Question

A circular disc having radius 3 cm is warmed and due to expansion its radius is increasing at the rate 0.05 cm/s. Find the increasing rate of its area when its radius is 3.2 cm.

Hint:

Differentiate \(A=\pi r^2\) w.r.t. time: \(dA/dt=2\pi r\,(dr/dt)\).

Answer 1

Step 1: Understanding the Concept:
This is a related-rates problem: relate area \(A\) to radius \(r\) via \(A=\pi r^2\), then differentiate both sides with respect to time \(t\).

Step 2: Setting up the relation:
\(A=\pi r^2\).

Step 3: Differentiating w.r.t. time:
\(\dfrac{dA}{dt}=2\pi r\dfrac{dr}{dt}\) (chain rule, since \(r\) itself depends on \(t\)).

Step 4: Substituting the given values:
At the instant in question, \(r=3.2\) cm and \(\dfrac{dr}{dt}=0.05\) cm/s. So \(\dfrac{dA}{dt}=2\pi(3.2)(0.05)=0.32\pi\).

Final Answer:
The area is increasing at \(\boxed{0.32\pi\ \text{cm}^2/\text{s}}\) (\(\approx1.005\ \text{cm}^2/\text{s}\)).