We must find the arc of a circle with arc 30pi and 225 degrees.
For this, we can use the equation for the length of a circle which says that:
[tex]C=2\pi r[/tex]
Where r is the radius of the circumference, however first we must find the length of the 360-degree arc of the circumference, for we do the following equality:
[tex]\frac{30\pi}{C}=\frac{225}{360}[/tex]
Now we clear C:
[tex]\begin{gathered} C=\frac{30\pi\cdot360}{225} \\ C=\frac{10800\pi}{225} \\ C=48\pi \end{gathered}[/tex]
The length of the 360-degree arc of the circumference is 48pi, we replace this in the first equation and solve for the radius
[tex]\begin{gathered} 48\pi=2\pi r \\ r=\frac{48\pi}{2\pi} \\ r=24 \end{gathered}[/tex]
In conclusion, the answer is that the radius measures 24
