Question

A traffic engineer monitors the rate at which cars enter the main highway during the afternoon rush hour. From her data she estimates that between 4:30 P.M. and 5:30 P.M. the rate R(t) at which cars enter the highway is given by the formula R(t) = 100(1 − 0.0001t2) cars per minute, where t is the time (in minutes) since 4:30 P.M. Find the average rate, in cars per minute, at which cars enter the highway during the first half hour of rush hour

Answer 1

Answer:

On average, cars enter the highway during the first half hour of rush hour at a rate 97 per minute.

Step-by-step explanation:

Given that, the rate R(t) at which cars enter the highway is given the formula

$R(t)= 100(1-0.0001t^2)$

The average rate of car enter the highway during first half hour of rush hour is the average value of R(t) from t=0 to t=30.

$\therefore \int_0^{30} 100(1-0.0001t^2)\ dt$

$=[100(t-0.0001\frac{t^3}{3})]_0^{30}$

$=100[(30-0.0001\frac{30^3}{3})-(0-0.0001\frac{0^3}{3})]$

=2901

The average rate of car is $=\frac{\textrm{The number of car}}{Time}$

                                           $=\frac{2910}{30}$

                                          =97

On average, cars enter the highway during the first half hour of rush hour at a rate 97 per minute.