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Zorluk: OrtaCircle Geometry: Arc Length and Sector Area

A circular stage turntable has a radius of 99 meters. A wedge-shaped performance zone on the turntable is formed by a sector with a central angle measuring 100∘100^\circ. A safety border is installed along the entire boundary of this performance zone, which consists of its curved arc and both straight radius edges. What is the total length, in meters, of the safety border?

  1. A
    5π5\pi
  2. B
    5π+95\pi + 9
  3. 5π+185\pi + 18Cevap
  4. D
    2.5π+182.5\pi + 18
  5. E
    22.5π22.5\pi

Cevap

The total length of the safety border is 5π+185\pi + 18 meters.
To find the total length of the border surrounding the sector, calculate the arc length of the sector using s=θ360∘×2πr=100360×18π=5πs = \frac{\theta}{360^\circ} \times 2\pi r = \frac{100}{360} \times 18\pi = 5\pi meters. Then, add the two straight radial sides (2r=2×9=182r = 2 \times 9 = 18 meters) to get the complete perimeter of 5π+185\pi + 18 meters.

Adım Adım Çözüm

1
Calculate the arc length (ss) of the circular sector.
s=100∘360∘×2π(9)=518×18π=5πs = \frac{100^\circ}{360^\circ} \times 2\pi(9) = \frac{5}{18} \times 18\pi = 5\pi meters.
The arc length formula for a sector measured in degrees is s=θ360∘×2πrs = \frac{\theta}{360^\circ} \times 2\pi r.
2
Calculate the total perimeter (PP) of the sector.
P=s+2r=5π+2(9)=5π+18P = s + 2r = 5\pi + 2(9) = 5\pi + 18 meters.
The boundary of a sector includes the curved arc length plus two straight radial sides.

Anahtar Kavram

Perimeter of a Circle Sector
Tahmini Süre:1m 0s
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Konular

Circle Geometry: Arc Length and Sector Area

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