A storage tank consists of a right circular cylindrical section surmounted by a…

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

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.