road at a given speed. He assumes that each car is 18 feet long, travels at speed , and follows the car in front of it at a safe distance for that speed. He finds that the number of cars that can pass a given spot per minute is modeled by the function N(s)=(83s)/(18+18(s/21)^2) At what speed can the greatest number of cars travel safely on that road?

• Answer:

  So I'm assuming that the variable s is speed. Then to maximize N with respect to s we need to find the derivative of N(s) = (83/18) * s/(1 + (s^2)/21^2) and set it equal to 0 to find the critical values. So dN/ds = (83/18) * (1 * (1 + (s^2)/21^2) - s * (2s/21^2)) / (1 + (s^2)/21^2))^2 after using the quotient rule. So this is equal to 0 when 1 + (s^2)/21^2 - 2*s^2/21^2 = 0 ---> 1 = s^2 / 21^2 ---> s = 21 (units?). dN/ds > 0 for s < 21 and dN/ds < 0 for s > 21 so N is maximized at s = 21 at a value of N = 48.4 or roughly 48 cars per minute. If the cars are 18 feet long and have two car lengths between them to be safe then that would work out to be about 30 miles/hour, so I'm wondering what the units of s are in this question.