Two connected gears, Gear A and Gear B, rotate together such that the belt connecting them… — SAT

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 $15$ centimeters and the radius of Gear B is $9$ centimeters. If Gear A rotates through a central angle of $\frac{4\pi}{15}$ radians, Gear B rotates through a central angle of $x$ degrees. What is the value of $x$ ?

Answer: 80

Answer

The arc length

s

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 \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

\theta = \frac{4\pi}{9}

radians. To convert this angle to degrees, multiply by

\frac{180}{\pi}

to get

\frac{4\pi}{9} \times \frac{180}{\pi} = 80

degrees.

Step-by-Step Solution

Calculate the arc length of the rotation for Gear A using the formula

s = r\theta

s = 4\pi

centimeters

To find the distance a point on the belt travels, which is shared by both gears.

Determine the rotation angle of Gear B in radians using the arc length and Gear B's radius.

\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.

Convert the angle of Gear B from radians to degrees by multiplying by

\frac{180}{\pi}

x = 80

To find the measure of the angle in degrees as requested by the question.

Key Concept

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

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