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Difficulty: MediumCircle Geometry: Arc Length and Sector Area

A coastal lighthouse beacon rotates through a central angle of 135∘135^\circ, sweeping across a circular sector of sea with a radius of 12 nautical miles12\text{ nautical miles}. What is the total perimeter, in nautical miles, of the region swept by the beacon?

  1. A
    9π9\pi
  2. B
    9π+129\pi + 12
  3. 9π+249\pi + 24Answer
  4. D
    54π54\pi
  5. E
    54π+2454\pi + 24

Answer

The total perimeter of the swept sector is 9π+249\pi + 24 nautical miles.
The perimeter of a sector consists of the curved arc length plus the lengths of the two straight radii bounding the sector. The arc length is calculated as 135360×2π(12)=38×24π=9π\frac{135}{360} \times 2\pi(12) = \frac{3}{8} \times 24\pi = 9\pi. Adding the two radii of 1212 nautical miles each gives 9π+12+12=9π+249\pi + 12 + 12 = 9\pi + 24 nautical miles.

Step-by-Step Solution

1
Calculate the arc length of the sector
Arc length = 135∘360∘×2π(12)=38×24π=9π\frac{135^\circ}{360^\circ} \times 2\pi(12) = \frac{3}{8} \times 24\pi = 9\pi nautical miles.
The curved outer boundary of a circular sector is determined by the fraction of the total circumference defined by the central angle.
2
Determine the total perimeter of the sector
Perimeter = Arc length+2r=9π+2(12)=9π+24\text{Arc length} + 2r = 9\pi + 2(12) = 9\pi + 24 nautical miles.
The total boundary of a sector consists of its outer curved arc plus its two straight radial line segments.

Key Concept

Perimeter of a Circular Sector
Estimated Time:1m 30s
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Topics

Circle Geometry: Arc Length and Sector Area

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