| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Logistic growth model |
| Difficulty | Standard +0.3 This is a straightforward logistic growth question requiring: (a) substitution of t=0, (b) solving an exponential equation using logarithms (standard technique), and (c) finding the horizontal asymptote by considering the limit as tââ. All parts use routine A-level techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.06g Equations with exponentials: solve a^x = b1.06i Exponential growth/decay: in modelling context |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(P_0 = 300\) | B1 | (1) |
| (b) \(420 = \frac{900e^{0.12t}}{2e^{0.12t}+1} \Rightarrow 60e^{0.12t} = 420\) | M1 A1 | |
| Correct use of lns \(\Rightarrow t = \frac{\ln 7}{0.12} = 16.22\) | dM1 A1 | (4) |
| (c) States that maximum number (upper limit) is 450 so cannot reach 500 | B1 | (1) |
**(a)** $P_0 = 300$ | B1 | (1)
**(b)** $420 = \frac{900e^{0.12t}}{2e^{0.12t}+1} \Rightarrow 60e^{0.12t} = 420$ | M1 A1 |
Correct use of lns $\Rightarrow t = \frac{\ln 7}{0.12} = 16.22$ | dM1 A1 | (4)
**(c)** States that maximum number (upper limit) is 450 so cannot reach 500 | B1 | (1)
**Total: 6 marks**
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\item A population of a rare species of toad is being studied.
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The number of toads, $N$, in the population, $t$ years after the start of the study, is modelled by the equation
$$N = \frac { 900 \mathrm { e } ^ { 0.12 t } } { 2 \mathrm { e } ^ { 0.12 t } + 1 } \quad t \geqslant 0 , t \in \mathbb { R }$$
According to this model,\\
(a) calculate the number of toads in the population at the start of the study,\\
(b) find the value of $t$ when there are 420 toads in the population, giving your answer to 2 decimal places.\\
(c) Explain why, according to this model, the number of toads in the population can never reach 500
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\hfill \mbox{\textit{Edexcel P3 2020 Q1 [6]}}