Question

1. determine the length of a 65° arc on a circle with a 4.2 - inch radius

Explanation:

Step1: Write the arc - length formula

The formula for the length of an arc \(s\) is \(s = r\theta\), where \(r\) is the radius of the circle and \(\theta\) is the central angle in radians. First, convert the angle from degrees to radians. We know that \(\theta\) (in radians) \(=\frac{\pi}{180}\times\theta\) (in degrees). Given \(\theta = 65^{\circ}\), then \(\theta=\frac{65\pi}{180}=\frac{13\pi}{36}\) radians.

Step2: Substitute the values of \(r\) and \(\theta\) into the arc - length formula

Given \(r = 4.2\) inches and \(\theta=\frac{13\pi}{36}\) radians. Substitute into \(s=r\theta\), we get \(s = 4.2\times\frac{13\pi}{36}\).

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Using \(\pi\approx3.14\), we have \(s=\frac{54.6\times3.14}{36}\)

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Answer:

The length of the arc is approximately \(4.8\) inches.