Question

A traffic expert wants to estimate the maximum number of cars that can safely travel on a particular road at a given speed. She assumes that each car is 20 feet long, travels at speed N, and follows the car in front of it at a safe distance for that speed. She finds that the number

of cars that can pass a given spot per minute is modeled by the function

N = 81s/20+20(s/24)^2

Answer 1

Answer: The given function is:

N = 81s / (20 + 20(s/24)^2)

where:

N = number of cars that can pass a given spot per minute

s = speed of the cars in miles per hour (mph)

To find the maximum number of cars that can safely travel on the road at a given speed, we need to find the value of s that maximizes the function N.

Taking the derivative of N with respect to s:

dN/ds = (81 / (20 + 20(s/24)^2)) * (20/24 - (2s/24^2) * (s/24))

Setting dN/ds to zero to find the critical point:

0 = (81 / (20 + 20(s/24)^2)) * (20/24 - (2s/24^2) * (s/24))

Simplifying and solving for s:

20/24 = (2s/24^2) * (s/24)

20 * 24 = 2s * s

s^2 = (20 * 24) / 2

s^2 = 240

s = sqrt(240)

s ≈ 15.4919 mph

Therefore, the maximum number of cars that can safely travel on the road at a speed of 15.4919 mph is:

N = 81s / (20 + 20(s/24)^2) = 81(15.4919) / (20 + 20((15.4919)/24)^2) ≈ 180.19 cars per minute.

Explanation: