4.1 Related rates

We have seen that for quantities that are changing over time, the rates at which these quantities change are given by derivatives. If two related quantities are changing over time, the rates at which the quantities change are related. For example, if a balloon is being filled with air, both the radius of the balloon and the volume of the balloon are increasing. In this section, we consider several problems in which two or more related quantities are changing and we study how to determine the relationship between the rates of change of these quantities.

Setting up related-rates problems

In many real-world applications, related quantities are changing with respect to time. For example, if we consider the balloon example again, we can say that the rate of change in the volume, $V,$ is related to the rate of change in the radius, $r.$ In this case, we say that $\frac{dV}{dt}$ and $\frac{dr}{dt}$ are related rates    because V is related to r . Here we study several examples of related quantities that are changing with respect to time and we look at how to calculate one rate of change given another rate of change.

Inflating a balloon

A spherical balloon is being filled with air at the constant rate of $2{\text{cm}}^3\text{/}\text{sec}$ ( [link] ). How fast is the radius increasing when the radius is $3\text{cm}?$

The volume of a sphere of radius $r$ centimeters is

$V=\frac{4}{3}\pi r^3{\text{cm}}^3.$

Since the balloon is being filled with air, both the volume and the radius are functions of time. Therefore, $t$ seconds after beginning to fill the balloon with air, the volume of air in the balloon is

$V(t)=\frac{4}{3}\pi {[r(t)]}^3{\text{cm}}^3.$

Differentiating both sides of this equation with respect to time and applying the chain rule, we see that the rate of change in the volume is related to the rate of change in the radius by the equation

$V\prime \left(t\right)=4\pi {\left[r\left(t\right)\right]}^2{r}^{\prime }\left(t\right).$

The balloon is being filled with air at the constant rate of 2 cm ³ /sec, so $V\prime (t)=2{\text{cm}}^3\text{/}\text{sec}.$ Therefore,

$2{\text{cm}}^3\text{/}\text{sec}=\left(4\pi {\left[r(t)\right]}^2{\text{cm}}^2\right)·\left(r\prime \left(t\right)\text{cm/s}\right)\text{,}$

which implies

$r\prime \left(t\right)=\frac{1}{2\pi {\left[r\left(t\right)\right]}^2}\text{cm/sec}.$

When the radius $r=3\text{cm,}$

$r\prime (t)=\frac{1}{18\pi }\text{cm/sec}.$

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Before looking at other examples, let’s outline the problem-solving strategy we will be using to solve related-rates problems.