Question

The length of the perimeter of a sector of a circle is 24 cm, the maximum area of the sector is ______.

पर्याय

• 36 cm²

• 20 cm²

• 40 cm²

• 25 cm²

Answer 1

The length of the perimeter of a sector of a circle is 24 cm, the maximum area of the sector is 36 cm².

Explanation:

Let OAB be a given sector of a circle of a radius r cm such that arc AB = l cm, and ∠ AOB = e radians.

Then

2r + l = 24    ...(i)

`l/"r" = theta`      ....(ii)

A = `1/2 "r"^2 theta`     ...(iii)

From (i), (ii) ,(iii), we get

A = `1/2 "r"^2 xx l/"r" = 1/2 "r" l = 1/2 "r" (24 - 2"r")`

⇒ A = 12r - r²

Now, `"dA"/"dr" = 12 - 2"r" = 0 => "r" = 6`

∴ `("d"^2"A")/"dr"^2 = - 2 < 0`

∴ A is maximum at r = 6

Hence, the maximum area

= 12 × 6 - 36 = 36 cm²     ...[From (iv)]