Mathematics

Problem

An industrial storage vessel has the shape of a frustum of a right circular cone. The lower radius is

2\,\text{m}

, the upper radius is

5\,\text{m}

, and the height is

8\,\text{m}

. A liquid is pumped into the vessel at a constant rate of

10\,\text{m}^3/\text{min}

. At the same time, a valve at the bottom removes liquid at a constant rate of

4\,\text{m}^3/\text{min}

. Let

h(t)

denote the depth of the liquid (in meters) measured from the bottom. a. Part 1: (a) Show that the radius

r

of the liquid surface can be written as a linear function of

h

. Clearly state this function. b. Part 2: (b) Derive a formula for the volume

V

of liquid in the vessel as a function of

h

. c. Part 3: (c) Calculate

\dfrac{dh}{dt}

when

h=6\,\text{m}

. Provide units and interpret whether the liquid level is rising or falling at this instant.