To solve this problem, we need to find the ratio of the radius of gyration of a solid sphere to that of a thin hollow sphere.$\\$The radius of gyration $k$ is related to the moment of inertia $I$ and mass $M$ by the formula:$\\ k = \sqrt{\frac{I}{M}} \\$Let's calculate the radius of gyration for each sphere:$\\$1. Solid Sphere:$\\$The moment of inertia of a solid sphere about its own axis is $I_{solid} = \frac{2}{5}MR^2. \\$Thus, the radius of gyration $k_{solid}$ is:$\\ k_{solid} = \sqrt{\frac{I_{solid}}{M}} = \sqrt{\frac{\frac{2}{5}MR^2}{M}} = \sqrt{\frac{2}{5}R^2} = \frac{R}{\sqrt{5}} \\$2. Thin Hollow Sphere:$\\$The moment of inertia of a thin hollow sphere about its own axis is $I_{hollow} = \frac{2}{3}MR^2. \\$Thus, the radius of gyration $k_{hollow}$ is:$\\ k_{hollow} = \sqrt{\frac{I_{hollow}}{M}} = \sqrt{\frac{\frac{2}{3}MR^2}{M}} = \sqrt{\frac{2}{3}R^2} = \frac{R}{\sqrt{3}} \\$Now, we find the ratio of the radius of gyration of the solid sphere to the hollow sphere:$\\$Ratio$= \frac{k_{solid}}{k_{hollow}} = \frac{\frac{R}{\sqrt{5}}}{\frac{R}{\sqrt{3}}} = \frac{\sqrt{3}}{\sqrt{5}} \\$Simplifying the ratio:$\\$Ratio$= \frac{\sqrt{3}}{\sqrt{5}} = \frac{\sqrt{3} \cdot \sqrt{5}}{5} = \frac{\sqrt{15}}{5} \\$To express this as a simple ratio, multiply both the numerator and denominator by $\sqrt{15}: \\$Ratio$= \frac{\sqrt{15} \cdot \sqrt{15}}{5 \cdot \sqrt{15}} = \frac{15}{5\sqrt{15}} = \frac{3}{\sqrt{15}} \\$Multiply numerator and denominator by $\sqrt{15}$ to rationalize:$\\$Ratio$= \frac{3\sqrt{15}}{15} = \frac{3}{5} \\$Therefore, the ratio of the radius of gyration of the solid sphere to the hollow sphere is $3:5. \\$This corresponds to Option 3.