Integrate[r/Sqrt[OmegaG+OmegaM r ],{r,0,R},GenerateConditions->False]         
Series[%,{R,0,3}]   
                2                  3
              R           OmegaM R         4
Out[4]= -------------- - ----------- + O[R]
        2 Sqrt[OmegaG]           3/2
                          6 OmegaG

out=%%/H0 /. {OmegaM->.306,H0->1/14.4,OmegaG->.000084}
FindRoot[out==.00038,{R,.001}] 
Out[4]= {R -> 0.000919458}
In[5]:= 2.725/R /. %                                                            

Out[5]= 2963.7

In[6]:= 1/R-1 /. %%                                                             

Out[6]= 1086.6

In[8]:= OmegaG/OmegaM /. {OmegaM->.306,OmegaG->.000084}                         

Out[8]= 0.00027451

In[10]:= %3 /. {R->0.00027451}                                                  

Out[10]= 0.0000462367 (* 46,000 years *)

* ? 3*60/(1e9*year)
    .5703976907766376E-14         !time in billion years
* ? sqrt(2*sqrt(.000084)*H0*180)  ! H0 was in 1/sec
    .2694515501504289E-08         !approx value of R)

FindRoot[out==.5703976907766376 10^(-14),{R,2.7 10^(-9)}]
                        -9
Out[11]= {R -> 2.6946 10  }

In[12]:= 2.725/R /. %                                                           

                   9
Out[12]= 1.01128 10