HW 31 - Applications of Definite Integrals Section 5.4: Problem 10
(1 point)
The traffic flow rate, in cars per hour, past a certain point on a highway is $q(t)=1000+2000 t-360 t^{2}$ where $t$ is in hours and $t=0$ is 8 AM. How many cars pass by during the time interval from 8 AM to 9 AM?
Answer: $\square$ cars.
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Step 1 ：The problem is asking for the total number of cars that pass by from 8 AM to 9 AM. This can be found by integrating the traffic flow rate function \(q(t)=1000+2000 t-360 t^{2}\) from \(t=0\) to \(t=1\) (since \(t=0\) is 8 AM and \(t=1\) is 9 AM).

Step 2 ：The integral of a rate function over an interval gives the total quantity accumulated over that interval. In this case, the rate function is the traffic flow rate and the interval is from 8 AM to 9 AM.

Step 3 ：So, the integral of \(q(t)\) from 0 to 1 will give the total number of cars that pass by during that time.

Step 4 ：By calculating the integral, we find that the total number of cars that pass by from 8 AM to 9 AM is 1880.

Step 5 ：Final Answer: \(\boxed{1880}\) cars.