3. Experiments show that the rate of change of the atmospheric pressure P(x) wit

Question

3. Experiments show that the rate of change of the atmospheric pressure P(x) with

altitude x is proportional to the pressure. Find the differential equation for P(x), and

solve it, assuming that the pressure at 6000 meters is half its value P0 at sea level.

Please show all work!

Explanation / Answer

dp/dx=k*p =>dp/p=k*dx so, log p =kx+c (eqn 1) now, log p0=c and log p0/2 = k*6000 +c =>logp0 - log2 = 6000k+c =>c-log 2=6000k+c =>2c=-log 2 -6000k c= -3000k -(log2)/2 so, c = -3000k -0.3465 hence, putting the val of c in eqn 1 log p = k*x - 3000k - 0.3465

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