Mathematics: Ahighway engineer wants to estimate the maximum number of carsthat can safely travel on a particular road at a given speed. heassumes that each car is 17 feet long, travels at speeds, and follows the car in front of it at a safedistance for that speed. he finds that the number n ofcars that can pass a given spot per minute is modeled by thefunction n (s) = (81s) /(17+17(s/20)^2) at what speed can the greatest number of cars travel safely on thatroad?

Step-by-step explanation:its too easy u can use factorization method or formula...

Ahighway engineer wants to estimate the maximum

Ответ:

duplessistoccara

09.01.2022

Mathematics

s = 20 ft/s

Step-by-step explanation:

Given:-

- The relationship between the N of cars that can pass a given spot per minute is modeled by the function:

$N(s) = \frac{81s}{17 + 17*(\frac{s}{20})^2 }$

Where , s = speed of the car.

Find:-

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

Solution:-

- To maximize the number of cars (N) that travel safely on the road we will take derivative of the given function as follows and find the critical value(s) at which (N) maximizes / minimizes.

$N(s) = 81*\frac{s}{17 + (\frac{17*s^2}{400}) }\\\\N'(s) = 81* [ \frac{( 17 + \frac{17s^2}{400}) - s*( \frac{2s*17}{400} + 0 )}{(\frac{17*s^2}{400} + 17 )^2} ]\\\\N'(s) = 81* [ \frac{( 17 - \frac{17s^2}{400}) }{(\frac{17*s^2}{400} + 17 )^2} ] = - \frac{32400(s^2 - 400)}{17(s^2 + 400)^2}$

- Now set the first derivative N'(s) equal to zero and solve for "s":

$N'(s) = - \frac{32400(s^2 - 400)}{17(s^2 + 400)^2} = 0\\\\(s^2 - 400) = 0\\\\s = \sqrt{400} = 20$

- So the maximum number of cars (N) that travel safely on the road we would have a speed of s = 20 ft/s.

0,0(0 оценок)

Ответ:

Ответ:

Ask your question to the AI advisor

More tips

Answers on questions: Mathematics

Ask an AI advisor a question