If 20 m of wire is available for fencing off a flower-bed

Question: 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

(a) 12.5 (b) 10

(c) 25 (d) 30

Solution:- The correct option is (C)

Let the radius of circular sector = r

Angle made by sector in centre $= \theta$

Total length $= 2r+r\theta=20$

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

Now area of flower bed $(A) = \dfrac{1}{2}r^2\theta$

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

$\Rightarrow A =10r-r^2$

differentiate with respect to r

$\frac{dA}{dr}=10-2r$

For max and minima, $\frac{dA}{dr}=0$

$\Rightarrow 10-2r =0$

$\Rightarrow r = 5$

Again differerntiate with respect to r

$\frac{d^2A}{dr^2}=-2<0$

Hence, the area is max when $r=5$

Max area $A_{max}=\frac{1}{2}(5)^2\left[\frac{20-2(5)}{5}\right]$

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

The correct option is (C)

Prove that the volume of the largest cone that can be