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Difficulty: MediumCircles, Arc Lengths, and Sector Areas

Two concentric circles centered at point OO have radii of 66 units and 1010 units. A central angle θ\theta defines a sector that intersects the region between the concentric circles, creating a region bounded by an outer arc, an inner arc, and two straight line segments. If the area of this bounded region is 16π16\pi square units, what is the perimeter of the region?

  1. A
    8π8\pi
  2. B
    4+8π4 + 8\pi
  3. 8+8π8 + 8\piAnswer
  4. D
    8+16π8 + 16\pi
  5. E
    20+8π20 + 8\pi

Answer

The perimeter of the region is 8+8π8 + 8\pi units.
The region between the concentric circles has an area equal to the fraction of the central angle times the area of the ring: θ360∘⋅π(102−62)=64π(θ360∘)\frac{\theta}{360^\circ} \cdot \pi (10^2 - 6^2) = 64\pi \left(\frac{\theta}{360^\circ}\right). Setting this equal to 16π16\pi gives θ360∘=14\frac{\theta}{360^\circ} = \frac{1}{4}. The outer arc length is 14(2π⋅10)=5π\frac{1}{4}(2\pi \cdot 10) = 5\pi, and the inner arc length is 14(2π⋅6)=3π\frac{1}{4}(2\pi \cdot 6) = 3\pi. The region is bounded on the sides by two segments of length 10−6=410 - 6 = 4 units each. The total perimeter is 5π+3π+4+4=8+8π5\pi + 3\pi + 4 + 4 = 8 + 8\pi.

Step-by-Step Solution

1
Express the area of the bounded region in terms of the central angle fraction f=θ360∘f = \frac{\theta}{360^\circ}.
Area=f⋅π(R2−r2)=f⋅π(102−62)=64πf\text{Area} = f \cdot \pi(R^2 - r^2) = f \cdot \pi(10^2 - 6^2) = 64\pi f.
The area of the region between two concentric sector arcs is the difference between the outer sector area and the inner sector area.
2
Solve for the fraction ff using the given area of 16π16\pi.
64πf=16π  ⟹  f=16π64π=1464\pi f = 16\pi \implies f = \frac{16\pi}{64\pi} = \frac{1}{4}.
Equating the algebraic expression for the region's area to 16π16\pi allows finding the proportion of the circle represented by the central angle.
3
Calculate the lengths of the outer arc, inner arc, and straight boundary segments.
Outer Arc=14(2π⋅10)=5π\text{Outer Arc} = \frac{1}{4}(2\pi \cdot 10) = 5\pi, Inner Arc=14(2π⋅6)=3π\text{Inner Arc} = \frac{1}{4}(2\pi \cdot 6) = 3\pi, Segment Length=10−6=4\text{Segment Length} = 10 - 6 = 4.
Arc lengths use the formula f⋅2πrf \cdot 2\pi r, and each straight boundary segment is the radial distance between the two circles.
4
Sum all boundary components to find the total perimeter.
Perimeter=5π+3π+4+4=8+8π\text{Perimeter} = 5\pi + 3\pi + 4 + 4 = 8 + 8\pi.
The complete perimeter of the bounded region consists of the outer arc, inner arc, and two radial segments.

Key Concept

Annular Sector Area and Perimeter
Estimated Time:1m 30s
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Topics

Circles, Arc Lengths, and Sector Areas

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