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Zorluk: OrtaCircle Geometry: Arc Length and Sector Area

A rotary lawn sprinkler sweeps out a sector-shaped region of a yard with a central angle of 150∘150^\circ. If the area of the irrigated sector is 60π60\pi square feet, what is the perimeter, in feet, of the irrigated lawn sector?

  1. A
    10π10\pi
  2. B
    10π+1210\pi + 12
  3. 10π+2410\pi + 24Cevap
  4. D
    5π+245\pi + 24
  5. E
    20π+2420\pi + 24

Cevap

The perimeter of the irrigated lawn sector is 10π+2410\pi + 24 feet.
The area of a sector with central angle 150∘150^\circ is 150∘360∘×πr2=512πr2\frac{150^\circ}{360^\circ} \times \pi r^2 = \frac{5}{12}\pi r^2. Setting this equal to 60π60\pi yields r2=144r^2 = 144, so the radius r=12r = 12 feet. The arc length is 512×2π(12)=10π\frac{5}{12} \times 2\pi(12) = 10\pi feet. The perimeter of a sector is the arc length plus two radii (2r2r), giving 10π+2(12)=10π+2410\pi + 2(12) = 10\pi + 24 feet.

Adım Adım Çözüm

1
Find the radius of the circle using the sector area formula.
r=12r = 12 feet
The formula for sector area is Area=θ360∘⋅πr2\text{Area} = \frac{\theta}{360^\circ} \cdot \pi r^2. Substituting θ=150∘\theta = 150^\circ and Area=60π\text{Area} = 60\pi gives 150360⋅πr2=60π\frac{150}{360} \cdot \pi r^2 = 60\pi, which simplifies to 512r2=60\frac{5}{12} r^2 = 60, so r2=144r^2 = 144 and r=12r = 12.
2
Calculate the arc length of the sector.
Arc length =10π= 10\pi feet
The formula for arc length is s=θ360∘⋅2πrs = \frac{\theta}{360^\circ} \cdot 2\pi r. Substituting θ=150∘\theta = 150^\circ and r=12r = 12 yields s=512⋅2π(12)=10πs = \frac{5}{12} \cdot 2\pi (12) = 10\pi.
3
Calculate the total perimeter of the sector.
Perimeter =10π+24= 10\pi + 24 feet
The perimeter of a sector consists of the curved arc length plus two straight radial edges: Perimeter=s+2r=10π+2(12)=10π+24\text{Perimeter} = s + 2r = 10\pi + 2(12) = 10\pi + 24.

Anahtar Kavram

Perimeter of a Circle Sector
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Konular

Circle Geometry: Arc Length and Sector Area

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