Ideal gas pressure question.

The Earth’s atmosphere may be treated as an ideal gas whose density, pressure and temperature all decrease with height. In 1924, Howard Somervell and Edward Norton set a new altitude record when attempting to climb Mount Everest. They managed to climb to a vertical height of 8570 m above sea level by breathing in natural air. At this height, the air pressure was 0.35 times the pressure at sea level and the temperature was −33 °C. At sea level, air has a temperature 20°C and density 1.3 kg m−3. (i) Calculate the density of the air at a height of 8570 m at the time the record was set.

Hey could anyone explain to me how I do this question, as I am completely stuck on it, thanks.

Hi

So the ideal gas law has $PV=NkT$ where $N$ is the number of atoms. We also have density as $\rho= \frac{m}{V}$ where $m$ is the mass of the $N$
atoms and $V$ is the volume they take up. Combining these gives:

$Pm=\rho NkT$

or

$\frac{P}{\rho T}=\frac{Nk}{m}=\text{constant}$

Since the right hand side is a constant, the left hand side must be independent of altitude and so

$\frac{P_1}{\rho_1 T_1}=\frac{P_2}{\rho_2 T_2}$

which can be used to find the required answer.