To solve this problem, we need to determine the final kinetic energy of the block after it has been subjected to a retarding force.$\\$Given:$\\$• Mass of the block $m = 10$ kg$\\$• Initial speed $v_i = 10$ m/s$\\$• Retarding force $F = 0.1 \times J$ m$^{-1} \\$The work done by the retarding force as the block moves from $x = 20$ m to $x = 30$ m is given by:$\\ W = \int_{20}^{30} F \, dx = \int_{20}^{30} 0.1 \times J \, dx \\$Since $F = 0.1 \times J$ is constant over the distance, the work done is:$\\ W = 0.1 \times J \times (30 - 20) = 0.1 \times J \times 10 = J \\$The initial kinetic energy $KE_i$ of the block is:$\\ KE_i = \frac{1}{2} m v_i^2 = \frac{1}{2} \times 10 \times (10)^2 = 500$ J$\\$The work-energy principle states that the work done on the block is equal to the change in its kinetic energy:$\\ W = KE_f - KE_i \\$Substitute the known values:$\\ J = KE_f - 500 \\$Rearrange to solve for the final kinetic energy $KE_f: \\ KE_f = J + 500 \\$Since the retarding force is $F = 0.1 \times J$ m$^{-1},$ and the work done is $J, \\$we can conclude that $J = 0$ because the force is a function of $J$ and not a specific value.$\\$Thus, the final kinetic energy $KE_f$ is:$\\ KE_f = 0 + 500 = 500$ J$\\$However, since the force is retarding, we need to consider the negative work done:$\\$The correct interpretation is that the work done is negative, hence:$\\ KE_f = 500 - J \\$Given the options, the closest correct answer is:$\\$Option 4: 475 J$\\$Therefore, the final kinetic energy of the block is $475$ J.