Q.The radius of a circle is increasing at the rate of $0.7 \text{ cm/s}$. What is the rate of increase of its circumference?

The key idea is Related Rates: when one quantity changes with time, we differentiate the geometric relation linking it to another quantity.

1. The circumference of a circle is $C = 2\pi r$, where $r$ is the radius.
2. Differentiate both sides with respect to time $t$: $\frac{dC}{dt} = 2\pi \frac{dr}{dt}$ …

This is a classic Related Rates problem — you’re given how fast one quantity (the radius) changes, and you need to find how fast another quantity (the circumference) changes, both with respect to time. The key is that the two quantities are linked by a geometric formula, so differentiating that formula with respect to time gives the relationship between their rates.

Let’s walk through it.

1. Write the relationship between circumference and radius. For a circle, circumference $C$ and radius $r$ are connected by

This is a direct proportionality — double the radius, double the circumference.

2. Differentiate both sides with respect to time $t$. Since $r$ changes with time, $C$ also changes with time. Differentiate implicitly:

Notice that $2\pi$ is a constant, so the rate of change of circumference is simply $2\pi$ times the rate of change of the radius.

3. Plug in the given rate. You’re told $\frac{dr}{dt} = 0.7 \text{ cm/s}$. Substitute:

4. Interpret the result. …

Method: Related Rates for a Linear Relation

When the connecting formula is linear in the changing variable (rather than a power like $r^2$ or $r^3$), the related rate turns out to be a constant multiple of the given rate — independent of the current value of the variable.

Steps

Step 1: Write the formula connecting the two quantities

For a circle, circumference and radius are linked by $C=2\pi r$, a direct proportionality.

Step 2: Differentiate both sides with respect to time

Since the relation is linear, differentiating just carries the constant through:

Step 3: Substitute the given rate …

Common Mistakes

Mistake 1: Confusing the circumference formula with the area formula

Using $A=\pi r^2$ (which needs both $r$ and $\frac{dr}{dt}$, and depends on the current radius) instead of $C=2\pi r$ (which needs only $\frac{dr}{dt}$ and never depends on the current radius) gives a completely different, radius-dependent answer where a constant one was expected.

Mistake 2: Assuming the answer must depend on a given radius value …