To solve this problem, we need to use the formula for the moment of inertia of a thin rod about an axis passing through its mid-point and perpendicular to the rod.$\\ \\$The formula for the moment of inertia $I$ of a rod of mass $m$ and length $L$ is:$\\ I = \frac{1}{12} m L^2 \\ \\$Given:$\\$• Moment of inertia $I = 2400$ g cm$^2 \\$• Mass of the rod $m = 400$ g$\\ \\$We need to find the length $L$ of the rod.$\\ \\$Substitute the given values into the formula:$\\ 2400 = \frac{1}{12} \times 400 \times L^2 \\ \\$Simplify the equation:$\\ 2400 = \frac{400}{12} \times L^2 \\ 2400 = \frac{100}{3} \times L^2 \\ \\$Multiply both sides by 3 to eliminate the fraction:$\\ 7200 = 100 \times L^2 \\ \\$Divide both sides by 100:$\\ L^2 = \frac{7200}{100} \\ L^2 = 72 \\ \\$Take the square root of both sides to solve for $L: \\ L = \sqrt{72} \\ \\$Calculate the square root:$\\ L \approx 8.5$ cm$\\ \\$Therefore, the length of the rod is approximately $8.5$ cm, which corresponds to Option 3.