Search any question & find its solution
Question:
Answered & Verified by Expert
Among all sectors of a fixed perimeter, choose the one with maximum area. Then the angle at the center of this sector (i.e., the angle between the bounding radii) is
Options:
Solution:
1416 Upvotes
Verified Answer
The correct answer is:
2
Perimeter $=2 \mathrm{r}+\mathrm{r} \theta=\mathrm{k}$ (const.) Area $=\frac{\theta}{2} \mathrm{r}^{2}=\frac{1}{2}(\mathrm{k}-2 \mathrm{r}) \cdot \mathrm{r}$ $\mathrm{A}=\frac{1}{2}\left(\mathrm{kr}-2 \mathrm{r}^{2}\right)$ $\frac{\mathrm{d} \mathrm{A}}{\mathrm{dr}}=\frac{1}{2}(\mathrm{k}-4 \mathrm{r})=0 \Rightarrow \mathrm{r}=\frac{\mathrm{k}}{4}$ $\frac{\mathrm{d}^{2} \mathrm{~A}}{\mathrm{dr}^{2}} < 0$ So, $2 \mathrm{r}+2 \theta=\mathrm{k}$
$$
\begin{array}{l}
\frac{\mathrm{k}}{2}+\frac{\mathrm{k} \theta}{4}=\mathrm{k} \\
\theta=2
\end{array}
$$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.