If the area of a circular sector of perimeter $60 \mathrm{~m}$ is to be maximized, then its radius must be......... m

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Aicharya · Accepted Answer

Given, the perimeter of circular sector $=60 \mathrm{~m}$ Let radius be $r \mathrm{~m}$. Length of $\operatorname{arc} \mathbf{A B}=l$

$\begin{aligned} \therefore \text { perimeter }(p) & =l+2 r \\ l & =p-2 r\end{aligned}$ $\begin{aligned}& \because \text { Area }=\frac{1}{2} l r \\& \Rightarrow A=\frac{(p-2 r) r}{2}=\frac{p r-2 r^2}{2}\end{aligned}$ For $A$ to be maximum, $\frac{d A}{d r}=0$ $\frac{d}{d r}\left(\frac{p r-2 r^2}{2}\right)=0$ $\begin{array}{rlrl}\Rightarrow & & \frac{p-4 r}{2} & =0 \\ \Rightarrow & p & =4 r\end{array}$ $60=4 r$ $\Rightarrow \quad r=15 \mathrm{~m}$

If the area of a circular sector of perimeter $60 \mathrm{~m}$ is to be maximized, then its radius must be..…

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