Applicatioins of the Chain Rule
The Chain Rule can be in many different ways. One of the most practical is to find the rate of change of some dimension of a solid when the volume of the solid is changing. For example, a spherical ballon is being blown up and we might want to find the rate of change of the radius, treating the balloon as a sphere, or oil might be poured onto a puddle of oil, and we might want to find the rate of increase of the radius of the oil puddle.
Suppose that a spherically shaped raindrop is falling. As it falls, water condenses on it. The rate of increase of the radius of the raindrop is 0.01 mm/s at a tuime when the radius of the raindrop is 3 mm. What is the rate at which the volume is increasing?
If the raindrop can be treated as a sphere, then the volume of the raindrop is
The chain rule may be used in the form
so when
mm,
We are told in the question that the rate of increase of
mm/s, so
mm 3 /s.
Suppose on the other hand that air is being blown into a balloon at the rate of 5 cm 3 /s at a point when the radius of the balloon is 6 cm. Find the rate at which the radius of the ballon is increasing.
Again using the relationship
withso when r=6 cm,
is the rate at which air is being blown into the balloon, so
cm/s.