Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq.m) of the flowerbed is

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Aicharya · Accepted Answer

Total length of wire $=r+r+r 	heta$ $\begin{aligned} & \Rightarrow 20=2 \mathrm{r}+\mathrm{r} 	heta \ & \Rightarrow 	heta=\frac{20-2 \mathrm{r}}{\mathrm{r}} \end{aligned}$ $\begin{aligned} & A=\frac{1}{2} r^2 	heta \ & =\frac{1}{2} r^2\left(\frac{20-2 r}{r}ight)=10 r-r^2 \ 	herefore \quad & \frac{d A}{d r}=10-2 r \end{aligned}$ For maximum area, $\frac{\mathrm{dA}}{\mathrm{dr}}=0$ $\begin{aligned} & \Rightarrow 10-2 \mathrm{r}=0 \ & \Rightarrow \mathrm{r}=5 \ & \frac{\mathrm{~d}^2 \mathrm{~A}}{\mathrm{dr}^2}=-2 \lt 0 \end{aligned}$ Area is maximum at $\mathrm{r}=5$ $\begin{aligned} 	herefore \quad 	ext { Maximum area } & =10(5)-5^2 \ & =50-25=25 \mathrm{sq} . \mathrm{m} \end{aligned}$

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the…

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