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 ?

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Answer 1 · 2017-03-14T23:53:32

It would take 12.8 years.

Let initial number of sheep $sf(=S_0)$

Let the number of sheep remaining after time t $sf(=S_t)$

The equation for exponential decay gives us:

$sf(S_t=S_0e^(-kt))$

If the number of sheep have declined by 15% then the number remaining must be 85% of the original total.

So $sf(S_t=085S_0)$

Putting in the numbers:

$sf(0.85cancel(S_0)=cancel(S_0)e^(-k3))$

Taking natural logs of both sides gives:

$sf(ln(0.85)=-k3)$

$:.$ $sf(-0.1625=-k3)$

$:.$ $sf(k=0.1625/3=0.05416color(white)(x)"yr"^-1)$

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:

$sf(t_(1/2)=0.693/k)$

$:.$ $sf(t_(1/2)=0.693/0.05416= 12.8color(white)(x)"yr")$