Answer
The height of the atmosphere would be $~~16~km$
Work Step by Step
If the air density decreases linearly with height, then the average air density of a column of air would be $\frac{1.3~kg/m^3}{2}$ which is $~~0.65~kg/m^3$
We can find the height of the atmosphere:
$P = \rho~g~h$
$h = \frac{P}{\rho~g}$
$h = \frac{1.01\times 10^5~N/m^2}{(0.65~kg/m^3)(9.8~m/s^2)}$
$h = 1.6\times 10^4~m$
$h = 16~km$
The height of the atmosphere would be $~~16~km$
