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)= (89s)/(14+14( (s)/(17))^2 ) \\ \\N'(s)= ((89s)/(14+14( (s)/(17))^2 ) )'= (89* 14(1+( (s)/(17))^2)-89s* (28)/(17) )/(14^2(1+( (s)/(17))^2)) \\ \\N'(s)=0 \\ \\89* 14(1+( (s)/(17))^2)-89s* (28)/(17)=0 \\ \\1+( (s)/(17))^2- (2s)/(17) =0 \\ \\289+s^2-34s=0 \\ \\s^2-34s+289=0 \\ \\(s-17)^2=0 \\ \\s=17`