RADIOACTIVE DECAY AND HALF-LIFE

H = Half-life of Carbon 14 isotope (C14)
  = 5730 +- 30 years

Q = Initial quantity of C14 that we would expect to find in a living
    specimen.

q = Quantity of isotope remaining at time interval T after death

T = Time interval since death of specimen
  = t * ln(q/Q) / -ln(2)
  = 5730 * ln(q/Q) / -0.6931471805599453


If an paleontologist found a fossil sample that contained 21.7% carbon-14
in comparison to a living sample, the time of the fossil sample's death
can be determined, if q, Q, and t are known.

H   = 5730 years
q/Q = (Remaining C14 at T > 0) / (Initial C14 at T = 0)
    = 0.217
    = 21.7%

T = H * ln(q/Q) / -ln(2)
  = H * ln(q/Q) / -0.6931471805599453

T = 5730 * ln(0.217) / -ln(2)
  = 12,630 years since death of specimen.

-------------------------------------------
Considering the uncertainty of +- 30 years:

T = (5730-30) * ln(0.217) / -ln(2)
to
T = (5730+30) * ln(0.217) / -ln(2)

  = 12,630 +- 66 years since death of specimen.
  = 12,564 to 12,696 years




/*
   ###########################################################################
   This PHP function computes the C14 age of a specimen given the Q ratio.

   Q = Amount of C14 in a living specimen  at T = 0
   q = Amount of C14 remaining in specimen at T > 0

   QRatio = q/Q

   At death, T = 0, the (Q) value begins to diminish or
   decay, reducing to (q) at time T > 0.

   T = Time interval since death of specimen.
     = To be computed.

     Half-life of C14 = 5730 +- 30 years.

     This is an idealized computation.    In practice, nuclear testing over
     several years has added C14 to the Earth so,  in some cases,  this may
     have to be taken into account. It can still be used to obtain a decent
     ball-park figure.
     #########################################################################
*/


   function C14_Age ($QRatio)
{
   $QRatio = trim($QRatio);

   $pm = 30;
   $k  = log($QRatio) / -0.6931471805599453;
   $t1 = (5730-$pm) * $k;
   $tm =  5730*$k;
   $t2 = (5730+$pm) * $k;

   $pm = floor(abs(($t1 - $t2)/2) + 0.5);

   $T1 = Number_Format($t1);
   $Tm = Number_Format($tm);
   $T2 = Number_Format($t2);

   return "T = $Tm &plus;&minus; $pm years<br>
   Specimen died between $T2 and $T1 years ago.";
}