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

Total length of wire \(=r+r+r \theta\)
\(\begin{aligned}
& \Rightarrow 20=2 \mathrm{r}+\mathrm{r} \theta \\
& \Rightarrow \theta=\frac{20-2 \mathrm{r}}{\mathrm{r}}
\end{aligned}\)

\(\begin{aligned}
& A=\frac{1}{2} r^2 \theta \\
& =\frac{1}{2} r^2\left(\frac{20-2 r}{r}\right)=10 r-r^2 \\
\therefore \quad & \frac{d A}{d r}=10-2 r
\end{aligned}\)
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 \(\mathrm{r}=5\)
\(\begin{aligned}
\therefore \quad \text { Maximum area } & =10(5)-5^2 \\
& =50-25=25 \mathrm{sq} . \mathrm{m}
\end{aligned}\)