Related Rates Math Example 3

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

Example 3

easy

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

r = 10

cm?

Solution

1
$A = \pi r^2 \Rightarrow \frac{dA}{dt} = 2\pi r \frac{dr}{dt}$ .
2
At $r=10$ : $\frac{dA}{dt} = 2\pi(10)(3) = 60\pi$ cm²/s.

Answer

60\pi \approx 188.5 \text{ cm}^2/\text{s}

Differentiate the area formula with respect to time using the chain rule. Substitute the known radius and rate.

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 1 easy

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

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 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