3.5. RELATED RATES
197
3.5 Related Rates
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• If two quantities that are related, such as the radius and volume of a spherical
balloon, are both changing as implicit functions of time, how are their rates of
change related? That is, how does the relationship between the values of the
quantities affect the relationship between their respective derivatives with respect
to time?
Introduction
In most of our applications of the derivative so far, we have worked in settings where one
quantity (often called y) depends explicitly on another (say x), and in some way we have
been interested in the instantaneous rate at which y changes with respect to x, leading us
to compute
dy
dx . These settings emphasize how the derivative enables us to quantify how
the quantity y is changing as x changes at a given x-value.
We are next going to consider situations where multiple quantities are related to one
another and changing, but where each quantity can be considered an implicit function of
the variable t, which represents time. Through knowing how the quantities are related,
we will be interested in determining how their respective rates of change with respect to
time are related. For example, suppose that air is being pumped into a spherical balloon
in such a way that its volume increases at a constant rate of 20 cubic inches per second. It
makes sense that since the balloon’s volume and radius are related, by knowing how fast
the volume is changing, we ought to be able to relate this rate to how fast the radius is
changing. More specifically, can we find how fast the radius of the balloon is increasing at
the moment the balloon’s diameter is 12 inches?
The following preview activity leads you through the steps to answer this question.
Preview Activity 3.5. 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 d = 12 or when
d = 16? Why?
(a) Draw several spheres with different radii, and observe that as volume changes, the
radius, diameter, and surface area of the balloon also change.
(b) Recall that the volume of a sphere of radius r is V =
4
3 πr 3 . Note well that in the
setting of this problem, both V and r are changing as time t changes, and thus
197
3.5 Related Rates
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• If two quantities that are related, such as the radius and volume of a spherical
balloon, are both changing as implicit functions of time, how are their rates of
change related? That is, how does the relationship between the values of the
quantities affect the relationship between their respective derivatives with respect
to time?
Introduction
In most of our applications of the derivative so far, we have worked in settings where one
quantity (often called y) depends explicitly on another (say x), and in some way we have
been interested in the instantaneous rate at which y changes with respect to x, leading us
to compute
dy
dx . These settings emphasize how the derivative enables us to quantify how
the quantity y is changing as x changes at a given x-value.
We are next going to consider situations where multiple quantities are related to one
another and changing, but where each quantity can be considered an implicit function of
the variable t, which represents time. Through knowing how the quantities are related,
we will be interested in determining how their respective rates of change with respect to
time are related. For example, suppose that air is being pumped into a spherical balloon
in such a way that its volume increases at a constant rate of 20 cubic inches per second. It
makes sense that since the balloon’s volume and radius are related, by knowing how fast
the volume is changing, we ought to be able to relate this rate to how fast the radius is
changing. More specifically, can we find how fast the radius of the balloon is increasing at
the moment the balloon’s diameter is 12 inches?
The following preview activity leads you through the steps to answer this question.
Preview Activity 3.5. 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 d = 12 or when
d = 16? Why?
(a) Draw several spheres with different radii, and observe that as volume changes, the
radius, diameter, and surface area of the balloon also change.
(b) Recall that the volume of a sphere of radius r is V =
4
3 πr 3 . Note well that in the
setting of this problem, both V and r are changing as time t changes, and thus