# 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 ?

Mar 14, 2017

It would take 12.8 years.

#### Explanation:

Let initial number of sheep $\textsf{= {S}_{0}}$

Let the number of sheep remaining after time t $\textsf{= {S}_{t}}$

The equation for exponential decay gives us:

$\textsf{{S}_{t} = {S}_{0} {e}^{- k t}}$

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

So $\textsf{{S}_{t} = 085 {S}_{0}}$

Putting in the numbers:

$\textsf{0.85 \cancel{{S}_{0}} = \cancel{{S}_{0}} {e}^{- k 3}}$

Taking natural logs of both sides gives:

$\textsf{\ln \left(0.85\right) = - k 3}$

$\therefore$$\textsf{- 0.1625 = - k 3}$

$\therefore$$\textsf{k = \frac{0.1625}{3} = 0.05416 \textcolor{w h i t e}{x} {\text{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:

$\textsf{{t}_{\frac{1}{2}} = \frac{0.693}{k}}$

$\therefore$$\textsf{{t}_{\frac{1}{2}} = \frac{0.693}{0.05416} = 12.8 \textcolor{w h i t e}{x} \text{yr}}$