6. (10 points) In winter, at an ambient air temperature of $t_{0}=-20^{\circ} \mathrm{C}$, each square meter of the lake gives off 300 kJ of heat to the air per hour. Estimate how long after the formation of ice a fisherman will be able to walk on the surface of the lake, if the safe thickness of ice is 10 cm? The temperature of the water $t_{\mathrm{B}}=0^{\circ} \mathrm{C}$. The specific latent heat of fusion of ice is 330 kJ/kg, its specific heat capacity is 2100 J/kg ${ }^{\circ} \mathrm{C}$, and the density of ice is 900 kg $/ \mathrm{m}^{3}$. Assume the rate of heat transfer is constant.
Answer: 105.3 hours.

Solution. Inside the formed ice, the temperature changes linearly from $0^{\circ} \mathrm{C}$ in depth to $-20^{\circ} \mathrm{C}$ on the surface. That is, the heat released can be considered to result from the crystallization of ice and its subsequent cooling to an average of $-10^{\circ} \mathrm{C}$.

The mass of a square meter of ice capable of supporting a person:

$m=\rho V=\rho S h=900 \cdot 1 \cdot 0.1=90$ kg.

We get $300000=\frac{Q}{t}=\frac{\lambda m+c m \Delta T}{t}$.

As a result: $t=\frac{\lambda m+c m \Delta T}{300000}=\frac{330000 \cdot 90+2100 \cdot 90 \cdot 10}{300000}=105.3$ hours.