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A circular sector of perimeter $60 \mathrm{~m}$ with maximum area is to be constructed, The radius of the circular arc in metre must be
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Verified Answer
The correct answer is:
15
Perimeter of sector $=2 r+r \theta$
$\Rightarrow \quad 60=2 r+r \theta$ (given)
$\Rightarrow \quad \theta=\frac{60-2 r}{r}$
Now, area of sector,
$(A)=\frac{\pi r^{2} \theta}{360^{\circ}}=\frac{\pi r^{2}(60-2 r)}{360 r}$
$\begin{aligned} &=\frac{\pi r}{180}(30-r) \\ \frac{d A}{d r} &=\frac{\pi}{180}(30-2 r) \end{aligned}$
For maximum area $\frac{d A}{d r}=0$
$\begin{aligned}
&\Rightarrow 30-2 r=0 \\
&\Rightarrow 2 r=30 \Rightarrow r=15
\end{aligned}$
and $\frac{d^{2} A}{d r^{2}}=\frac{\pi}{180^{\circ}}(-2)=-\frac{\pi}{90} < 0$
Hence, it is maximum at $r=15 \mathrm{~m}$
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