A cylinder gets taller at a rate of 3 inches per second, but the radius shrinks at a rate of 1 inch per second. How fast is the volume of the cylinder changing when the height is 20 inches and the radius is 10 inches?

2 Answers
Nov 20, 2017

Volume of a cylinder is #V = pir^2h# where

#r# is the radius and #h# is the height of the cylinder,

We are given some rates of change:
#(dh)/(dt) = 3# #"in"#/#"s"# and #(dr)/(dt) = -1# #"in"#/#"s"#.

We seek #(dV)/dt# when #h=20# and #r = 10#.

Differentiate both sides of #V = pir^2h# with respect to #t#. We'll need the product rule on the right.

#d/dt(V) = d/dt(pir^2)h+pir^2 d/dt(h)#

#(dV)/dt = 2pir (dr)/dt h +pir^2(dh)/dt#

At the instant of interest,

#(dV)/dt = 2pi(10) (-1) (20) +pi(10)^2 (3) = -100pi# #"in"^3#/#"s"#

The volume is decreasing at a rate of #100pi# #"in"^3#/#"s"#

Nov 20, 2017

Volume is decreasing at a rate of #100pi \ i n^3s^(-1)#,
or approximately #314\ i n^3s^(-1)#

Explanation:

Let us setup the following variables:

# { (r, "Radius of cylinder at time t","(in)"), (h, "Height of cylinder at time t","(in)"), (V, "Volume of cylinder at time t", "(in"^3")"), (t, "time", "(sec)") :} #

Using the standard formula for volume of a cylinder:

# V = pir^2h #

Where both #r# and #h# are functions of #t#. Implicitly differentiating wrt #t#, we have via the product rule:

# d/dt V = (pir^2)(d/dt h) + (d/dt pir^2)(h) #

Then by the chain rule, we have:

# (dV)/dt = (pir^2)( (dh)/dt d/(dh) h) + ( (dr)/dt d/(dr) pir^2)(h) #
# \ \ \ \ \ \ = (pir^2)( (dh)/dt 1) + ( (dr)/dt 2pir)(h) #
# \ \ \ \ \ \ = pir^2 (dh)/dt + 2pirh (dr)/dt #

And we are given that

# (dh)/dt=3# and #(dr)/dt = -1#

So we have:

# (dV)/dt = 3pir^2 - 2pirh #

When #h=20# and #r=10# we have:

# [(dV)/dt]_(h=20, r=10) = 3pi(100) - 2pi(10)(20) #
# " " = 300pi - 400pi #
# " " = -100pi #

Hence, the volume is decreasing at a rate of #100 \ i n^3s^(-1)#