# The rate of change of atmospheric pressure P with respect to altitude h is proportional to P, provided that the temperature is constant.The rate of change of atmospheric pressure P with...

The rate of change of atmospheric pressure P with respect to altitude h is proportional to P, provided that the temperature is constant.

The rate of change of atmospheric pressure P with respect to altitude h is proportional to P, provided that the temperature is constant. At a specific temperature the pressure is 102.2 kPa at sea level and 87.9 kPa at h = 1,000 m. (Round your answers to one decimal place.)

(a) What is the pressure at an altitude of 25000 m?

______________kPa

(b) What is the pressure at the top of a mountain that is 6361 m high?

______________kPa

embizze | High School Teacher | (Level 1) Educator Emeritus

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The rate of change of atmospheric pressure P with respect to altitude h is proportional to P.

Then `(dP)/(dh)=kP` for some constant k.

`(dP)/P=kdh`       Integrating both sides we get:

`lnP=kh+C_1` Then

`e^(lnP)=e^(kh+C_1)` ** `e^(kh+C_1)=e^(kh)e^(C_1)` ;Let `C=e^(C_1)`

`P=Ce^(kh)`

Using `(h,P)=(0,102.2)` we find `102.2=Ce^(0)==>C=102.2`

Using `(1000,87.9)` and C=102.2 we get:

`87.9=102.2e^(1000k)`

`.8601=e^(1000k)`

`ln(.8601)=1000k ==>k=-.00015`

Thus we get `P=102.2e^(-.00015h)`

(a) For h=25000 we get `P=102.2e^(-.00015(25000))~~2.4` kPa

(b) For h=6361 we get `P=102.2e^(-.00015(6361))~~39.4` kPa