The density of air changes with height. Under some conditions density p. depends on height z, and temperature T according to the following equation where Po and À e both constants. A meteorological balloon ascends (i.e., starts at z =1 and gains height) over the course of several hours. Complete parts (a) and (b) below. iz P(z.T) = Po e dz (a) Assuming that the balloon ascends at a speed v (i.e., = v) and that the temperature changes over time (i.e., that T is given by a function T(t)), derive, using the chain rule, an expression for the rate of change of air density, dt as measured by the weather balloon. Choose the correct answer below. dp iz dT O A. = p dt và dp OB. dt dp vÀ iz dT Oc. dt v. dp OD. dt (b) Assume that v = 1, Po = 1, and =1 and that when t=0, T = 1. Are there any conditions under which the density, as measured by the balloon will not change in time? That is, find a differential equation that T must satisfy, if dp = =0, and solve this differential equation. Choose the correct equation for T below. O. T(t) = vt +C O B. T(t) = C(1 + vt) Oc. T(t) = 1+ vt O D. T(t) =vt