It is well-known that the temperature of the atmosphere varies with altitude. In the troposphere, which extends to an altitude of $11 \mathrm{km},$ for example, the variation of temperature can be approximated by $T=T_{0}-\beta z,$ where $T_{0}$ is the temperature at sea level, which can be taken to be $288.15 \mathrm{K},$ and $\beta=$ $0.0065 \mathrm{K} / \mathrm{m} .$ The gravitational acceleration also changes with altitude as $g(z)=g_{0} /(1+z / 6,370,320)^{2}$ where $g_{0}=9.807 \mathrm{m} / \mathrm{s}^{2}$ and $z$ is the elevation from sea level in $\mathrm{m}$. Obtain a relation for the variation of pressure in the troposphere ( $a$ ) by ignoring and (b) by considering the variation of $g$ with altitude.