Question

The radius of a right circular cylinder is given by sqr( t + 6) and its height is 1/6 sqr(t)
, where t is time in seconds and the dimensions are in inches. Find the rate of change of the volume with respect to time.

Answer 1

v = 1/3 pi r^2 h
= pi/3 (t+6)(1/6 √t)
= (pi/18)t^(3/2) + (pi/3)√t

dv/dt = pi/12 √t + pi/(6√t)

Or, using the chain rule,

v = pi/3 r^2 h
dv/dt = 2pi/3 rh dr/dt + pi/3 r^2 dh/dt
= (2pi/3)√(t+6)(1/6 √t)(1/(2√(t+6)) + (pi/3)(t+6)(1/(12√t))
= 2pi/3 √t/6 + pi/36 √t + pi/(6√t)
= (pi/12)√t + pi/(6√t)

or (pi/12√t)(t+2)