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Chapter 5: Problem 1

A carnival swing is fixed on the end of an 8.0 -m-long beam. If the swing and beam sweep through an angle of \(120^{\circ},\) what is the distance through which the riders move?

Short Answer

Expert verified
Answer: The distance the riders move through on the swing is \(\frac{16}{3}\pi\) meters.

Step by step solution

01

Convert the angle to radians

Since the given angle is in degrees, we need to convert it to radians to use it in the arc length formula. To convert degrees to radians, we can use the formula: $$Radians = \frac{Degrees}{180} \cdot \pi$$ Convert 120 degrees to radians: $$Radians = \frac{120}{180} \cdot \pi = \frac{2}{3}\pi$$
02

Calculate the arc length

Now that we have the angle in radians, we can find the arc length using the formula: $$Arc\,Length = Radius \cdot Angle$$ In our problem, the length of the beam is the radius (8 meters) and the angle is \(\frac{2}{3}\pi\) radians. Plug in the values and calculate the arc length: $$Arc\,Length = 8 \cdot \frac{2}{3}\pi = \frac{16}{3}\pi$$
03

Determine the distance

Since we have the arc length, which represents the distance the riders move through, we can write the answer: The distance through which the riders move is \(\frac{16}{3}\pi\) meters.

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