In this paper, a nonlinear problem for combined convective and radiative cooling of a spherical body is considered. This problem represents a strong nonlinearity in both the governing equation and the boundary condition. Analytic approximations for the solution of this problem are obtained using the homotopy analysis method and via a polynomial exponential basis. Also, the effect of the radiation-conduction parameter $Nrc$ and the Biot number Bi for the temperature on the surface of the spherical body is investigated and discussed.