MHT CET · Maths · Application of Derivatives
A stone is dropped in a quiet lake and it is observed that waves move in circles, If the radius of a circular wave increases at the rate \(2 \mathrm{~cm} / \mathrm{sec}\), then the rate of increase in its area at the instant when its radius is \(10 \mathrm{~cm}\), is \(\mathrm{cm}^2 / \mathrm{sec}\).
- A \(40 \pi\)
- B \(80 \pi\)
- C \(10 \pi\)
- D \(20 \pi\)
Answer & Solution
Correct Answer
(A) \(40 \pi\)
Step-by-step Solution
Detailed explanation
Given \(\frac{d r}{d t}=2 \mathrm{~cm} / \mathrm{sec} r=10 \mathrm{~cm}\)
We have \(A=\pi r^2\)
Diff. w.r.t. \(\mathrm{t}\)
\(\begin{aligned} & \frac{d A}{d t}=2 \pi r \frac{d r}{d t} \\ & =2 \pi \times 10 \times 2 \\ & =40 \pi\end{aligned}\)
We have \(A=\pi r^2\)
Diff. w.r.t. \(\mathrm{t}\)
\(\begin{aligned} & \frac{d A}{d t}=2 \pi r \frac{d r}{d t} \\ & =2 \pi \times 10 \times 2 \\ & =40 \pi\end{aligned}\)
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