Discussion Forum

If 20 m 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 

Answer:

Explanation:

   Total length = 2r+rθ =20

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

 Now, area of flower-bed,

          $A= \frac{1}{2}r^{2}\theta \Rightarrow A =\frac{1}{2}r^{2}(\frac{20-2r)}{r}$

  $\Rightarrow A=10r-r^{2}$

  $\therefore  \frac{\text{d}A}{\text{d}r}=10-2r$

For maxima or minima , put  $\therefore  \frac{\text{d}A}{\text{d}r}=0$

$\therefore 10-2r=0\Rightarrow r=5$

  $\therefore  A_{max}= \frac{1}{2}(5)^{2}[\frac{20-2(5)}{5}]$

               $= \frac{1}{2}\times25\times2=25 sq.m$