1 Answer

Napoli · Answer 1 · 2023-06-19T20:59:07+0000

We have to predict the number of African cheetahs living in the wild in t = 13 years.

We know that the current population is 7476 and it declines at a rate of 8% per year.

We can express this with an exponential model:

$P(t)=A\cdot b^t$

We can find the parameter A knowing that P(0) = 7476.

Then, we will have:

$\begin{gathered} P(0)=7476 \\ A\cdot b^0=7476 \\ A=7476 \end{gathered}$

Now, we can use the decrease rate r = 0.08 (or 8%) to find the parameter b.

If the population decrease by 8%, the population after one year will be 8% less. We can express this as:

$\begin{gathered} M(t+1)=M(t)-0.08\cdot M(t) \\ M(t+1)=0.92\cdot M(t) \\ (M(t+1))/(M(t))=0.92 \end{gathered}$

If we replace with the model definition we obtain:

$\begin{gathered} (M(t+1))/(M(t))=0.92 \\ (A\cdot b^(t+1))/(A\cdot b^t)=0.92 \\ b^(t+1-t)=0.92 \\ b=0.92 \end{gathered}$

We now have the model for the population after t years:

$P(t)=7476\cdot0.92^t$

We can now calculate the population after t = 13 years as:

$\begin{gathered} P(13)=7476\cdot0.92^(13) \\ P(13)\approx7476\cdot0.338253 \\ P(13)\approx2529 \end{gathered}$

Answer: the number of African cheetahs living in the wild in 13 years is expected to be 2529.

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