Q.A stone is dropped into a quiet lake and waves move in circles at the speed of $5 \text{ cm/s}$. At the instant when the radius of the circular wave is $8 \text{ cm}$, how fast is the enclosed area increasing?

Concept: Related Rates — we relate the rate of change of the radius to the rate of change of the area using differentiation with respect to time.

Step 1: Area of a circle: $A = \pi r^2$. Differentiate both sides with respect to time $t$:

$\frac{dA}{dt} = 2\pi r \cdot \frac{dr}{dt}.$ …

This is a classic related rates problem. The key idea: when two quantities change with time and are linked by a formula, we differentiate that formula with respect to time to find how fast one changes given the other.

Here, the wave spreads as a circle whose radius grows at a constant speed. The area enclosed depends on the radius, so the rate of area increase depends on both the current radius and how fast the radius is growing.

1. Identify the variables and given rates.

   Let $r$ be the radius (in cm) of the circular wave at time $t$ seconds.

   The wave speed is $\frac{dr}{dt} = 5$ cm/s (constant).

   We want $\frac{dA}{dt}$ when $r = 8$ cm, where $A$ is the enclosed area.

2. Write the relationship between area and radius.

   For a circle, $A = \pi r^2$.

3. Differentiate both sides with respect to time $t$.

   Since $A$ and $r$ are functions of $t$, we use the chain rule:

Method: Translating a Word Problem into a Related-Rates Setup

Word problems describing a physical scenario (a spreading wave, an inflating balloon, a growing shadow) first need to be converted into the standard geometric relation before the calculus begins.

Steps

Step 1: Identify the geometric shape and name its variables

Read the physical description and recognize the underlying shape — here, the spreading ripple is a circle, so let $r$ be its radius and $A$ its enclosed area, both functions of time.

Step 2: Identify which rate is given and which is asked for

"Waves move ... at the speed of $5\text{ cm/s}$" is the given rate of the radius, $\frac{dr}{dt}$; "how fast is the area increasing" is the rate being asked for, $\frac{dA}{dt}$. …

Common Mistakes

Mistake 1: Misreading the wave speed as the rate of change of area instead of radius

The "$5\text{ cm/s}$" describes how fast the radius of the circular wavefront grows, not directly the area — plugging it straight into a formula for $\frac{dA}{dt}$ without first relating $A$ and $r$ skips the actual related-rates step.

Mistake 2: Substituting the radius before differentiating …