Avalanche forecasting Avalanche forecasters measure the temper...

Question

Avalanche forecasting 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 -16° 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?

a. An avalanche forecaster digs a snow pit and takes two temperature measurements. At the surface (h = 0), the temperature is -16° 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?

Accepted Answer

Welcome back, everyone. In this problem, an environmental scientist is examining the temperature change in our soil profile. At the surface, where depth is zero, the temperature is 18 °C. At the depth of 1.5 m, the temperature is 30 °C. A temperature gradient exceeding a threshold of 6 °C per meter indicates the potential for rapid soil moisture evaporation. Using the mean value theorem. Find the temperature gradient and determine if it exceeds the critical threshold. A says the temperature gradient is 8 °C per meter indicating the potential for rapid evaporation. B says it's 7 °C per meter indicating the potential for rapid evaporation. C says it's 5 °C per meter indicating no potential for rapid evaporation, and the D says it's 4 °C per meter indicating no potential for rapid evaporation. Now before we apply the mean value theorem, let's make a note of what we already know here. Now, so far we know that the surface temperature, let's call that T1 is 18 °C, OK? And as the surface temperature, this means that it's at a depth, D1, OK. Of 0 m. Now, we also know at a depth of 1.5 m, the temperature is 30 °C. So we can say that T2 equals 30 °C, OK, at a depth D2 of 1.5 m. So how can we use this information to help us figure out the temperature gradient through the mean value theorem? Well, recall, OK, that the mean value theorem states that for a continuous function F, OK, let's make a note here, a continuous function F on the interval A and B inclusive, there exists at least one point C in the interval such that F of the derivative of FF C equals F of B minus F of A, OK, divided by B minus A, right? And as we said, this is on the interval. Between A and B. Now if we think about it here, that means if we apply that to the temperature gradient, then our temperature gradient G. Yeah, let me, let's define that. The temperature gradient in G. G for gradient, of course. Um. By the mean value theorem then is going to be the difference between the temperatures or function T2 minus T1 divided by the difference between the depths D2 minus D1. So in this case, then that means the gradient, let me put this in red. It's gonna be equal to 1830 °C rather. -18 °C, OK. Divided by 1 here divided by 1.5 m minus 0, and when we were at all then we get our temperature gradient to be equal to 8 °C per meter. So what does this mean? Well, remember we were trying to figure out if our temperature gradient is going to exceed the threshold of 6 °C per meter. So since it equals 8 °C per meter, it exceeds the threshold which indicates potential for rapid soil moisture evaporation. If we look back on our answer choices, that means A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Mean Value Theorem

Temperature Gradient

Weak Layer Formation

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