The so$-$called barometric equation gives the pressure variations as a function of height throughout the atmosphere, assumed to be isothermal with temperature $,$ as $,$ where is the Boltzmann constant $($ J$/$molec$/$K$),$ the other symbols have their usual meaning and atm and K$.($a$)$ Calculate the air pressure at km and $($b$)$ the height at which the air pressure has dropped by a factor of $($ is known as the scale height$).($c$)$ Write an equation for the vertical dependence of the air density in terms of $.($d$)$ The cruising altitudes of subsonic and supersonic aircrafts are $12$ km and $20$ km$,$ respectively. What is the relative difference in air density between these two altitudes?