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) 10 

(B) 25 

(C) 30 

(D) 12.5

Answer 1

Answer is (B) 25

It is given that r + r + rθ = 20 meters.

Therefore,

Now, the area is

1/2(r²θ) = 1/2r²((20 - 2r)/r)

That is

z = 1/2(20r - 2r²)

Differentiating w.r.t. r, we get

dz/dr = 1/2(20 - 4r) = 0

⇒ r = 5

At r = 5, we get θ = 2; therefore, d²z/dt² < 0 (hence, it is maxima).

Therefore, the maximum area is

z = 1/2(r²θ) = 1/2 x 5² x 2 = 25m²