Question

A flower bed is made in the shape of sector of a circle. $20m$ of wire is available to make a fence for the flower bed. Find the radius of the circle so that area of the flower bed is maximum.

Answer 1

Total length $=r+r=r\theta =20$

$\Rightarrow \theta =\frac{20-2r}{r}$

Area$=\frac{1}{2}{r}^2\theta =\frac{1}{2}{r}^2\left(\frac{20-2r}{r}\right)$

$A=10r-r^2$

$\Rightarrow \frac{dA}{dr}=0$

$\Rightarrow 10-2r=0$

$\Rightarrow 2r=10$ or $r=\frac{10}{2}=5$m

$\therefore A=50-25=25$m