Question

Poaching is causing a population of elephants to decline by 5% per year. Use the approximate half-lifeformula to determine the number that remains in 59 years if there are 8661 elephants today.

Answer 1

To find the half-life formula we will use the rule

$t_{\frac{1}{2}}=\frac{1}{\log _{\frac{1}{2}}(1-r)}$

Since the population of elephants decline by 5%, then

$\begin{gathered} r=\frac{5}{100} \\ r=0.05 \end{gathered}$

Then, substitute r in the rule above by 0.05

$\begin{gathered} t_{\frac{1}{2}}=\frac{1}{\log _{\frac{1}{2}}(1-0.05)} \\ t_{\frac{1}{2}}=\frac{1}{\log _{\frac{1}{2}}(0.95)} \\ t_{\frac{1}{2}}=13.5134\text{ years} \end{gathered}$

Now, to find the new value we will use the rule

$N=N_0(\frac{1}{2})^{\frac{t}{t_{_0}}_{}}$

N(0) is the initial value

$N_0=8661$

t is the time

$t=59$

Substitute these values in the rule above

$N=8661(\frac{1}{2})^{\frac{59}{13.5134}}$

Find the answer

$N=420.0103933$

Round it to the whole number, then

The number of elephants will be 420