# Differential Equations: Calculating Barometric Pressure at an Altitude

I'm trying to solve a differential equation problem that has to do with barometric pressure at a certain altitude.

The rate at which barometric pressure decreases with altitude is proportional to the pressure at that altitude. If the altitude in feet, then the constant of proportionality is 3.7 x 105. The barometric pressure at sea level is 29.92 inches of mercury.

How would I calculate the barometric pressure, in inches of mercury, at any altitude using the information above?

The differential equation that I came up with looks like this:
Let $P=$ Barometric Pressure
Let $a=$ Altitude
$\dfrac{dP}{da}=-k(29.92-P)$
But when I solve it, it doesn't look like it's the right answer.

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Pressure decreases exponentially with altitude and thats what you will get provided you fix your signs in the equation. Your equation will look like:

$\frac{dP}{da} = -kP$

where $k$ is some positive constant.

Solve this to get $P(a) = P(0) e^{-ka}$.

and at $a = 0$, $P(0) = 29.92$. Make sure to keep track of all your units.

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That's the differential equation I was looking for. I guess I was thinking too much into it. Thank you! –  Zanneth Nov 25 '10 at 1:07

If your constant k doesn't allow the organic factors to coexist within this specific equation Just use a new one with your original formula you should get the right answer accurate to 6% of the correct answer because it was never officially empirically determined.

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