Question

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a circle has an arc of length 48π that is intercepted by a central angle of 120°.
what is the radius of the circle?
enter your answer in the box
units

Explanation:

Step1: Write the arc - length formula

The formula for the length of an arc \(s\) is \(s=\frac{\theta}{360^{\circ}}\times2\pi r\), where \(\theta\) is the central angle in degrees and \(r\) is the radius of the circle.
Given \(s = 48\pi\) and \(\theta=120^{\circ}\).
Substitute the values into the formula: \(48\pi=\frac{120^{\circ}}{360^{\circ}}\times2\pi r\).

Step2: Simplify the equation

Simplify \(\frac{120^{\circ}}{360^{\circ}}=\frac{1}{3}\).
The equation becomes \(48\pi=\frac{1}{3}\times2\pi r\).
Multiply both sides by \(3\) to get \(144\pi = 2\pi r\).

Step3: Solve for \(r\)

Divide both sides of the equation \(144\pi = 2\pi r\) by \(2\pi\).
\(r=\frac{144\pi}{2\pi}\).
Cancel out \(\pi\) (since \(\pi
eq0\)), and \(r = 72\).

Answer:

\(72\)