The population of a herd of sheep declines exponentially. If it takes 3 years for 15% to decline then how long will it take for half the population to decrease ?
It would take 12.8 years.
Let initial number of sheep
Let the number of sheep remaining after time t
The equation for exponential decay gives us:
If the number of sheep have declined by 15% then the number remaining must be 85% of the original total.
Putting in the numbers:
Taking natural logs of both sides gives:
I won't go into the derivation here but it can be shown that the expression for 1/2 life in terms of the decay constant k is given by: