Q.A stone is dropped into a quiet lake and waves move in circles at a speed of $4$ cm per second. At the instant, when the radius of the circular wave is $10$ cm, how fast is the enclosed area increasing?

The key idea is Related Rates: we connect the rate of change of the radius $\frac{dr}{dt}$ to the rate of change of the area $\frac{dA}{dt}$ using the formula for the area of a circle.

Step 1: Area of a circle: $A = \pi r^2$.

Step 2: Differentiate both sides with respect to time $t$:

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

When a stone hits still water, it creates a circular ripple that expands outward. The speed given — 4 cm per second — is the rate at which the radius of that circle grows. The question asks: at the moment the radius is 10 cm, how fast is the area inside the circle increasing?

This is a related rates problem. The key idea: the area $A$ and the radius $r$ are linked by a geometric formula. If we know how fast $r$ changes, we can find how fast $A$ changes by differentiating that formula with respect to time $t$. Both $A$ and $r$ are functions of time, so we use the chain rule.

Let’s work through it step by step.

1. Write the relationship between area and radius. For a circle,

This is the static formula. But here, $r$ is changing with time, so $A$ changes too.

2. Differentiate both sides with respect to time $t$. Since $r$ is a function of $t$, we apply the chain rule:

This is the core related rates equation. It tells us: the rate of change of area depends on the current radius and the rate of change of the radius.

3. Plug in the known values.

   We are given:

   • $\frac{dr}{dt} = 4$ cm/s (the speed at which the radius increases),
   • at the instant of interest, $r = 10$ cm.

   Substituting:

4. Interpret the result. The units: $r$ is in cm, $\frac{dr}{dt}$ in cm/s, so $\frac{dA}{dt}$ comes out in cm²/s. …

Method: The Standard Related-Rates Recipe — Differentiate Before You Substitute

This method is the general five-step recipe for any related-rates problem where a single geometric formula links two quantities that both change with time — here, a circle's radius and its enclosed area.

Steps

Step 1: Identify the quantities changing with time and the rate wanted

Here the radius $r$ of the ripple grows with a known rate $\frac{dr}{dt}$, and the question wants $\frac{dA}{dt}$ at the instant $r$ takes a specific value.

Step 2: Write the static formula linking the quantities

Step 3: Differentiate both sides with respect to time, treating every variable as a function of $t$

This is the crucial ordering: differentiate the general relation first, while $r$ is still a variable — this is what produces the $\frac{dr}{dt}$ factor at all.

Step 4: Substitute the known numerical values only after differentiating …

Common Mistakes

Mistake 1: Substituting $r=10$ into $A=\pi r^2$ before differentiating

A student computes $A=100\pi$ first and then tries to differentiate that number, getting zero. Why it's wrong: once a specific numerical value is substituted for $r$, it is no longer a variable, so its rate of change $\frac{dr}{dt}$ vanishes from the expression entirely — the derivative describes behaviour at an instant, not a static snapshot. Correct approach: differentiate the general formula $A=\pi r^2$ with respect to $t$ first to get $\frac{dA}{dt}=2\pi r\frac{dr}{dt}$, and only then substitute the specific $r=10$ and $\frac{dr}{dt}=4$.

Mistake 2: Confusing the given wave speed with the rate of change of area …