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Twenty metres of wire is available to fence off a flower bed in the form of a circular sector. What must the radius of the circle be, if the area of the flower bed be greatest?
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$5 \mathrm{~m}$
Let, radius $=r$, arc length $=\ell$
$2 \mathrm{r}+\ell=20 \Rightarrow \ell=20-2 \mathrm{r}$
$\Rightarrow \mathrm{A}=\frac{1}{2}\mathrm{r}^{2} \theta = \frac{1}{2} \ell \mathrm{r} \Rightarrow \frac{\mathrm{dA}}{\mathrm{dr}}=\frac{1}{2}(20-4 \mathrm{r})=0 \Rightarrow \mathrm{r}=5 \quad \therefore \mathrm{r}=5 \mathrm{~m}\right.$
$2 \mathrm{r}+\ell=20 \Rightarrow \ell=20-2 \mathrm{r}$
$\Rightarrow \mathrm{A}=\frac{1}{2}\mathrm{r}^{2} \theta = \frac{1}{2} \ell \mathrm{r} \Rightarrow \frac{\mathrm{dA}}{\mathrm{dr}}=\frac{1}{2}(20-4 \mathrm{r})=0 \Rightarrow \mathrm{r}=5 \quad \therefore \mathrm{r}=5 \mathrm{~m}\right.$
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