To solve this problem, we need to calculate the Schwarzschild radius, which is the radius at which an object becomes a black hole.$\\$The formula for the Schwarzschild radius $(R_s)$ is given by:$\\ R_s = \frac{2GM}{c^2} \\$where:$\\$• $G$ is the gravitational constant, $G = 6.674 \times 10^{-11} \, \mathrm{m^3 \, kg^{-1} \, s^{-2}} \\$• $M$ is the mass of the object, in this case, Earth's mass $M = 5.98 \times 10^{24} \, \mathrm{kg} \\$• $c$ is the speed of light, $c = 3 \times 10^8 \, \mathrm{m/s} \\$Substitute the known values into the formula:$\\ R_s = \frac{2 \times (6.674 \times 10^{-11}) \times (5.98 \times 10^{24})}{(3 \times 10^8)^2} \\$Calculate the numerator:$\\ 2 \times 6.674 \times 10^{-11} \times 5.98 \times 10^{24} = 7.982 \times 10^{14} \\$Calculate the denominator:$\\ (3 \times 10^8)^2 = 9 \times 10^{16} \\$Now, calculate $R_s: \\ R_s = \frac{7.982 \times 10^{14}}{9 \times 10^{16}} \\ R_s = 8.869 \times 10^{-3} \, \mathrm{m} \\$Therefore, the approximate radius to which Earth would have to be compressed to become a black hole is $8.869 \times 10^{-3} \, \mathrm{m}, \\$which is approximately $10^{-2} \, \mathrm{m}. \\$This corresponds to Option 3.