Edexcel IAL Mathematics P3.4.4 exponential models

Practise exponential growth and decay models for temperatures, medicines, populations and rates, including initial values and limits.

Exam points

use t = 0 or a given observation to find constants in an exponential model
differentiate an exponential model to calculate a rate at a specified time

[Maximum number: 5]

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).

(a)

Find the upper limit to the number of fish in the lake.

[ 1 ]

(b)

Given that when t=T, $\frac{dN}{dt}=8$ ,
find the value of T to one decimal place.

(Solutions relying entirely on calculator technology are not acceptable.)

[ 4 ]

$8=\frac{360e^{0.3t}}{(2+e^{0.3t})^2}$ $8(e^{0.3t})^2-328(e^{0.3t})+32=0$ $e^{0.3t}=\frac{41+\sqrt{1665}}{2}$ $t=\frac{\ln\left(\frac{41+\sqrt{1665}}{2}\right)}{0.3}$ $T\approx12.4$

M1 dM1 ddM1 A1

Marking guidance:
M1 for setting dN/dt=8 and obtaining a quadratic in $e^{0.3t}$ .
dM1 for a correct attempt to solve the quadratic in $e^{0.3t}$ .
ddM1 for correct log work for $e^{0.3t}=k$ , k>0.
A1 for awrt 12.4.

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P3.4.4 - exponential question 1

Question (a)

Question (b)