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Zorluk: ZorRadians and Degrees

Two connected gears, Gear A and Gear B, rotate together such that the belt connecting them does not slip. The radius of Gear A is 1515 centimeters and the radius of Gear B is 99 centimeters. If Gear A rotates through a central angle of 4π15\frac{4\pi}{15} radians, Gear B rotates through a central angle of xx degrees. What is the value of xx?

Cevap: 80

Cevap

80
The arc length ss that a point on the belt travels is given by the product of the radius and the angle in radians of Gear A: s=15×4π15=4πs = 15 \times \frac{4\pi}{15} = 4\pi cm. Since the belt does not slip, Gear B rotates through the same arc length. The angle of Gear B in radians is θ=4π9\theta = \frac{4\pi}{9} radians. To convert this angle to degrees, multiply by 180π\frac{180}{\pi} to get 4π9×180π=80\frac{4\pi}{9} \times \frac{180}{\pi} = 80 degrees.

Adım Adım Çözüm

1
Calculate the arc length of the rotation for Gear A using the formula s=rθs = r\theta.
s=4πs = 4\pi centimeters
To find the distance a point on the belt travels, which is shared by both gears.
2
Determine the rotation angle of Gear B in radians using the arc length and Gear B's radius.
θB=4π9\theta_B = \frac{4\pi}{9} radians
Because the belt does not slip, Gear B must rotate by the same linear arc length as Gear A.
3
Convert the angle of Gear B from radians to degrees by multiplying by 180π\frac{180}{\pi}.
x=80x = 80
To find the measure of the angle in degrees as requested by the question.

Anahtar Kavram

Converting central angles between radians and degrees in the context of arc lengths of connected circles.
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Konular

Radians and Degrees

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