Carbon dating exponential
This means that given a statistically large sample of carbon 14, we know that if we sit it in a box, go away, and come back in 5730 years, half of it will still be carbon 14, and the other half will have decayed.
How am I supposed to figure out what the decay constant is?
In equation form we have \[ \dfrac=ky.\] If we multiply both sides by \(dt\) and integrate, we get \[ \int \dfrac = k dt \] or \[ \ln y = kt C_0.
\] Exponentiating both sides to get rid of the \(\ln\) function gives \[ y = e^ = e^ e^. \] Then \[ y = Ce^ \] where \(C\) and \(k\) are constants.
\] To keep things compact we are still writing \(k\) instead of -0.000121.
Now divide by \(C\): \[ 0.09 = e^.\] Take ln of both sides at divide by \(k\) to get \[ t =\dfrac = \dfrac = 19,905.\] So the skull is about 20,000 years old.