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 radius of a right circular cylinder is increasing at a rate of 2 units per second. The height of the cylinder is decreasing at a rate of 5 units per second. Which of the following expressions gives the rate at which the volume of the cylinder is changing with respect to time in terms of the radius $r$ and height $h$ of the cylinder? (The volume $V$ of a cylinder with radius $r$ and height $h$ is $V = \pi r ^ { 2 } h$.)
(A) $- 20 \pi r$
(B) $- 2 \pi r h$
(C) $4 \pi r h - 5 \pi r ^ { 2 }$
(D) $4 \pi r h + 5 \pi r ^ { 2 }$

A container has the shape of an open right circular cone. The height of the container is 10 cm and the diameter of the opening is 10 cm. Water in the container is evaporating so that its depth $h$ is changing at the constant rate of $\frac { - 3 } { 10 } \text{ cm/hr}$. (Note: The volume of a cone of height $h$ and radius $r$ is given by $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$.)
(a) Find the volume $V$ of water in the container when $h = 5 \text{ cm}$. Indicate units of measure.
(b) Find the rate of change of the volume of water in the container, with respect to time, when $h = 5 \text{ cm}$. Indicate units of measure.
(c) Show that the rate of change of the volume of water in the container due to evaporation is directly proportional to the exposed surface area of the water. What is the constant of proportionality?

The inside of a funnel of height 10 inches has circular cross sections, as shown in the figure above. At height $h$, the radius of the funnel is given by $r = \frac { 1 } { 20 } \left( 3 + h ^ { 2 } \right)$, where $0 \leq h \leq 10$. The units of $r$ and $h$ are inches.
(a) Find the average value of the radius of the funnel.
(b) Find the volume of the funnel.
(c) The funnel contains liquid that is draining from the bottom. At the instant when the height of the liquid is $h = 3$ inches, the radius of the surface of the liquid is decreasing at a rate of $\frac { 1 } { 5 }$ inch per second. At this instant, what is the rate of change of the height of the liquid with respect to time?

The inside of a funnel of height 10 inches has circular cross sections, as shown in the figure above. At height $h$, the radius of the funnel is given by $r = \frac { 1 } { 20 } \left( 3 + h ^ { 2 } \right)$, where $0 \leq h \leq 10$. The units of $r$ and $h$ are inches.
(a) Find the average value of the radius of the funnel.
(b) Find the volume of the funnel.
(c) The funnel contains liquid that is draining from the bottom. At the instant when the height of the liquid is $h = 3$ inches, the radius of the surface of the liquid is decreasing at a rate of $\frac { 1 } { 5 }$ inch per second. At this instant, what is the rate of change of the height of the liquid with respect to time?

Water falls from a tap of circular cross section at the rate of 2 metres/sec and fills up a hemispherical bowl of inner diameter 0.9 metres. If the inner diameter of the tap is 0.01 metres, then the time needed to fill the bowl is
(A) 40.5 minutes
(B) 81 minutes
(C) 60.75 minutes
(D) 20.25 minutes

A pond has been dug at the Indian Statistical Institute as an inverted truncated pyramid with a square base. The depth of the pond is 6 m. The square at the bottom has side length 2 m and the top square has side length 8 m. Water is filled in at a rate of $\frac { 19 } { 3 }$ cubic meters per hour. At what rate is the water level rising exactly 1 hour after the water started to fill the pond?

A spherical balloon is filled with 4500$\pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72\pi$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is
(1) $\frac{9}{7}$
(2) $\frac{7}{9}$
(3) $\frac{2}{9}$
(4) $\frac{9}{2}$

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is $\tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)$. Water is poured into it at a constant rate of $5$ cubic $\mathrm { m } / \mathrm { min }$. Then the rate (in $\mathrm { m } / \mathrm { min }$), at which the level of water is rising at the instant when the depth of water in the tank is $10 m$; is:
(1) $\frac { 1 } { 10 \pi }$
(2) $\frac { 1 } { 15 \pi }$
(3) $\frac { 1 } { 5 \pi }$
(4) $\frac { 2 } { \pi }$

Water is being filled at the rate of $1 \mathrm{~cm}^3 \mathrm{sec}^{-1}$ in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the height of the water level is 10 cm, the rate (in $\mathrm{cm}^2 \mathrm{sec}^{-1}$) at which the wet conical surface area of the vessel increases is
(1) 5
(2) $\frac{\sqrt{21}}{5}$
(3) $\frac{\sqrt{26}}{5}$
(4) $\frac{\sqrt{26}}{10}$