v1 Algebra, STD2 EQ-Bank 29

The number \((N)\) of native birds in a wildlife reserve is decreasing. The number of birds is modelled using

\(N=k(1.08)^{-t}\) for \( t \geq 0,\)

where \(t\) is time in years.

The number of birds in the reserve today is 8000.

Use the model to estimate the number of birds in the reserve 12 years from today. (2 marks)

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The number of birds in the reserve needs to always be above 1500.
Explain why this model is NOT appropriate to use in this case. (1 mark)

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Show Answers Only

a. \(N=8000(1.08)^{-12}=3176.91…=3177\ \text{(nearest bird)}\)

b. \(\text{When}\ \ t=25:\)

\(N=8000(1.08)^{-25}=1168.14…\)

\(\text{Since the number of birds needs to always be over 1500, it is not appropriate.}\)

Show Worked Solution

a. \(\text{When}\ \ t=0,\ \ N=8000:\)

\(8000=k(1.08)^{0}\ \ \Rightarrow\ \ k=8000\)

\(\text{Find}\ N\ \text{when}\ \ t=12:\)

\(N=8000(1.08)^{-12}=3176.91…=3177\ \text{(nearest bird)}\)

b. \(\text{When}\ \ t=25:\)

\(N=8000(1.08)^{-25}=1168.14…\)

\(\text{Since the number of birds needs to always be over 1500, it is not appropriate.}\)