Answer
The height of the atmosphere would be 16.0 km
Work Step by Step
The density of air at sea level is $1.29~kg/m^3$. Half of this value is $0.645~kg/m^3$. We know that the pressure at the Earth's surface is $1.01\times 10^5~Pa$. We can find the height of the atmosphere.
$P = \rho~g~h$
$h = \frac{P}{\rho~g}$
$h = \frac{1.01\times 10^5~N/m^2}{(0.645~kg/m^3)(9.80~m/s^2)}$
$h = 1.60 \times 10^4~m = 16.0~km$
The height of the atmosphere would be 16.0 km
