The atmospheric pressure P(h) (in pounds per square inch) at a height h (in miles) above sea level on Earth satisfies a differential equation P' = -kP for some positive constant k.

Question

(a) Measurements with a barometer show that P(0) = 1.2 and P(9) = 0.7. What is the decay constant k?
(b) Determine the atmospheric pressure 14 miles above sea level.

Answer 1

P' = -kP
dP/P = -k dh
lnP = -kh + c

P = c e^-kh
P(0) = 1.2, so c = 1.2

P = 1.2 e^-kh
1.2 e^(-9k) = 0.7
k = .06

so,

P(h) = 1.2 e^-.06h