Question

Civil engineer 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 14 feet long, travels at speed S, and follows the car in front of it at a safe distance for that speed. He finds that the number N of cars that can pass a given spot per minute is modeled by the function
N=(89s)/(14+14(s/17)^2))

At what speed can the greatest number of cars travel safely on that road? Assume that the maximum possible speed of a car is less than 300.

Answer 1

$N(s)= \frac{89s}{14+14( \frac{s}{17})^2 } \\ \\N'(s)= (\frac{89s}{14+14( \frac{s}{17})^2 } )'= \frac{89\times 14(1+( \frac{s}{17})^2)-89s\times \frac{28}{17} }{14^2(1+( \frac{s}{17})^2)} \\ \\N'(s)=0 \\ \\89\times 14(1+( \frac{s}{17})^2)-89s\times \frac{28}{17}=0 \\ \\1+( \frac{s}{17})^2- \frac{2s}{17} =0 \\ \\289+s^2-34s=0 \\ \\s^2-34s+289=0 \\ \\(s-17)^2=0 \\ \\s=17$