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
Volume of a cylinder is
We are given some rates of change:
We seek
Differentiate both sides of
At the instant of interest,
The volume is decreasing at a rate of
Volume is decreasing at a rate of
or approximately
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
# 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
# [(dV)/dt]_(h=20, r=10) = 3pi(100) - 2pi(10)(20) #
# " " = 300pi - 400pi #
# " " = -100pi #
Hence, the volume is decreasing at a rate of