Edexcel A-Level Mathematics A2 P3.3.2 Function Ln X and Its Graph Questions

Practise applying ln to isolated exponential terms in Edexcel IAL models, then use the logarithmic relationship to identify curve intercepts.

Exam points

Use ln after isolating an exponential term to find model parameters or times.
Use the logarithmic relationship to identify an x-intercept of a curve.

[Maximum number: 4]

A scientist is studying a population of fish in a lake. The number of fish, N, in the population, t years after the start of the study, is modelled by the equation

N=\frac{600 \mathrm{e}^{0.3 t}}{2+\mathrm{e}^{0.3 t}} \quad t \geqslant 0

Use the equation of the model to answer parts (a), (b), (c), (d) and (e).

Find the time, after the start of the study, when there are predicted to be 500 fish in the lake. Give your answer in years and months to the nearest month.

$500=\frac{600e^{0.3t}}{2+e^{0.3t}}\Rightarrow 100e^{0.3t}=1000\Rightarrow e^{0.3t}=10$ $t=\frac{\ln 10}{0.3}=7\text{ years }8\text{ months}$

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Edexcel A-Level Mathematics A2 P3.3.2 Function Ln X and Its Graph Questions question 1