# #
Root mean square velocity #= sqrt{\ bar{c^2}\ }\ =\ sqrt{{3P}/d}#
Put the given values, to get:
#{183000\ cm}/sec\ =\ sqrt{{3(101325\ Pa)}/d}#
# #
Convert the root mean square velocity from #\ \ {cm}/s\ \ # to #\ \ m/s\ \ # and remember that, #\ \ 1\ Pa\ =\ {1\ N}/m^2\ \ \ # and #\ \ 1\ N\ =\ 1\ Kg{m}/s^2#
#{183000\ cm}/s\ =\ {1830\ m}/s#
# #
Square both sides replace the units:
#\frac{(1830)^2\ m^2}{s^2}\ =\ \frac{3\cdot 101325\ N}{m^2\cdot d}#
#\frac{(1830)^2\ m^2}{color(red){cancel{s^2}}# #=# #\frac{3\cdot 101325\ Kg*m}{m^2*\color(red){cancel{s^2}}\cdot d}#
#\frac{1}{(1830)^2\ m^2}\ =\ \frac{m^2\cdot d}{3\cdot 101325\ Kg\cdot m}#
# #
Solve for #\ \ d\ \ # to get:
#\frac{3\cdot 101325\ Kg\cdot color(blue){cancel m}}{(1830)^2\ \color(blue){cancelm^{4}##\ =\ ##d#
#d\ =\ \frac{3\cdot 101325\ Kg}{(1830)^2\ m^3}#
# #
#d\ =\ 0.091\ {Kg}/m^3#
# #
That's it!
