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

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

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)

 

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