A 12,000-liter tank of water is filled to capacity. At time $t = 0$, water begins to drain out of the tank at a rate modeled by $r(t)$, measured in liters per hour, where $r$ is given by the piecewise-defined function $$r(t) = \begin{cases} \dfrac{600t}{t+3} & \text{for } 0 \leq t \leq 5 \\ 1000e^{-0.2t} & \text{for } t > 5 \end{cases}$$ (a) Is $r$ continuous at $t = 5$? Show the work that leads to your answer. (b) Find the average rate at which water is draining from the tank between time $t = 0$ and time $t = 8$ hours. (c) Find $r^{\prime}(3)$. Using correct units, explain the meaning of that value in the context of this problem. (d) Write, but do not solve, an equation involving an integral to find the time $A$ when the amount of water in the tank is 9000 liters.
A 12,000-liter tank of water is filled to capacity. At time $t = 0$, water begins to drain out of the tank at a rate modeled by $r(t)$, measured in liters per hour, where $r$ is given by the piecewise-defined function
$$r(t) = \begin{cases} \dfrac{600t}{t+3} & \text{for } 0 \leq t \leq 5 \\ 1000e^{-0.2t} & \text{for } t > 5 \end{cases}$$
(a) Is $r$ continuous at $t = 5$? Show the work that leads to your answer.
(b) Find the average rate at which water is draining from the tank between time $t = 0$ and time $t = 8$ hours.
(c) Find $r^{\prime}(3)$. Using correct units, explain the meaning of that value in the context of this problem.
(d) Write, but do not solve, an equation involving an integral to find the time $A$ when the amount of water in the tank is 9000 liters.