In areas where the air temperature remains below $0^{\circ} \mathrm{C}$ for prolonged periods of time, the freezing of water in underground pipes is a major concern. Fortunately, the soil remains relatively warm during those periods, and it takes weeks for the subfreezing temperatures to reach the water mains in the ground. Thus, the soil effectively serves as an insulation to protect the water from the freezing atmospheric temperatures in winter.
The ground at a particular location is covered with snow pack at $-8^{\circ} \mathrm{C}$ for a continuous period of 60 days, and the average soil properties at that location are $k=0.35 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$ and $\alpha=0.15 \times 80^{-6} \mathrm{~m}^2 / \mathrm{s}$. Assuming an initial uniform temperature of $8^{\circ} \mathrm{C}$ for the ground, determine the minimum burial depth to prevent the water pipes from freezing.