jmc

A central angle of $75^{\circ }$ cuts off an arc that is $\frac{75}{360}=\frac{5}{24}$ of the circumference of the circle. Since $\frac{5}{24}$ of the circle's circumference is $10\pi$, the entire circumference of the circle is $10\pi \cdot \frac{24}{5}=48\pi$. Therefore, the diameter of the circle is 48, and the radius of the circle is $\boxed{24}$.