Question

exercises 1.4 integration formulas and the net change theorem
score: 70/140 answered: 8/14
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question 9
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the traffic flow rate (cars per hour) across an intersection is $r(t) = 300 + 700t - 90t^2$, where $t$ is in hours, and $t = 0$ is 6am. how many cars pass through the intersection between 6 am and 10 am?
cars
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Explanation:

Step1: Define time interval

From 6 am to 10 am, $t$ ranges from $0$ to $4$ hours.

Step2: Set up integral for total cars

Total cars = $\int_{0}^{4} r(t) dt = \int_{0}^{4} (300 + 700t - 90t^2) dt$

Step3: Find antiderivative

Antiderivative: $300t + 350t^2 - 30t^3$

Step4: Evaluate from 0 to 4

At $t=4$: $300(4) + 350(4)^2 - 30(4)^3 = 1200 + 5600 - 1920$
At $t=0$: $300(0) + 350(0)^2 - 30(0)^3 = 0$
Subtract: $(1200 + 5600 - 1920) - 0 = 4880$

Answer:

4880 cars