Transportation engineers use traffic flow data to design the timing of traffic lights at intersections. At a certain intersection, the traffic flow, F(t), is modeled by the function
F(t)=82+4sin(t/2) when 0¡Üt¡Ü30 minutes, where F(t), in vehicles per minute, is the rate at which vehicles pass through the intersection.
a) How many vehicles (to the nearest whole number) pass through the intersection over the 30-minute period?
b) Is the traffic flow increasing or decreasing at t=7? Include a reason with your answer
c) What is the average value of the traffic flow durign the 30-minute time period?
d) What is the average rate of change of the traffic flow during the 30-minute time period?
Hannah, please use Western ISO8859-1 encoding when you post mathematical symbols. It would be much easier for everyone. Thanks.
When appropriate encoding is used, the second paragraph reads:
F(t)=82+4sin(t/2) when 0≤t≤30 minutes, where F(t), in vehicles per minute, is the rate at which vehicles pass through the intersection.
To answer these questions, we can use the given function for traffic flow, F(t) = 82 + 4sin(t/2). Let's go through each question one by one.
a) To find the number of vehicles that pass through the intersection over the 30-minute period, we need to find the total accumulated flow. We can do this by finding the definite integral of the flow function over the interval [0, 30].
∫[0,30] (82 + 4sin(t/2)) dt
Integrating, we get:
[82t - 8cos(t/2)] from 0 to 30
Substituting the upper and lower limits:
[82(30) - 8cos(30/2)] - [82(0) - 8cos(0/2)]
Simplifying further:
(2460 - 8cos(15)) - (0 - 8cos(0))
Calculating cos(15) and cos(0):
(2460 - 8cos(15)) - (0 - 8)
Now, calculate the final answer:
2460 - 8cos(15) - 8
Round the answer to the nearest whole number to account for vehicles:
2401 vehicles (approximately) pass through the intersection over the 30-minute period.
b) To determine if the traffic flow is increasing or decreasing at t = 7, we need to examine the derivative of the flow function at that specific time. If the derivative is positive, the flow is increasing. If the derivative is negative, the flow is decreasing.
Taking the derivative of the flow function:
F'(t) = 0 + 4/2 · cos(t/2)
Simplifying:
F'(t) = 2cos(t/2)
Now, substitute t=7 into the derivative:
F'(7) = 2cos(7/2)
Evaluating the cosine value:
F'(7) = 2cos(3.5)
Since the value of cos(3.5) is positive, it means that F'(7) is positive as well. Hence, the traffic flow is increasing at t=7.
c) To find the average value of the traffic flow during the 30-minute time period, we can find the average of the flow function over the interval [0, 30]. This can be calculated by finding the definite integral of the flow function divided by the length of the interval (30).
(1/30) * ∫[0,30] (82 + 4sin(t/2)) dt
Integrating, we get:
(1/30) * [82t - 8cos(t/2)] from 0 to 30
Substituting the upper and lower limits:
(1/30) * [(82(30) - 8cos(30/2)) - (82(0) - 8cos(0/2))]
Simplifying further:
(1/30) * (2460 - 8cos(15) - 8)
Now, calculate the final answer:
(1/30) * (2460 - 8cos(15) - 8)
This will give you the average value of the traffic flow during the 30-minute time period.
d) To find the average rate of change of the traffic flow during the 30-minute time period, we need to calculate the change in flow divided by the change in time over the interval [0, 30]. This can be calculated using the following formula:
Average rate of change = (F(30) - F(0)) / (30 - 0)
Substituting the values:
Average rate of change = (F(30) - F(0)) / 30
Substituting the flow function into the formula:
Average rate of change = ([82 + 4sin(30/2)] - [82 + 4sin(0/2)]) / 30
Simplifying and evaluating the sin values:
Average rate of change = (86 - 82) / 30
Now, calculate the final answer:
Average rate of change = 4 / 30
Simplify the fraction:
Average rate of change = 2/15
This will give you the average rate of change of the traffic flow during the 30-minute time period.