APPENDIX B GENERALIZED EQUATIONS FOR ESTIMATING ALIQUOT WEIGHT AND NUMBER OF ALIQUOTS PER SAMPLE Results for Am Suppose that the relationship between the standard deviation (s) counts and aliquot weight (w) in grams is s = aw? . for Am Ge(Li) (Bl) In addition, assume the total cost C for Ge(Li) counting n aliquots of any size w is C = nk, (B2) where K is the cost per aliquot. Since the variance of the estimated mean Am concentration is Va = s‘*/n, we see from Equation Al and A2 that VA = at /nweP = a-K/CweP . (B3) This can be solved for w to give 1 1 w= (a°/V,n)?? = (a’K/cv,)P , (B4) Hence, for fixed counting costs C and K, and a desired precision V, of xX we can estimate both the number (n) and weight (w) of aliquots that are required by solving Equations A2 and A4&. This assumes, of course, that we have previously obtained data that give reliable estimates of a and b in Equation Bl. For a specified precision Va the cost C will be minimized by taking n = 1. In that case, Equation A4 gives i we (a°/v,)7? : (B5) Conversely, if C and K are fixed (equivalent to fixing n) then Vag will be minimized by using the largest possible aliquot size w up to the limit where the flat rate of K dollars per aliquot no longer applies. This can be seen from Equation B3. 443