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A.

**P = ρg(z**

_{o}-z)P = atmospheric pressure, ρ = the density of the air, g = gravitational constant, z

_{o}= "max" altitude, z = altitude variable.B.

**ρ = PM/RT**

from the ideal gas law and ρ = Mn/V, M = molar mass of air.

To simplify the constants let H = RT/Mg ( so ρ = Pg/H). The answer at the end of this analysis is

**P = P**

_{o}Exp(-z/H).However, this seems conditioned on the approach. For example, taking the Derivative of A then substituting in B produces

1.

**dP = -ρgdz**(derivative of A)

2.

**dP = -P/Hdz**(substituting in B)

Collecting variables and integrating gives the answer,

**P = P**

_{o}Exp(-z/H).Now for the part that is weird to me, if instead we substitute first and then take the derivative then we get something that is (seemingly) impossible to reduce to the answer.

3.

**P = P/H(z**(substituted B into A)

_{o}-z)4.

**dP = 1/H d[P(z**

_{o}-z)]5.

**dP = 1/H (-Pdz + (z**

.

.

_{o}-z)dP).

.

6.

**dP/P = -dz/(H - (z**

_{o}-z))Which is about the cleanest that I could get it, but it doesn't seem reducible to the answer above...

Both ways seem logical, but they don't seem to reconcile. Now in the first approach (which gets the right answer) we assume that ρ is independent of z in line 1. (but is it? z doesn't appear in B., but ρ depends on pressure P, which depends on z, so does that make ρ independent or dependent of z?). In the second approach, when we take the derivative (line 4.) we assume that P depends on z, forcing us to use the product rule and mess things up.

Is there a simple answer that reconciles these approaches? I.e. how can ρ be independent of z (line 1.) given equations B and A (they seem recursively dependent upon each other) and how can P be independent of z (line 4.) so that both approaches give the same answer?