A cylindrical barrel with a diameter of 2 feet contains collected rainwater. The water drains out through a valve (not shown) at the bottom of the barrel. The rate of change of the height $h$ of the water in the barrel with respect to time $t$ is modeled by $\dfrac{dh}{dt} = -\dfrac{1}{10}\sqrt{h}$, where $h$ is measured in feet and $t$ is measured in seconds. (The volume $V$ of a cylinder with radius $r$ and height $h$ is $V = \pi r^2 h$.) (a) Find the rate of change of the volume of water in the barrel with respect to time when the height of the water is 4 feet. Indicate units of measure. (b) When the height of the water is 3 feet, is the rate of change of the height of the water with respect to time increasing or decreasing? Explain your reasoning. (c) At time $t = 0$ seconds, the height of the water is 5 feet. Use separation of variables to find an expression for $h$ in terms of $t$.
A cylindrical barrel with a diameter of 2 feet contains collected rainwater. The water drains out through a valve (not shown) at the bottom of the barrel. The rate of change of the height $h$ of the water in the barrel with respect to time $t$ is modeled by $\dfrac{dh}{dt} = -\dfrac{1}{10}\sqrt{h}$, where $h$ is measured in feet and $t$ is measured in seconds. (The volume $V$ of a cylinder with radius $r$ and height $h$ is $V = \pi r^2 h$.)
(a) Find the rate of change of the volume of water in the barrel with respect to time when the height of the water is 4 feet. Indicate units of measure.
(b) When the height of the water is 3 feet, is the rate of change of the height of the water with respect to time increasing or decreasing? Explain your reasoning.
(c) At time $t = 0$ seconds, the height of the water is 5 feet. Use separation of variables to find an expression for $h$ in terms of $t$.