Question

the formula for the length of an arc is ( s = r\theta ). ( mangle qsr = 150^{circ} ), ( ru = 16 mathrm{cm} ), and ( pu ) and ( qt ) are diameters of circle ( s ). what is the measure of arc ( qr ) in terms of ( pi )?
a ( \frac{40}{3}mathrm{cm} )
b ( \frac{44}{3}mathrm{cm} )
c ( \frac{120}{3}mathrm{cm} )
d ( \frac{480}{3}mathrm{cm} )

Explanation:

Step1: Find the radius of the circle

Since \(RU = 16\) cm and \(RU\) is a diameter of the circle \(S\), the radius \(r=\frac{RU}{2}=\frac{16}{2} = 8\) cm.

Step2: Use the arc - length formula

The formula for the length of an arc is \(s = r\theta\) (where \(\theta\) is in radians). First, convert the central angle \(\angle QSR\) from degrees to radians. We know that \(\angle QSR = 150^{\circ}\), and \(\theta=150\times\frac{\pi}{180}=\frac{5\pi}{6}\) radians.
Substitute \(r = 8\) cm and \(\theta=\frac{5\pi}{6}\) into the arc - length formula \(s=r\theta\). Then \(s=8\times\frac{5\pi}{6}=\frac{40\pi}{6}=\frac{20\pi}{3}\) cm.

Answer:

C \(\frac{20\pi}{3}\) cm