A rotating beacon rotates at a constant rate of 160^\circ per second. Through how many rad… — SAT

A rotating beacon rotates at a constant rate of $160^\circ$ per second. Through how many radians does the beacon rotate in $4.5$ seconds?

The correct answer is the value of

4\pi

radians.

The total angle of rotation in degrees is found by multiplying the rate of rotation by the time elapsed:

160^\circ/\text{second} \times 4.5\text{ seconds} = 720^\circ

. To convert this angle from degrees to radians, multiply the degree measure by the conversion factor

\frac{\pi}{180^\circ}

. This yields

720 \times \frac{\pi}{180} = 4\pi

radians.

Calculate the total angle of rotation in degrees by multiplying the rate of rotation by the time.

160^\circ \times 4.5 = 720^\circ

To find the total angular displacement, we multiply the constant angular speed by the duration of the rotation.

Convert the total angle from degrees to radians by multiplying by the conversion factor

\frac{\pi}{180^\circ}

720^\circ \times \frac{\pi}{180^\circ} = 4\pi

radians

Since

180^\circ

is equivalent to

\pi

radians, multiplying by

\frac{\pi}{180^\circ}

converts the angle to radians.

Radian-degree conversion

Convert the speed of rotation to radians per second first:

160^\circ \times \frac{\pi}{180^\circ} = \frac{8\pi}{9}

radians per second. Then multiply by the time elapsed:

\frac{8\pi}{9} \times 4.5 = 4\pi

radians.

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