The height of the water in a conical storage tank is modeled by a differentiable function $h$, where $h ( t )$ is measured in meters and $t$ is measured in hours. At time $t = 0$, the height of the water in the tank is 25 meters. The height is changing at the rate $h ^ { \prime } ( t ) = 2 - \frac { 24 e ^ { - 0.025 t } } { t + 4 }$ meters per hour for $0 \leq t \leq 24$. (a) When the height of the water in the tank is $h$ meters, the volume of water is $V = \frac { 1 } { 3 } \pi h ^ { 3 }$. At what rate is the volume of water changing at time $t = 0$ ? Indicate units of measure. (b) What is the minimum height of the water during the time period $0 \leq t \leq 24$ ? Justify your answer. (c) The line tangent to the graph of $h$ at $t = 16$ is used to approximate the height of the water in the tank. Using the tangent line approximation, at what time $t$ does the height of the water return to 25 meters?
The height of the water in a conical storage tank is modeled by a differentiable function $h$, where $h ( t )$ is measured in meters and $t$ is measured in hours. At time $t = 0$, the height of the water in the tank is 25 meters. The height is changing at the rate $h ^ { \prime } ( t ) = 2 - \frac { 24 e ^ { - 0.025 t } } { t + 4 }$ meters per hour for $0 \leq t \leq 24$.\\
(a) When the height of the water in the tank is $h$ meters, the volume of water is $V = \frac { 1 } { 3 } \pi h ^ { 3 }$. At what rate is the volume of water changing at time $t = 0$ ? Indicate units of measure.\\
(b) What is the minimum height of the water during the time period $0 \leq t \leq 24$ ? Justify your answer.\\
(c) The line tangent to the graph of $h$ at $t = 16$ is used to approximate the height of the water in the tank. Using the tangent line approximation, at what time $t$ does the height of the water return to 25 meters?