Related rates

Using the chain rule to link rates of change of related quantities, such as an inflating balloon, a filling conical tank and an expanding ripple, and the steps for solving related rates problems.

The method

Example 1: a balloon●●

A spherical balloon is inflated at $100$ cm$^3$/s. How fast is the radius increasing when $r = 5$ cm?

$V = \tfrac43\pi r^3$, so $\dfrac{dV}{dr} = 4\pi r^2$.

$100 = 4\pi \times 25 \times \dfrac{dr}{dt}$, so $\dfrac{dr}{dt} = \dfrac{1}{\pi} \approx 0.318$ cm/s.

Example 2: a conical tank●●●

Water flows at $2$ m$^3$/min into a cone with its vertex down, where the radius of the water surface is half its depth. How fast is the depth rising when it is $4$ m?

$r = \tfrac12 h$, so $V = \tfrac13\pi\big(\tfrac12 h\big)^2 h = \dfrac{\pi h^3}{12}$ and $\dfrac{dV}{dh} = \dfrac{\pi h^2}{4}$.

At $h = 4$: $2 = 4\pi \times \dfrac{dh}{dt}$, so $\dfrac{dh}{dt} = \dfrac{1}{2\pi} \approx 0.159$ m/min.

Flashcards

Chain rule for rates

$\dfrac{dV}{dt} = \dfrac{dV}{dr} \times \dfrac{dr}{dt}$

Sphere

$V = \tfrac43\pi r^3$, $\dfrac{dV}{dr} = 4\pi r^2$

When to substitute values

After differentiating.

Practice

A balloon is inflated at $100$ cm$^3$/s. Find $\dfrac{dr}{dt}$ when $r = 5$ cm, correct to three decimal places.
In Example 2, find $\dfrac{dh}{dt}$ when $h = 4$ m, correct to three decimal places.
A ripple's radius grows at $2$ m/s. Find $\dfrac{dA}{dt}$ when $r = 5$ m, correct to three decimal places.
A cube's volume grows at $48$ cm$^3$/s. Find the rate of increase of the edge when the edge is $2$ cm.
A square's side grows at $3$ cm/s. Find the rate of increase of its area when the side is $4$ cm.
A circle's area grows at $2$ cm$^2$/s. Find $\dfrac{dr}{dt}$ when $r = \dfrac{25}{\pi}$ cm.
For $V = \dfrac{\pi h^3}{12}$, $\dfrac{dV}{dh} = $
The rate $\dfrac{dr}{dt}$ for a balloon inflated at a constant rate: