Question

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 103.6 kPa at sea level and 89.1 kPa at h = 1,000 m. (Round your answers to one decimal place.)
(a) What is the pressure at an altitude of 3000 m? kPa
(b) What is the pressure at the top of a mountain that is 6358 m high? kPa

Answer 1

Answer:

Explanation:

Rate of change of pressure with respect to heigh

$\frac{dp}{dh}$ = k p ( given )

$\frac{dp}{p}$ = k dh

Integrating on both sides

∫$\frac{dp}{p}$ = ∫ k dh

lnp = kh + c , c is a constant

when h = 1000 m , p = 89.1 k Pa

ln 89.1 = 1000 k + c -------------- (1)

when h = 0 , p = 103.6 k Pa

ln 103.6 = 0 x  k + c

c = ln 103.6

Putting it in quation (1)

ln 89.1 = 1000 k + ln103.6

ln $\frac{89.1}{103.6}$ = 1000k

k = $\frac{-0.15}{1000}$

= - .15 x 10⁻³

lnp = - .15 x 10⁻³ h +  ln 103.6

when h = 3000

lnp = - .15 x 10⁻³ x 3000 +  ln 103.6

= - .45 +4.64

= 4.2

p = 66.68 k Pa

b )

when h = 6358

lnp =  - .15 x 10⁻³ x 6358 +  ln 103.6

= - .9537 + 4.64

= 3.6863

39.9 k Pa.