Question

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 flower-bed, is:

• A. $25$
• B. $12.5$
• C. $30$
• D. $10$

Answer 1

The correct option is A $25$

Area $=\left(\frac{r^2\theta }{2}\right)$
and $20=2r+r\theta \cdots \left(i\right)$
So, Area $=f\left(r\right)=\frac{r^2}{2}\left(\frac{20-2r}{r}\right)=10r-r^2$
$\Rightarrow f^{\prime }\left(r\right)=10-2r$
For area $f\left(r\right)$ to be maximum $f^{\prime }\left(r\right)=0$
$\Rightarrow r=5$
$f^{\prime \prime }\left(r\right)=-2<0$
$\Rightarrow$Area is maximum for $r=5$
$\Rightarrow \theta =2$ radians ,from $\left(i\right)$
$\Rightarrow$Area of circular sector$=\frac{r^2\theta }{2}=25$sq. m