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Difficulty: MediumCircle Geometry: Arc Length and Sector Area

A robotic arm mounted on a flat work surface pivots through a central angle of θ\theta degrees, sweeping out a sector of a circle with a radius of 1515 centimeters. If the total perimeter of the swept sector is (30+5π)(30 + 5\pi) centimeters, what is the value of θ\theta?

  1. A
    8∘8^\circ
  2. B
    30∘30^\circ
  3. C
    45∘45^\circ
  4. 60∘60^\circAnswer
  5. E
    120∘120^\circ

Answer

The measure of the central angle θ\theta is 60∘60^\circ.
The perimeter of a sector consists of two straight radii of length rr plus the arc length ss, so Perimeter=2r+s\text{Perimeter} = 2r + s. Given r=15r = 15 cm and a perimeter of (30+5π)(30 + 5\pi) cm, we find 2(15)+s=30+5π2(15) + s = 30 + 5\pi, which means s=5πs = 5\pi cm. Using the arc length formula s=θ360∘×2πrs = \frac{\theta}{360^\circ} \times 2\pi r, we set 5π=θ360∘×30π5\pi = \frac{\theta}{360^\circ} \times 30\pi. Simplifying yields θ12∘=5\frac{\theta}{12^\circ} = 5, giving θ=60∘\theta = 60^\circ.

Step-by-Step Solution

1
Express the perimeter of a sector in terms of radius rr and arc length ss.
Perimeter=2r+s\text{Perimeter} = 2r + s
A sector's boundary consists of two straight radii and one curved arc.
2
Substitute r=15r = 15 cm and the given perimeter (30+5π)(30 + 5\pi) cm to solve for the arc length ss.
30+s=30+5π  ⟹  s=5π cm30 + s = 30 + 5\pi \implies s = 5\pi \text{ cm}
Subtracting the combined length of the two straight edges (2×15=302 \times 15 = 30) isolates the arc length.
3
Set up the arc length formula using degrees: s=θ360∘×2πrs = \frac{\theta}{360^\circ} \times 2\pi r.
5π=θ360∘×2π(15)=30πθ360∘=πθ12∘5\pi = \frac{\theta}{360^\circ} \times 2\pi(15) = \frac{30\pi \theta}{360^\circ} = \frac{\pi \theta}{12^\circ}
The arc length represents the fraction θ360∘\frac{\theta}{360^\circ} of the circle's total circumference 2πr2\pi r.
4
Solve the equation for θ\theta.
5π=πθ12∘  ⟹  5=θ12∘  ⟹  θ=60∘5\pi = \frac{\pi \theta}{12^\circ} \implies 5 = \frac{\theta}{12^\circ} \implies \theta = 60^\circ
Dividing both sides by π\pi and multiplying by 12∘12^\circ isolates θ\theta.

Key Concept

Perimeter of a Sector and Arc Length
Estimated Time:1m 15s
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Topics

Circle Geometry: Arc Length and Sector Area

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