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Zorluk: OrtaCircles

A searchlight beam sweeps horizontally across a central angle of 72∘72^\circ. If the maximum effective range of the searchlight is 20 meters20 \text{ meters}, what is the length of the arc formed by the outer edge of the beam? (Leave your answer in terms of π\pi).

  1. A
    4π meters4\pi \text{ meters}
  2. 8π meters8\pi \text{ meters}Cevap
  3. C
    40π meters40\pi \text{ meters}
  4. D
    80π meters80\pi \text{ meters}

Cevap

8π meters8\pi \text{ meters}
The length of an arc is calculated by multiplying the full circumference of the circle (2πr2\pi r) by the fraction of the circle represented by the central angle (θ360∘\frac{\theta}{360^\circ}). In this scenario, the full circumference is 2π(20)=40π meters2\pi(20) = 40\pi \text{ meters}. The fraction representing the 72∘72^\circ sweep is 72360\frac{72}{360}, which simplifies to 15\frac{1}{5}. Multiplying the full circumference by this fraction gives 15×40π=8π meters\frac{1}{5} \times 40\pi = 8\pi \text{ meters}.

Adım Adım Çözüm

1
Identify the formula for the arc length of a circle.
Arc Length=(θ360∘)×2πr\text{Arc Length} = (\frac{\theta}{360^\circ}) \times 2\pi r
The arc length represents a specific fraction of the entire circumference based on the central angle.
2
Determine the fraction of the circle covered by the searchlight beam.
72∘360∘=15\frac{72^\circ}{360^\circ} = \frac{1}{5}
Simplifying the angle fraction makes the subsequent multiplication easier.
3
Calculate the full circumference of the circle using the given radius (range).
C=2π(20)=40π metersC = 2\pi(20) = 40\pi \text{ meters}
The full circumference is necessary to find the length of the fractional arc.
4
Multiply the full circumference by the angle fraction.
15×40π=8π meters\frac{1}{5} \times 40\pi = 8\pi \text{ meters}
This calculation applies the proportional fraction to the total boundary of the circle, yielding the arc length.

Anahtar Kavram

Arc Length of a Circle
Tahmini Süre:1m 0s
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Circles

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