Related Rates Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy

A spherical balloon is being inflated so its radius increases at 2 cm/s. How fast is the volume increasing when the radius is 5 cm?

Solution

1
Volume of sphere: $V = \frac{4}{3}\pi r^3$ .
2
Differentiate with respect to time: $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$ .
3
Given: $\frac{dr}{dt} = 2$ cm/s, $r = 5$ cm.
4
$\frac{dV}{dt} = 4\pi(25)(2) = 200\pi$ cm³/s.

Answer

\frac{dV}{dt} = 200\pi \approx 628 \text{ cm}^3/\text{s}

The chain rule links rates through their geometric relationship. Write the geometric formula, differentiate both sides with respect to

t

, then substitute the known values.

About Related Rates

Problems where two or more quantities change with time and are related by an equation. Differentiate the equation with respect to time $t$ and use known rates to find an unknown rate.

Learn more about Related Rates →

More Related Rates Examples

Example 2 hard

A 13-ft ladder rests against a wall. The base slides away at 5 ft/s. How fast is the top sliding dow

Example 3 easy

A circle's radius grows at 3 cm/s. How fast is the area increasing when [formula] cm?

Example 4 medium

Water fills a cone (apex down) of radius 3 m and height 6 m at 2 m³/min. How fast is the water level

Related Concepts

Chain Rule Implicit Differentiation Rate of Change