A population of \(200\) rabbits was introduced to an island. One week later the number of rabbits was \(210\). The number of rabbits, \(N\) , can be modelled by the function

\[N(t) = 200 \times {b^t},\,\,t \geqslant 0\,,\]

where \(t\) is the time, in weeks, since the rabbits were introduced to the island.

Find the value of \(b\) .

Calculate the number of rabbits on the island after 10 weeks.

An ecologist estimates that the island has enough food to support a maximum population of 1000 rabbits.

Calculate the number of weeks it takes for the rabbit population to reach this maximum.

\(210 = 200 \times {b^1}\)        (M1)

Note: Award (M1) for correct substitution into equation.

\((b = )\,\,1.05\)               (A1)      (C2)

\(200 \times {1.05^{10}}\)          (M1)

Note: Award (M1) for correct substitution into formula. Follow through from part (a).

\( = 325\)               (A1)(ft)    (C2)

Note: The answer must be an integer.

\(200 \times {1.05^t} = 1000\)       (M1)

\(t = 33.0\,\,\,(32.9869...)\)     (A1)(ft)     (C2)

Note: Award (M1) for setting up the equation. Accept alternative methods such as \(t = \frac{{\log \,(5)}}{{\log \,(1.05)}}\) , or a sketch of \(y = 200 \times {1.05^t}\)  and \(y = 1000\)  with indication of point of intersection. Follow through from (a).

Question 14: Exponential Function
This question was well-answered by the majority of candidates.
Part (a) was generally accessible, unless subtraction was used in the rearrangement of the formula.

Part (b) required an integer value.

Part (c) saw good use of the GDC – with many sketch graphs being shown on paper (this is to be encouraged) – as well as attempts using logarithms. Use of the GDC is to be encouraged as its efficient use is a mandatory part of the course. Logarithms are not discouraged – but they are not a necessary component of the course and it is easy to construct equations that are not accessible to solution by logarithms.