Related Rates — Question 5

Question 5 - Solution

Let $r(t)$ be the radius of the ripple at time $t$, in meters.

(a) Rate of Change of Area

The area of a circle is: $$A = \pi r^2$$

Differentiate both sides with respect to time $t$: $$\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$$

We’re given: $$\frac{dr}{dt} = 0.05 \, \text{m/s}, \quad r = 0.3 \, \text{m}$$

Substitute: $$\frac{dA}{dt} = 2\pi (0.3)(0.05) = 0.03\pi \approx \boxed{0.0942 \, \text{m}^2/\text{s}}$$

(b) Rate of Change of Circumference

Circumference is given by: $$C = 2\pi r$$

Differentiate: $$\frac{dC}{dt} = 2\pi \frac{dr}{dt}$$

Substitute: $$\frac{dC}{dt} = 2\pi (0.05) = \boxed{0.1\pi \approx 0.3142 \, \text{m/s}}$$