Question

A wire of length 20 m is available to fence off a flower bed in the form of a sector of a circle. What must be the radius of the circle, if we wish to have a flower bed with the greatest possible area?

Answer 1

Let OAB be a given sector of a circle of radius r with are AB = l m and ∠AOB = θ radians.

Then,

2r + l = 20 m ....(1)

\(\frac{l}{r}=\theta\) ....(2)

Area \(\frac{1}{2}r^ 2\theta\) ....(3)

[Using (1), (2), (3)]

For maximum or minimum area 

\(\frac{dA}{dr}=0 \) ⇒ 10 -2r = 0

r = 5

\(\frac{d^2A}{dr^2}=-2\) (negative) at r = 5

∴ A is maximum at r = 5m