Algebra, STD2 EQ-Bank 29

The population ( \(P\) ) of a town is reducing. The population is modelled using

\(P=k(1.1)^{-t}\) for \( t \geq 0,\)

where \(t\) is time in years.

The population of the town today is 5000 .

Use the model to estimate the population of the town 15 years from today. (2 marks)

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The population of the town needs to always be above 500 .
Explain why this model is NOT appropriate to use in this case. (1 mark)

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Show Answers Only

a. \(P=5000(1.1)^{-15}=1196.96…=1197\ \text{(nearest person)}\)

b. \(\text{When}\ \ t=30\ \ \Rightarrow\ \ P \approx 287\)

\(\text{Since the population needs to always be over 500, it is not appropriate.}\)

Show Worked Solution

a. \(\text{When}\ \ t=0,\ \ P=5000:\)

\(5000=k(1.1)^{0}\ \ \Rightarrow\ \ k=5000\)

\(\text{Find}\ P\ \text{when}\ \ t=15:\)

\(P=5000(1.1)^{-15}=1196.96…=1197\ \text{(nearest person)}\)

b. \(\text{When}\ \ t=30\ \ \Rightarrow\ \ P \approx 287\)

\(\text{Since the population needs to always be over 500, it is not appropriate.}\)