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: Approximation-based cross-check:
Over a very short time \(\Delta t\), \(\Delta r\approx0.05\,\Delta t\), and the corresponding change in area is approximately \(\Delta A\approx\dfrac{dA}{dr}\Delta r=2\pi r\,\Delta r\).

Step 2: Dividing by the time step:
\(\dfrac{\Delta A}{\Delta t}\approx2\pi r\cdot\dfrac{\Delta r}{\Delta t}=2\pi(3.2)(0.05)\) in the limit \(\Delta t\to0\), which is exactly the same chain-rule expression as before.

Final Answer:
Same result: \(\boxed{0.32\pi\ \text{cm}^2/\text{s}}\), confirming the direct differentiation method.