Question

Assume the tabLe is exponential There was a release of European wild rabbits in Victoria, Australia in December 1858. without natural predators, the population grew. Round to the nearest whole number

Answer 1

The population of rabbits (P) can be expressed with an exponential model of the form:

$P=a\times e^{b\times t}$

Then, we need to find the values of a and b in order to calculate the population at the time t.

Let's take the population at time t1 to be P1 and the population at time t2 to be P2. then:

$\begin{gathered} P1=a\times e^{b\times t1} \\ P2=a\times e^{b\times t2} \end{gathered}$

By taking the ratio of P1 to P2, we get:

$\begin{gathered} \frac{P1}{P2}=\frac{a\times e^{b\times t1}}{a\times e^{bt1}} \\ \frac{P1}{P2}=\frac{a}{a}\times\frac{e^{b\times t1}}{e^{bt\times1}} \\ \frac{P1}{P2}=\frac{a}{a}\times e^{b\times t1-b\times t2} \\ \frac{P1}{P2}=e^{b\times(t1-t2)} \end{gathered}$

We can solve for b by taking the natural logarithm on both sides of this expression:

$\begin{gathered} \frac{P1}{P2}=e^{b\times(t1-t2)} \\ \ln (\frac{P1}{P2})=\ln (e^{b\times(t1-t2)}) \\ \ln (\frac{P1}{P2})=b\times(t1-t2)\ln (e^{}) \\ b=\frac{\ln(\frac{P1}{P2})}{(t1-t2)} \end{gathered}$

By taking the first and the second row of the table, we can calculate b like this:

$b=\frac{\ln (\frac{250}{375})}{(12-13)}\approx0.4$

Now that we know the value of b, we only need to calculate a, we can do it like this:

$\begin{gathered} P=a\times e^{b\times t} \\ a=\frac{P}{e^{b\times t}} \end{gathered}$

Taking the data of the third row, we get:

$a=\frac{563}{e^{0.4\times14}}\approx1.92$

Then, for the time 15 and 16, we get:

$\begin{gathered} P=1.92\times e^{0.4\times15}\approx844.5 \\ P=1.92\times e^{0.4\times16}\approx1266.75 \end{gathered}$