Q101 / 170 · Manipal (MET) 2018 · Mathematics
If the rate of change in the circumference of a circle of $0.3 \mathrm{~cm} / \mathrm{s}$, then the rate of change in the area of the circle when the radius is 5 cm , is
- Correct answerA$1.5 \mathrm{sq} \mathrm{cm} / \mathrm{s}$
- B$0.5 \mathrm{sq} \mathrm{cm} / \mathrm{s}$
- C$5 \mathrm{sq} \mathrm{cm} / \mathrm{s}$
- D$3 \mathrm{sq} \mathrm{cm} / \mathrm{s}$
Solution
Circumference of circle, $C=2 \pi r$
$\begin{aligned}
& \Rightarrow \quad \frac{d C}{d t}=2 \pi \frac{d r}{d t} \\
& \Rightarrow \quad \frac{0.3}{2 \pi}=\frac{d r}{d t}
\end{aligned} \quad\left(\because \frac{d C}{d t}=0.3 \mathrm{~cm} / \mathrm{s} \text { given }\right) \$ 1$<br/>Now, $\quad A=\pi r^2$
$\begin{array}{cc}
& \frac{d A}{d t}=2 \pi r \frac{d r}{d t} \Rightarrow \frac{d A}{d t}=r \times 0.3 \\
\Rightarrow & {\left[\frac{d A}{d t}\right]_{r=5}=5 \times 0.3=1.5 \mathrm{sq} \mathrm{~cm} / \mathrm{s}}
\end{array}$
Hence, the correct answer is option: 'A'.