R = (2πaL)/(4πr**2)    cylinder to sphere surface area ratio 
                         where "a" is cylinder radius,
                         L is cylinder length, 
                         and r is sphere radius.

  πa**2*L = (4π/3)*r**3        equating cylinder volume to sphere volume 
                               to solve for sphere radius r.

  r = [(3/4)*a**2*L]**(1/3)    The solution for sphere radius r.
 
  R = [(1/2)*(4/3)**(2/3)]         cylinder to sphere surface area ratio 
            *[aL/(a**2*L)**(2/3)]  with r eliminated. 

  R = [(1/2)*(16/9)**(1/3)]*(L/a)**(1/3)  simplified cylinder 
                                          to sphere surface area ratio.

  R = [(2/9)**(1/3)]*(L/a)**(1/3)         exact formula.  QED
 
  R  = (0.6057 ...)(L/a)**(1/3)           exact formula 
                                          with decimal fraction coefficient .