Avalanche forecasters measure the temperature gradient dT/dh, which is the rate at which the temperature in a snowpack T changes with respect to its depth h. A large temperature gradient may lead to a weak layer in the snowpack. When these weak layers collapse, avalanches occur. Avalanche forecasters use the following rule of thumb: If dT/dh exceeds 10°C/m anywhere in the snowpack, conditions are favorable for weak-layer formation, and the risk of avalanche increases. Assume the temperature function is continuous and differentiable. 

a. An avalanche forecaster digs a snow pit and takes two temperature measurements. At the surface (h = 0), the temperature is -12°C. At a depth of 1.1 m, the temperature is 2°C. Using the Mean Value Theorem, what can he conclude about the temperature gradient? Is the formation of a weak layer likely?

b. One mile away, a skier finds that the temperature at a depth of 1.4 m is -1°C, and at the surface, it is -12°C. What can be concluded about the temperature gradient? Is the formation of a weak layer in her location likely?

c. Because snow is an excellent insulator, the temperature of snow covered ground is often near 0°C. Furthermore, the surface temperature of snow in a particular area does not vary much from one location to the next. Explain why a weak layer is more likely to form in places where the snowpack is not too deep. 

d. The term isothermal is used to describe the situation where all layers of the snowpack are at the same temperature (typically near the freezing point). Is a weak layer likely to form in isothermal snow? Explain.