Question

what is the arc length of an arc with a central angle of \frac{11}{6}\pi radians in a circle with a radius of 14 cm? enter your answer to the nearest whole centimeter in the box. cm

Explanation:

Step1: Recall the arc - length formula

The formula for the arc length \(s\) of a circle is \(s = r\theta\), where \(r\) is the radius and \(\theta\) is the central angle in radians.

Step2: Substitute the given values into the formula

Given \(r = 14\) cm and \(\theta=\frac{11}{6}\pi\) radians. Then \(s=14\times\frac{11}{6}\pi\).
First, calculate \(14\times\frac{11}{6}=\frac{14\times11}{6}=\frac{154}{6}=\frac{77}{3}\). So \(s = \frac{77}{3}\pi\).

Step3: Approximate the value

Using \(\pi\approx3.14\), we have \(s=\frac{77}{3}\times3.14\). \(\frac{77}{3}\times3.14=\frac{77\times3.14}{3}=\frac{241.78}{3}\approx80.59\).
Or \(s = 14\times\frac{11}{6}\times3.14=\frac{14\times11\times3.14}{6}=\frac{483.56}{6}\approx80.59\).
Rounding to the nearest whole number, \(s\approx81\).

Answer:

\(81\)