Author 
Comment/Response 
Bill Simpson

09/26/12 1:59pm
Use Reduce to solve for R
In[1]:= Reduce[700000*2*R/(Pi*(R^4r^4))==318&&R>r>0,R]
Out[1]= r > 0 && R == Root[159*Pi*r^4  700000*#1 + 159*Pi*#1^4 & , 2]
Scrape that Root object and paste into ToRadicals
In[2]:= x = ToRadicals[Root[159*Pi*r^4  700000 #1 + 159*Pi*#1^4 &, 2]];
Try to minimize
In[3]:= NMinimize[{x,r>0},r]
From In[3]:= NMinimize::nrnum: The function value 5.59526+9.69128I is not a real number at {r} = {0.0637073}
and your function is complex in some places.
Table[r = RandomReal[{0, 1000}]; {r, x}, {100}]
seems to show me that it is complex lots of places and perhaps almost everywhere.
Is there some mistake in something I've done or can you perhaps revise your question so that a domain where it is real can be isolated?
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