Answer
The answer is below.
Work Step by Step
a) We are told to integrate the following equation, so we find:
$p = \rho h_0g\\ \int dP = \int -P\frac{dh}{h_0}$
We know that the bounds of the left hand integral are from $p_0$ to p, and the bounds of the right integral are from $h_0$ to h. Thus, we find:
$ln(\frac{p}{p_0})=\frac{-h}{h_0}\\ p = p_0e^{\frac{-h}{h_0}}$
b) We find:
$.5p_0=p_0e^{\frac{-h}{h_0}}$
$ h = -h_0ln(.5) = -(8.2)(ln.5) = 5.68 \ km$
