Edexcel IAL Mathematics P3.3.2 natural logarithm graphs

Practise using ln x as the inverse of e^x to solve exponential equations, find exact constants and interpret logarithmic graph features.

Exam points

take ln after isolating an exponential term to find times or distances

[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.

$500=\frac{600e^{0.3t}}{2+e^{0.3t}}\Rightarrow 100e^{0.3t}=1000\Rightarrow e^{0.3t}=10$
M1 A1
$t=\frac{\ln 10}{0.3}=7\text{ years }8\text{ months}$
dM1 A1

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P3.3.2 - function ln x and its graph question 1