The traffic flow rate (cars per hour) across an intersection is $r(t)=500+800 t-240 t^{2}$, where $t$ is in hours, and $t=0$ is $6 \mathrm{am}$. How many cars pass through the intersection between 6 am and 9 am?
cars
Answer
Final Answer: The total number of cars that pass through the intersection between 6 am and 9 am is \(\boxed{2940}\).
Step 1 ：The problem is asking for the total number of cars that pass through the intersection between 6 am and 9 am. This is equivalent to finding the area under the curve of the function \(r(t)=500+800 t-240 t^{2}\) from \(t=0\) to \(t=3\). This can be done by integrating the function over this interval.
Step 2 ：Let's denote the time as \(t\) and the traffic flow rate as \(r\). So, \(r = -240*t**2 + 800*t + 500\).
Step 3 ：By integrating the function from \(t=0\) to \(t=3\), we find that the total number of cars that pass through the intersection between 6 am and 9 am is 2940.
Step 4 ：Final Answer: The total number of cars that pass through the intersection between 6 am and 9 am is \(\boxed{2940}\).
