To solve this problem, we need to find the height $h$ from which a piece of ice falls and melts completely.$\\$Given:$\\$• Latent heat of ice $L = 3.4 \times 10^5 \, \mathrm{J/kg} \\$• Acceleration due to gravity $g = 10 \, \mathrm{N/kg} \\$• Only one-quarter of the heat produced is absorbed by the ice.$\\$Let's denote:$\\$• $m$ as the mass of the ice.$\\$• $h$ as the height from which the ice falls.$\\$The potential energy lost by the ice when it falls from height $h$ is given by:$\\ mgh \\$According to the problem, only one-quarter of this energy is absorbed by the ice to melt it.$\\$Thus, the energy absorbed by the ice is:$\\ \frac{1}{4} mgh \\$This absorbed energy is used to melt the ice, which is equal to the latent heat required:$\\ \frac{1}{4} mgh = mL \\$Cancel $m$ from both sides (assuming $m \neq 0$):$\\ \frac{1}{4} gh = L \\$Rearrange to solve for $h: \\ h = \frac{4L}{g} \\$Substitute the given values:$\\ h = \frac{4 \times 3.4 \times 10^5}{10} \\ h = \frac{13.6 \times 10^5}{10} \\ h = 1.36 \times 10^5 \, \mathrm{m} \\$Convert meters to kilometers:$\\ h = \frac{1.36 \times 10^5}{1000} \, \mathrm{km} \\ h = 136 \, \mathrm{km} \\$Therefore, the value of $h$ is $136 \, \mathrm{km}$, which corresponds to Option 1.