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## Homework Statement

At the top of a mountain the temperature is -5 degree C and a mercury barometer reads 566 mm, whereas the reading at the foot of the mountain is 749 mm.

## Homework Equations

Assuming a temperature lapse rate of 0.0065 K/m and R = 287 J/kg K, calculate the height of the mountain.

## The Attempt at a Solution

The lapse rate tells me how much temperature increases or decreases with height. So I can say that Temperature is a function of height T(d). And I'm assuming it to be linear function so I used the equation of a line which is y=mx+c.

where

y=T(d)

m=0.0065K/m

x=d

and c is the y intercept which is the temperature when height is 0 so the equation will be

T(d)= -0.0065d+c

I've taken slope as negative because temperature decreases with change in height which is obvious.

The pressure at the peak will be (from the mercury barometer reading)

P1=ρgh1

P1=13560x9.81x566x10^-3

P1= 75291.3576m

The pressure at the foot of the mountain is: (also from the mercury barometer reading)

P2=ρgh2

P2=13560x9.81x749x10^-3

P2= 99634.6764m

From the formula PV=mRT

we get P=ρRT

I assumed that the density of air will remain constant with change in height.

so what I did was I made rho subject.

ρ=P/RT

so at peak

ρ=P1/RT1

and at the foot

ρ=P2/RT2

ρ=ρ

P1/RT1=P2/RT2

Cancelling R on both sides will give us

P1/T1=P2/T2

making T2 subject will give us the temperature at the foot of the mountain.

T2=P2T1/P1

T2=(99634.67640)*(273-5)/(75291.3576)

T2=354.65

Which is the temperature at the foot of the mountain. Its also the y intercept c of the linear function T(d)= -0.0065d+c because it tells us the temperature when height is zero.

so the equation we'll end up with will be

T(d)= -0.0065d+354.65

To find the height of the peak from the foot of the mountain we'll put T(d) = 273-5 = 268K

268 = -0.0065d+354.65

after making d subject and solving further we'll get d=13330.76923m

Can anyone tell me what I did wrong in my attempt to solve this question? I know this is not the correct answer.