Question

The perimeter of a sector is p. The area of the sector is maximum when its radius is

(A) √p

(B) 1/√p

(C) p/2

(D) p/4

Answer 1

Answer is (D) p/4

See Fig.

 

The perimeter of a sector is p. Let AOB be the sector with radius r. If the angle of the sector be θ radians, then the area of the sector is

A = 1/2r²θ   (1)

Length of the arc is

s = rθ or θ = s/r

Therefore, the perimeter of the sector is

p = r + s + r = 2r + s    (2)

Substituting θ = s/r = in Eq. (1), we have

A = (1/2r²)(s/r) = (1/2)rs ⇒ s = 2A/r

Now, substituting the value of s in Eq. (2), we get

p = 2r + (2A/r) or 2A = pr - 2r²

Differentiating w.r.t. r, we get

2(dA/dr) = p - 4r

We know that for the maximum value of area is

dA/dr = 0 or p - 4r = 0 or r = p/4