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DETAILS SCALC9 4.4.066.
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Water flows from the bottom of a storage tank at a rate of $r(t)=200-4 t$ liters per minute, where $0 \leq t \leq 50$. Find the amount of water (in liters) that flows from the tank during the first 30 minutes.
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Step 1 ：The problem is asking for the total amount of water that flows from the tank during the first 30 minutes. This can be found by integrating the rate function from 0 to 30. The integral of a rate function over an interval gives the total amount of change over that interval. In this case, the rate function is \(r(t) = 200 - 4t\), and we are integrating from 0 to 30.

Step 2 ：Let's denote time as \(t\) and the rate of water flow as \(r\), which is given by \(r = 200 - 4*t\).

Step 3 ：By integrating the rate function from 0 to 30, we find that the total amount of water that flows from the tank during the first 30 minutes is 4200 liters.

Step 4 ：Final Answer: The total amount of water that flows from the tank during the first 30 minutes is \(\boxed{4200}\) liters.