Single Variable Calculus, Chapter 5, 5.4, Section 5.4, Problem 60
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Find the amount of $H_2 O$ that flows from the tank during the first 10min. Suppose that the $H_2 O$ flows from the bottom of a storage tank at a rate of $r(t) = 200 - 4t$ $\displaystyle \frac{\text{liters}}{\text{min}}$, where $0 \leq t \leq 50$.
Using Net Change Theorem
$ \begin{equation} \begin{aligned} \int^b_a r(t) dt &= \int^{10}_0 (200 -4t) dt\\ \\ \int^b_a r(t) dt &= \left[ 200t - 2t^2 \right]^{10}_0\\ \\ \int^b_a r(t) dt &= 200(10) - 2(10)^2 - \left[ 200(0) - 2(0)^2 \right]\\ \\ \int^b_a r(t) dt &= 2000 - 200 - 0\\ \\ \int^b_a r(t) dt &= 1800 \text{ liters} \end{aligned} \end{equation} $
This means that during the first 10 mins. the amount of water that flows inside the tank is 1800 liters.
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