Single Variable Calculus Questions and Answers

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Single Variable Calculus, Chapter 3, 3.7, Section 3.7, Problem 18

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The equation $\displaystyle V = 5000 \left( 1- \frac{t}{40} \right)^2 \, 0 \leq t \leq 40$ represents the Torricelli's Law that states the volume $V$ of water remaining in the tank after $t$ minutes. Suppose that a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes. Find the rate at which water is draining from the tank after (a) 5 min, (b) 10 min, (c) 20 min, and (d) 40 min. At what time is the water flowing out the fastest? The slowest? Summarize your findings.

Using Chain Rule,

$ \begin{equation} \begin{aligned} \frac{dV}{dt} &= 5000 \frac{d}{dt} \left( 1 - \frac{t}{40} \right)^2 \cdot \frac{d}{dt} \left( 1 - \frac{t}{40} \right)\\ \\ \frac{dV}{dt} &= 5000 (2) \left( 1 - \frac{t}{40} \right) \left( \frac{-1}{40} \right)\\ \\ \frac{dV}{dt} &= -250 \left( 1 - \frac{t}{40} \right) \end{aligned} \end{equation} $

a.) when $t = 5$ min

$ \begin{equation} \begin{aligned} \frac{dV}{dt} &= -250 \left( 1 - \frac{5}{40}\right)\\ \frac{dV}{dt} &= -218.75 \frac{\text{volume}}{\text{min}} \end{aligned} \end{equation} $

b.) when $t = 10$ min

$ \begin{equation} \begin{aligned} \frac{dV}{dt} &= -250 \left( 1 - \frac{10}{40}\right)\\ \frac{dV}{dt} &= -187.5 \frac{\text{volume}}{\text{min}} \end{aligned} \end{equation} $

c.) when $t = 20$ min

$ \begin{equation} \begin{aligned} \frac{dV}{dt} &= -250 \left( 1 - \frac{20}{40}\right)\\ \frac{dV}{dt} &= -125 \frac{\text{volume}}{\text{min}} \end{aligned} \end{equation} $

d.) when $t = 40$ min

$ \begin{equation} \begin{aligned} \frac{dV}{dt} &= -250 \left( 1 - \frac{40}{40}\right)\\ \frac{dV}{dt} &= 0 \end{aligned} \end{equation} $

Based from the values we obtain,

The time at which the water is flowing out fastest is when $t = 5$min at $\displaystyle 218.75 \frac{\text{volume}}{\text{min}}$. On the other hand, the flow rate is slowest when $t = 20$ min at $\displaystyle 125 \frac{\text{volume}}{\text{min}}$. We never consider the slowest flow rate at 0 because that's the moment where the water is all drained from the tank. Hence, no flow rate. The flow rates represents how fast the water is flowing out from the tank, until such time that the water is all drained from the tank, that is at $t = 40$ min.