A traffic engineer monitors the rate at which cars enter a freeway onramp during rush hour. From her data, she estimates that between 4:30pm and 5:30pm the rate R(t) at which cars enter is given by:
R(t)= 100(1-0.00012t^2)
car per min., where t is time in min. since 4:30.
a. find the average rate, in cars per min, at which cars enter the highway the 1st half hour.
b. find the average wight, in cars per min., at which cars enter the highway during the 2nd half hour
c. find the total number of cars that enter the highway during that hour.
d. cars also exit the freeway at a rate of
E(t)= 50 (1-e^(-t))
find the total number of cars on the highway for the first 30 min. assume no cars initially.
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a. To find the average rate at which cars enter the highway in the first half hour, we need to find the average value of the rate function R(t) over the interval [0, 30], where t represents time in minutes.
The average value of a function f(x) over an interval [a, b] is given by the formula:
Average value = (1 / (b - a)) * Integral[a to b] f(x) dx
In our case, the average rate at which cars enter the highway in the first half hour is given by:
Average rate = (1 / (30 - 0)) * Integral[0 to 30] R(t) dt
Substituting the given rate function R(t) into the integral:
Average rate = (1 / 30) * Integral[0 to 30] 100(1 - 0.00012t^2) dt
Integrating the function with respect to t:
Average rate = (1 / 30) * [100t - 0.00004t^3] evaluated from 0 to 30
Simplifying further:
Average rate = (1 / 30) * [(100(30) - 0.00004(30^3)) - (100(0) - 0.00004(0^3))]
Calculate the expression inside the brackets to find the average rate at which cars enter the highway in the first half hour.
b. To find the average rate at which cars enter the highway during the second half hour, we consider the interval [30, 60] instead of [0, 30] in the previous part. Repeat the same steps as in part (a) but with the new interval.
c. To find the total number of cars that enter the highway during the hour, we have the rate function R(t) that gives the rate at which cars enter the highway at each minute. We need to find the integral of this rate function over the interval [0, 60] to find the total number of cars.
Total number of cars = Integral[0 to 60] R(t) dt
Substitute the rate function R(t) into the integral and evaluate it over the interval [0, 60] to find the total number of cars.
d. To find the total number of cars on the highway for the first 30 minutes, we have the exit rate function E(t) that gives the rate at which cars exit the freeway at each minute. We need to find the integral of this rate function over the interval [0, 30] to find the total number of cars that exited the freeway.
Total number of cars = Integral[0 to 30] E(t) dt
Substitute the exit rate function E(t) into the integral and evaluate it over the interval [0, 30] to find the total number of cars that exited the freeway.