In order to find the length of the arc, first let's find the central angle corresponding to the arc.
To find it, let's use the formula for the area of a sector:
[tex]A=\frac{r^2\theta}{2}[/tex]
Using A = 5pi and r = 6, we have:
[tex]\begin{gathered} 5\pi=\frac{6^2\theta}{2}\\ \\ 36\theta=10\pi \\ \theta=\frac{10}{36}\pi \end{gathered}[/tex]
Now, to find the length of the arc, we have the formula below:
[tex]\begin{gathered} l=\theta\cdot r\\ \\ l=\frac{10\pi}{36}\cdot6\\ \\ l=\frac{10}{6}\pi=\frac{5}{3}\pi \end{gathered}[/tex]
This is the smaller arc. To find the greater arc, we subtract the circumference by the smaller arc:
[tex]\begin{gathered} arc=2\pi r-\frac{5}{3}\pi\\ \\ arc=12\pi-\frac{5}{3}\pi\\ \\ arc=\frac{36}{3}\pi-\frac{5}{3}\pi\\ \\ arc=\frac{31}{3}\pi \end{gathered}[/tex]
Therefore the arc is 31/3 pi.
