A spherical balloon is filled through an area, A1, with air flowing at velocity, V1, and constant density, p1. The radius of the balloon, R(t), can change with time, t. The average density within the balloon at any given time is pb(t). Determine the relationship between the rate of change of the density within the balloon and the rest of the variables. What the steps to get this final answer..? Solution density = mass/vol mass of air enterning = rho1 * v1*A1 mass inside baloon changing = 4pi*rhob*R^2*dR/dt net change of mass inside baloon = rho1 * v1*A1 - 4pi*rhob*R^2*dR/dt vol at that time = 4/3piR^3 d(rhob) / dt = rho1 * v1*A1 - 4pi*rhob*R^2*dR/dt / 4/3piR^3 .