A circular sector of perimeter 60 meter with maximum area is to be constructed. The radius of the circular…
A circular sector of perimeter 60 meter with maximum area is to be constructed. The radius of the circular are in meter must be
$5 \mathrm{~m}$
$15 \mathrm{~m}$
$10 \mathrm{~m}$
$20 \mathrm{~m}$
Solution
Perimeter $=2 r+r \theta=60$
$\Rightarrow \theta=\frac{60-2 r}{r}$
Area $=\mathrm{A}(\mathrm{r})=\frac{\pi \mathrm{r}^2 \theta}{360}=\frac{\pi \mathrm{r}^2\left(\frac{60-2 \mathrm{r}}{\mathrm{r}}\right)}{360}=\frac{\pi}{360}\left(60 r-2 \mathrm{r}^2\right)$
$A^{\prime}(r)=\frac{\pi}{360}(60-4 r)$
For area to be maximum $A^{\prime}(r)=0 \Rightarrow r=15$