Question

The stingray population in a particular reef is decreasing at a rate of 1.4% per year. The population was 3800 in the year 2008. What is the best prediction of the population in the year 2013? Round to the nearest whole number, Show all work for full credit.

Answer 1

We have a initial population P(0) of 3800.

We know that the population is decreasing by 1.4% each year, so the population at year 1 (2009) we will have a population of:

`\begin{gathered} P(1)=P(0)-0.014\cdot P(0) \\ P(1)=(1-0.014)\cdot P(0) \\ P(1)=0.986\cdot P(0) \end{gathered}`

We can generalize this for P(n), where n is the number of years from 2008.

`\begin{gathered} P(2)=0.986\cdot P(1)=0.986\cdot(0.986\cdot P(0))=0.986^2\cdot P(0) \\ \Rightarrow \\ P(n)=0.986^n\cdot P(0)=0.986^n\cdot3800=3800\cdot0.986^n \end{gathered}`

From 2008 to 2013 we have 2013-2008=5 years, so for the year 2013, the value of n is n=5.

Then, the population for 2013 will be P(5) and can be calculated as:

`P(5)=3800\cdot0.986^5\approx3800\cdot0.932\approx3541`

Answer: the predicted population for the year 2013 is 3541.