Calculus

The rate of change of atmospheric pressure P with respect to the altitude h is proportional to P provided that the temperature is consistent. At 15 degrees Celsius, the pressure is 101.3 pounds per square inch (psi) at sea level and 87.1 psi at height 1000m. Find the pressure in psi at the top of a mountain with an altitude of 8,200 meters. Round to the nearest integer.

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  1. dp/dh = kp
    dp/p = k dh
    ln p = kh+c
    p = c*e^(kh)
    p(0)=101.3, so c=101.3

    p(h) = 101.3 e^(kh)
    p(1000) = 87.1, so

    101.3 e^(1000k) = 87.1
    1000k = ln(87.1/101.3)
    k = -.000151

    p(h) = 101.3 e^(-.000151h)

    so, now solve for h when p = 8200

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  2. I'm still confused to where to plug in 8200. Do i plug it into P=Ce^kh where C=101.3?

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  3. actually, I misspoke. You want p when h=8200, so

    p(8200) = 101.3 e^(-.000151*8200)

    You should have caught that typo...

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