Question

what is the radius of a circle if an arc of 90° has a length of 5π meters?
a. 20 meters
b. 10 meters
c. 5π meters
d. 5 meters

Explanation:

Step1: Recall the arc - length formula

The formula for the arc length \(L\) of a circle is \(L=\frac{\theta}{360^{\circ}}\times2\pi r\), where \(\theta\) is the central angle in degrees and \(r\) is the radius.
Given \(\theta = 90^{\circ}\) and \(L = 5\pi\) meters.
Substitute the values into the formula: \(5\pi=\frac{90^{\circ}}{360^{\circ}}\times2\pi r\).

Step2: Simplify the equation

First, simplify \(\frac{90^{\circ}}{360^{\circ}}=\frac{1}{4}\).
The equation becomes \(5\pi=\frac{1}{4}\times2\pi r\).
Simplify \(\frac{1}{4}\times2\pi r=\frac{\pi r}{2}\).
So, \(5\pi=\frac{\pi r}{2}\).
Multiply both sides of the equation by \(2\) to get \(10\pi=\pi r\).
Divide both sides by \(\pi\) (since \(\pi
eq0\)), we get \(r = 10\) meters.

Answer:

b. 10 meters