A traffic engineer monitors the rate at which cars enter a freeway onramp during rush hour. From her data, she estimates that between 4:30 pm and 5:30 pm, the rate R(t) at which cars enter is given by:
R(t)=100(1-0.00012t^2)
cars per minute, where t is time in minutes since 4:30 pm.
A) find the average rate in cars per minute, at which cars enter the highway during the first half hour.
B) find the average rate, in cars per minute, at which cars enter the highway during the second 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 minutes. Assume no cars initially.
A) To find the average rate at which cars enter the highway during the first half hour, we need to calculate the total number of cars that enter the highway and divide it by the duration of half an hour.
The given function R(t) represents the rate at which cars enter the highway in cars per minute. To find the total number of cars that enter the highway during the first half hour, we integrate the function R(t) over the interval [0, 30] (since t represents time in minutes since 4:30 pm).
Using the formula for indefinite integration, ∫(1 - 0.00012t^2) dt, we can apply the power rule to integrate term by term:
∫(1 - 0.00012t^2) dt = t - (0.00012/3)t^3 + C
Substituting the limits of integration (0 and 30) for t, we get:
∫[0,30] (1 - 0.00012t^2) dt = (30 - (0.00012/3)(30)^3) - (0 - (0.00012/3)(0)^3)
= (30 - 0.36) - (0 - 0)
= 29.64
So, the total number of cars that enter the highway during the first half hour is 29.64 cars.
To find the average rate, we divide the total number of cars by the duration of half an hour:
Average rate = Total number of cars / Duration = 29.64 cars / 30 minutes = 0.988 cars per minute
Therefore, the average rate at which cars enter the highway during the first half hour is approximately 0.988 cars per minute.
B) To find the average rate at which cars enter the highway during the second half hour, we follow the same steps as in part A, but this time integrating over the interval [30, 60].
∫[30,60] (1 - 0.00012t^2) dt = (60 - (0.00012/3)(60)^3) - (30 - (0.00012/3)(30)^3)
= (60 - 0.432) - (30 - 0.108)
= 29.676
The total number of cars that enter the highway during the second half hour is 29.676 cars.
Average rate = Total number of cars / Duration = 29.676 cars / 30 minutes = 0.9892 cars per minute
Therefore, the average rate at which cars enter the highway during the second half hour is approximately 0.9892 cars per minute.
C) To find the total number of cars that enter the highway during the hour, we need to sum the total number of cars for each half hour.
Total number of cars = Number of cars in the first half hour + Number of cars in the second half hour
= 29.64 cars + 29.676 cars
= 59.316 cars
Therefore, the total number of cars that enter the highway during the hour is approximately 59.316 cars.
D) To find the total number of cars on the highway for the first 30 minutes, we integrate the exit rate function E(t) over the interval [0, 30].
Using the formula for indefinite integration, ∫50(1 - e^(-t)) dt, we can integrate term by term:
∫[0,30] 50(1 - e^(-t)) dt = 50(t + e^(-t)) |[0,30]
= 50(30 + e^(-30)) - 50(0 + e^(-0))
= 1500 + 50e^(-30)
Therefore, the total number of cars on the highway for the first 30 minutes is 1500 + 50e^(-30) cars.