Help with a related rates problem?
As a balloon in the shape of a sphere is being blown up, the radius is increasing #1/pi# inches per second. at what rate is the volume increasing when the radius is 1 inch
As a balloon in the shape of a sphere is being blown up, the radius is increasing
1 Answer
Mar 5, 2017
when
Explanation:
Let us set up the following variables:
# { (r, "radius of balloon at time "t, "inches"), (V, "Volume at time " t,"inches"^3s^-1) :} #
We are told that:
# (dr)/dt = 1/pi # (constant)
And we aim to find:
# (dV)/dt # when#r=1#
As the balloon is a sphere, then:
# V=4/3pir^3 #
Differentiating wrt
# (dV)/(dr) = 4pir^2 #
And applying the chain rule we have:
# (dV)/dt = (dV)/(dr) * (dr)/dt #
# " " = (4pir^2) * (1/pi) #
# " " = 4r^2 #
So when