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Zorluk: ZorRadians and Degrees

In the xyxy-plane, a particle moves along a circular path centered at the origin. The particle starts at the point (3,0)(3, 0) and travels counterclockwise along the circle. If the particle stops after traveling an arc length of 5π5\pi units, what is the measure of the central angle subtended by the particle's path, in degrees?

  1. A
    150∘150^\circ
  2. B
    225∘225^\circ
  3. 300∘300^\circCevap
  4. D
    600∘600^\circ

Cevap

300∘300^\circ
The radius of the circular path is 33 units since it starts at (3,0)(3,0) and is centered at the origin. Using the arc length formula s=rθs = r\theta where s=5πs = 5\pi is the arc length and r=3r = 3 is the radius, the central angle is θ=5π3\theta = \frac{5\pi}{3} radians. To convert this angle to degrees, we multiply by 180∘π\frac{180^\circ}{\pi}, which yields 5π3×180∘π=300∘\frac{5\pi}{3} \times \frac{180^\circ}{\pi} = 300^\circ.

Adım Adım Çözüm

1
Determine the radius of the circular path.
r=3r = 3
Since the path is centered at the origin (0,0)(0, 0) and starts at (3,0)(3, 0), the distance from the center to the path (the radius) is 33 units.
2
Calculate the central angle in radians using the arc length formula.
θ=5π3\theta = \frac{5\pi}{3} radians
The relationship between arc length ss, radius rr, and central angle θ\theta in radians is given by s=rθs = r\theta. Substituting s=5πs = 5\pi and r=3r = 3 gives 5π=3θ5\pi = 3\theta, which simplifies to θ=5π3\theta = \frac{5\pi}{3}.
3
Convert the central angle from radians to degrees.
300∘300^\circ
To convert radians to degrees, multiply the angle in radians by 180∘π\frac{180^\circ}{\pi}. This gives θ=5π3×180∘π=5×60∘=300∘\theta = \frac{5\pi}{3} \times \frac{180^\circ}{\pi} = 5 \times 60^\circ = 300^\circ.

Anahtar Kavram

Using the arc length formula s=rθs = r\theta to find an angle in radians and converting that angle to degrees.
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Konular

Radians and Degrees

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