Question

On your most recent mountain climbing adventure, you recorded a barometric pressure of 1000 mb at the base of a mountain. When you reached the summit (top of the mountain), you measured a barometric pressure of 950 mb. Approximately how tall is the mountain?
A) 50 meters

B) 5000 meters

C) 500 meters

Answer 1

Answer:

B) 5000 meters

Explanation:

Barometric pressure depends on the height, as of course there is more air above an area as the air column is higher. Usually, it is defined as a exponential relation which comes as follows:

1. $P=P_{0}*e^{(\alpha*h) }$

Being:

P the pressure at the determined height. In our case, would be the 950 mb at the summit

$P_{0}$ the initial pressure. In our case, would be the 1000 mb at the base.

α a constant value, which is usually found on literature as $-1.19*10^{-5} m^{-1}$ . It depends on the air density and the estimations on how much air do we have above.

h is the height which we want to determine the pressure at

Of course, we need to clear the height on the previous equation, so we need to seek help on the natural logarythm. First of all, let's clear the exponential expression from 1 :

2. $\frac{P}{P_{0}}=e^{\alpha*h}$

After that, apply the natural logarythm at both sides of the equation 2:

3. $ln(\frac{P}{P_{0}})=ln(e^{\alpha *h})$

Computing 3:

4. $ln(\frac{P}{P_{0}})=\alpha*h$

And clearing h in 4:

5. $\frac{ln\frac{P}{P_{0}} }{\alpha } =h$

Replacing known values in 5. Please have in mind, when replacing, both of the pressure values need to be in the same unit (in this case, millibars)

$h=\frac{ln\frac{950 mb}{1000 mb} }{-1,19*10^{-5} m^{-1}}=4310,36 m$

As this is an estimate, we would say that mountain is approximately 5000 m. Quite far from the result, maybe the source has different α constant values.