The temperature of the atmosphere varies with altitude. In the troposphere, which extends to an altitude of 11 km, for example, temperature variation 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 K, beta = 0.0065 K/m and z is the vertical height (in meters) above sea-level. The variation of pressure with altitude in a fluid can be shown to follow the differential equation: dp/dz = p rho g - Thermodynamics Obtain a relation for the variation of pressure, p, with vertical height, z, in the troposphere: a) by ignoring the variation of the temperature, b) by taking into account the variation of temperature with height. c) An airplane is flying at 9 km above sea level. What is the temperature at this height? Use the expression you found in part (b) to determine the atmospheric pressure and air density at this altitude. Compare these with values you get by using the expression you found in (a). In solving the differential equation you may find this helpful: p = rho R_g T, where R_g is the gas constant, and here it is 287 J/kg. K for air