Mathematics

Problem

A right circular cylinder has fixed volume

V=1000\text{ cm}^3

. At time

t

, its radius is

r(t)

centimeters and its height is

h(t)

centimeters. a. Express

h

as a function of

r

using the volume constraint. b. When

r=5\text{ cm}

, the radius is increasing at

\dfrac{dr}{dt}=0.020\text{ cm/s}

. Calculate

\dfrac{dh}{dt}

at that instant. Show each derivation step clearly. c. The lateral surface area of the cylinder is

S=2\pi r h

. Determine

\dfrac{dS}{dt}

at the same instant (

r=5\text{ cm}

\dfrac{dr}{dt}=0.020\text{ cm/s}

) using your result from part (b). Include units and interpret the physical meaning of the numerical value you obtain.