Question

Twenty meter wire is available to fence off a flower bed in the form of a sector. If the flower bed has the maximum surface then radius is

• A. $10$
• B. $\frac{5}{2}$
• C. $5$
• D. $\frac{15}{2}$

Answer 1

The correct option is C $5$

Let the radius of the sector of the flower bed be $r$ and the angle subtended be $\theta$

$\therefore S=$ Surface Area of Sector

$\Rightarrow S=\frac{\theta }{360}\pi r^2$

Also, $\theta =\frac{LengthofArea}{Radius}=\frac{20-2r}{r}$

$\therefore S=\frac{\theta }{2\pi }\left(\pi r^2\right)=\frac{20-2r}{2r}\Rightarrow S=(10-r)r$

Now, $\frac{dS}{dr}=(10-r)+r(-1)$

For surface area to be maximum

$\frac{dS}{dr}=0\Rightarrow r=5.$