Barometric Pressure Variation

The problem is to determine the pressure at a  particular height "h", given that the barometric pressure at the sea level is 29.92 of Hg.

The pressure decreases with the height and small change in height results in small change in pressure given by:

The change in pressure depends on density (ρ), but ρ depends on the pressure as follows:

This can be written as:

dP(h)/dh = -kP

which is the first order Ordinary Differential Equation (ODE). 

Let's solve for P:

dP/P = -kdh

ln (P) = -kP + B, where B is an unknown constant

P = eBe-kh

At height = 0, P(0) = 29.92 inches of Hg which is the Initial Boundary Condition (IBC)

P(0) = eB = 29.92

This gives the final solution of 

P(h) = 29.92e-kh