The radius of the circle is found to be 3 by setting a proportion between the arc length and the angle of the sector. The area of sector LMN is then calculated as 1/12th of the area of a full circle. The correct answer is 9π, option A.
To find the area of sector LMN, we must first determine the radius of the circle. We have been given that the arc length MN is 6π and that the angle MLN is 30 degrees.
From the information provided, it is understood that the whole circumference of the circle would be associated with a 360-degree angle. Therefore, we can set up a proportion:
The arc length for a full circle is the circumference, which is 2πr.
The arc length MN (6π) corresponds to the 30-degree angle.
So, we can say that if 2πr (full circumference) corresponds to 360 degrees, then 6π (arc length MN) corresponds to 30 degrees.
Now we can solve for the radius (r):
(6π) / (30 degrees) = (2πr) / (360 degrees)
Which simplifies to:
r = 3
With the radius found, we can now calculate the area of the sector.
The area of a circle is πr², and since the sector is 1/12th of the full circle (30/360), we can find the area of sector LMN by multiplying the area of the full circle by 1/12th:
(1/12) × (π × 3²) = (1/12) × (9π) = ¾π
The area of sector LMN is ¾π, which corresponds to option A: 9π.