The Solution.
First, we shall find the radius,r, of the given circle by using the formula below:
[tex]\begin{gathered} C=2\pi r \\ C=144\pi,r=? \end{gathered}[/tex]
Substituting these values in the formula above, we get
[tex]144\pi=2\pi r[/tex]
Dividing both sides by 2 pi, we get
[tex]r=\frac{144\pi}{2\pi}=72[/tex]
Now, to find the length of arc AB, we shall use the formula below:
[tex]L=\frac{\theta}{360}\times2\pi r[/tex]
In this case,
[tex]\theta=120,r=72,L=?[/tex]
Substituting these values in the formula above, we have
[tex]\begin{gathered} L=\frac{120}{360}\times2\times\pi\times72 \\ \\ L=\frac{1}{3}\times2\times\pi\times72=2\times\pi\times24=48\pi \end{gathered}[/tex]
Hence, the correct answer is option 3.
