Mathematics

Problem

A storage tank consists of a right circular cylindrical section surmounted by a hemispherical dome. At time

t

(in minutes): •

r

is the common radius (in meters) of both the cylinder and the hemisphere, •

h

is the height (in meters) of the cylindrical section. The radius is increasing at a rate of

0.02\text{ m/min}

, while the height of the cylindrical section is decreasing at a rate of

0.05\text{ m/min}

. At the instant when

r=2\text{ m}

and

h=6\text{ m}

, determine how fast the total volume of the tank is changing. (The volume of a hemisphere of radius

r

\dfrac{2}{3}\pi r^{3}

, and the volume of a right circular cylinder of radius

r

and height

h

\pi r^{2}h

.) (a) Write an explicit formula for the total volume

V

of the tank in terms of

r

and

h

. Simplify your expression. (b) Differentiate your formula from part (a) with respect to time to obtain an expression for

\dfrac{dV}{dt}

in terms of

r

h

\dfrac{dr}{dt}

, and

\dfrac{dh}{dt}

. Show all steps clearly. (c) Evaluate

\dfrac{dV}{dt}

at the given instant, including correct units, and state whether the total volume of the tank is increasing or decreasing at that moment.

Intuitive explanation

The total volume is the sum of two parts—a cylinder and a hemisphere—each depending on $r$ and $h$ . To find how fast the total volume changes, you'll use the chain rule to differentiate the volume formula with respect to time, then substitute the given rates $\frac{dr}{dt}$ and $\frac{dh}{dt}$ and the current values of $r$ and $h$ . The key is recognizing that both $r$ and $h$ are functions of $t$ , so each term in the volume formula contributes to $\frac{dV}{dt}$ .

Intuitive explanation

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