This problem will lead you through the steps to answer this question:
A spherical balloon is being inflated at a constant rate of 20 cubic
inches per second. How fast is the radius of the balloon changing at the instant the
balloon's diameter is 12 inches? Is the radius changing more rapidly when or when
? Why?
Draw several spheres with different radii, and observe that as volume changes, the
radius, diameter, and surface area of the balloon also change. Recall that the volume
of a sphere of radius is .
Note that in the
setting of this problem, both and are changing as time changes,
and thus
both and may be viewed as implicit functions of ,
with respective derivatives
and .
Differentiate both sides of the equation
with respect to (using the chain
rule on the right) to find a formula for
that depends on both and
.
At this point in the problem, by differentiating we have "related the rates" of
change of and . Recall that we are given in the problem that the balloon is
being inflated at a constant rate of 20 cubic inches per second. To which
derivative does this rate correspond?
From the above discussion, we know the value of
at every value of . Next, observe that
when the diameter of the balloon is 12, we know the value of the radius. In the
equation ,
substitute these values for the relevant quantities and solve
for the remaining unknown quantity, which is
.
How fast is the radius changing at the instant when ?
How fast is the radius changing at the instant when ?
When is the radius changing more rapidly, when or when ?
You can earn partial credit on this problem.