When the piece of ice falls from a height $h$, the loss in its potential energy is equal to the heat produced. Heat produced, $Q = mgh$ Given that only one-quarter of the heat produced is absorbed by the ice to melt completely. Heat absorbed by ice $= \frac{1}{4}Q = \frac{1}{4}mgh$ The heat required to completely melt the ice is $Q' = mL$, where $L$ is the latent heat of fusion of ice. Equating the heat absorbed to the heat required for melting: $\frac{1}{4}mgh = mL$ $\implies h = \frac{4L}{g}$ Substituting the given values ($L = 3.4 \times 10^5 \text{ J/kg}$ and $g = 10 \text{ N/kg}$): $h = \frac{4 \times 3.4 \times 10^5}{10} \text{ m}$ $h = 4 \times 3.4 \times 10^4 \text{ m} = 13.6 \times 10^4 \text{ m} = 136000 \text{ m} = 136 \text{ km}$.