Q.A circular disc of radius $3$ cm is being heated. Due to expansion, its radius increases at the rate of $0.05$ cm/s. Find the rate at which its area is increasing when radius is $3.2$ cm.

Concept: Related Rates — we connect 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 $t$:

Step 2: Given $\frac{dr}{dt} = 0.05$ cm/s and we need $\frac{dA}{dt}$ when $r = 3.2$ cm.

Step 3: Substitute: …

This is a classic Related Rates problem — a staple in calculus and a favourite in Indian board exams (CBSE, ISC, etc.). The core idea is simple: when two quantities are linked by a formula, their rates of change are also linked. Here, area $A$ depends on radius $r$, so how fast $A$ grows depends on how fast $r$ grows and on the current size of $r$.

The key is to differentiate the relationship with respect to time, not space. That’s the whole trick — treat $r$ as a function of time $r(t)$, then use the chain rule.

Let’s walk through it step by step.

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

Here $A$ is in cm² and $r$ is in cm. Both change with time, so we write $A(t)$ and $r(t)$.

2. Differentiate both sides with respect to time $t$. Using the chain rule:

This is the rate equation — it tells us how the area’s growth rate depends on the radius and its growth rate at any instant.

3. Identify the given data at the instant of interest.

   • Radius at that moment: $r = 3.2$ cm
   • Rate of increase of radius: $\frac{dr}{dt} = 0.05$ cm/s (Note: the initial radius of 3 cm is irrelevant — we only need the radius at the instant we’re evaluating.)

4. Substitute into the rate equation.

Compute stepwise:

So …

Method: Related Rates for a Simple Area–Radius Relationship

The most basic related-rates pattern, worth mastering as a template before tackling more geometrically involved versions (cones, spheres, similar triangles).

Steps

Step 1: Write the formula connecting the two quantities

Identify the fixed geometric formula linking the quantity whose rate is given and the quantity whose rate is required — here, $A = \pi r^2$.

Step 2: Differentiate both sides with respect to time

Treat both quantities as functions of $t$ and apply the chain rule:

Step 3: Identify the radius value at the specific instant asked about …

Common Mistakes

Mistake 1: Substituting the original radius ($r=3$) instead of the instant's radius ($r=3.2$)

Why it's wrong: the problem states the disc started at radius $3$ cm and is now expanding, but asks for the rate of area increase specifically when the radius is $3.2$ cm — using the starting value instead of the current one gives a numerically different, wrong answer for $\frac{dA}{dt}$. Correct approach: identify precisely which value of $r$ corresponds to the instant the question is actually asking about, and ignore earlier values given only for context.

Mistake 2: Dropping the factor of $2$ when differentiating $r^2$ …