The length of an arc, the radius of a circle and the central angle that contains that arc are related as follows:
[tex]s=r\theta[/tex]
From the figure, we have that:
[tex]\begin{gathered} s=17\pi \\ \theta=\frac{4\pi}{3} \end{gathered}[/tex]
Substitute those values into the equation to find the value of r:
[tex]\begin{gathered} 17\pi=r\times\frac{4\pi}{3} \\ \Rightarrow17\pi\times\frac{1}{\pi}=r\times\frac{4\pi}{3}\times\frac{1}{\pi} \\ \Rightarrow17=r\times\frac{4}{3} \\ \Rightarrow17\times\frac{3}{4}=r\times\frac{3}{4}\times\frac{4}{3} \\ \Rightarrow r=17\times\frac{3}{4} \\ \Rightarrow r=\frac{17\times3}{4} \\ \therefore r=\frac{51}{4} \end{gathered}[/tex]
Therefore, the radius of the circle is:
[tex]\frac{51}{4}[/tex]
