Logistic growth model

1. A population of a rare species of toad is being studied.

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,

1. calculate the number of toads in the population at the start of the study,
2. find the value of \(t\) when there are 420 toads in the population, giving your answer to 2 decimal places.
3. Explain why, according to this model, the number of toads in the population can never reach 500

4. The growth of a weed on the surface of a pond is being studied. The surface area of the pond covered by the weed, \(A \mathrm {~m} ^ { 2 }\), is modelled by the equation $$A = \frac { 80 p \mathrm { e } ^ { 0.15 t } } { p \mathrm { e } ^ { 0.15 t } + 4 }$$ where \(p\) is a positive constant and \(t\) is the number of days after the start of the study.
Given that

• \(30 \mathrm {~m} ^ { 2 }\) of the surface of the pond was covered by the weed at the start of the study
• \(50 \mathrm {~m} ^ { 2 }\) of the surface of the pond was covered by the weed \(T\) days after the start of the study
  1. show that \(p = 2.4\)
  2. find the value of \(T\), giving your answer to one decimal place.
     (Solutions relying entirely on graphical or numerical methods are not acceptable.)

The weed grows until it covers the surface of the pond.

Find, according to the model, the maximum possible surface area of the pond.

10. A population of insects is being studied. The number of insects, \(N\), in the population, is modelled by the equation $$N = \frac { 300 } { 3 + 17 \mathrm { e } ^ { - 0.2 t } } \quad t \in \mathbb { R } , t \geqslant 0$$ where \(t\) is the time, in weeks, from the start of the study.
Using the model,

1. find the number of insects at the start of the study,
2. find the number of insects when \(t = 10\),
3. find the time from the start of the study when there are 82 insects. (Solutions based entirely on graphical or numerical methods are not acceptable.)
4. Find, by differentiating, the rate, measured in insects per week, at which the number of insects is increasing when \(t = 5\). Give your answer to the nearest whole number.

1. A particular species of orchid is being studied. The population \(p\) at time \(t\) years after the study started is assumed to be

$$p = \frac { 2800 a \mathrm { e } ^ { 0.2 t } } { 1 + a \mathrm { e } ^ { 0.2 t } } , \text { where } a \text { is a constant. }$$ Given that there were 300 orchids when the study started,

1. show that \(a = 0.12\),
2. use the equation with \(a = 0.12\) to predict the number of years before the population of orchids reaches 1850.
3. Show that \(p = \frac { 336 } { 0.12 + \mathrm { e } ^ { - 0.2 t } }\).
4. Hence show that the population cannot exceed 2800.

8. A rare species of primrose is being studied. The population, \(P\), of primroses at time \(t\) years after the study started is modelled by the equation $$P = \frac { 800 \mathrm { e } ^ { 0.1 t } } { 1 + 3 \mathrm { e } ^ { 0.1 t } } , \quad t \geqslant 0 , \quad t \in \mathbb { R }$$

1. Calculate the number of primroses at the start of the study.
2. Find the exact value of \(t\) when \(P = 250\), giving your answer in the form \(a \ln ( b )\) where \(a\) and \(b\) are integers.
3. Find the exact value of \(\frac { \mathrm { d } P } { \mathrm {~d} t }\) when \(t = 10\). Give your answer in its simplest form.
4. Explain why the population of primroses can never be 270

1. A scientist is studying a population of mice on an island.

The number of mice, \(N\), in the population, \(t\) months after the start of the study, is modelled by the equation $$N = \frac { 900 } { 3 + 7 \mathrm { e } ^ { - 0.25 t } } , \quad t \in \mathbb { R } , \quad t \geqslant 0$$

1. Find the number of mice in the population at the start of the study.
2. Show that the rate of growth \(\frac { \mathrm { d } N } { \mathrm {~d} t }\) is given by \(\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N ( 300 - N ) } { 1200 }\) The rate of growth is a maximum after \(T\) months.
3. Find, according to the model, the value of \(T\). According to the model, the maximum number of mice on the island is \(P\).
4. State the value of \(P\).

4. A population of deer is introduced into a park. The population \(P\) at \(t\) years after the deer have been introduced is modelled by \(P = \frac { 2000 a ^ { t } } { 4 + a ^ { t } }\), where \(a\) is a constant. Given that there are 800 deer in the park after 6 years,

1. calculate, to 4 decimal places, the value of \(a\),
2. use the model to predict the number of years needed for the population of deer to increase from 800 to 1800.
3. With reference to this model, give a reason why the population of deer cannot exceed 2000.

7 A biologist is investigating the growth of a population of a species of rodent. The biologist proposes the model $$N = \frac { 500 } { 1 + 9 \mathrm { e } ^ { - \frac { t } { 8 } } }$$ for the number of rodents, \(N\), in the population \(t\) weeks after the start of the investigation. Use this model to answer the following questions.

1. 1. Find the size of the population at the start of the investigation.
   2. Find the size of the population 24 weeks after the start of the investigation. your answer to the nearest whole number.
   3. Find the number of weeks that it will take the population to reach 400 . Give your answer in the form \(t = r \ln s\), where \(r\) and \(s\) are integers.
2. 1. Show that the rate of growth, \(\frac { \mathrm { d } N } { \mathrm {~d} t }\), is given by $$\frac { \mathrm { d } N } { \mathrm {~d} t } = \frac { N } { 4000 } ( 500 - N )$$
   2. The maximum rate of growth occurs after \(T\) weeks. Find the value of \(T\).

9. A population of deer is introduced into a park. The population \(P\) at \(t\) years after the deer have been introduced is modelled by $$P = \frac { 2000 a ^ { t } } { 4 + a ^ { t } }$$ where \(a\) is a constant. Given that there are 800 deer in the park after 6 years,

1. calculate, to 4 decimal places, the value of \(a\),
2. use the model to predict the number of years needed for the population of deer to increase from 800 to 1800.
3. With reference to this model, give a reason why the population of deer cannot exceed 2000.
   [0pt] [P2 June 2001 Question 6]

7. A particular species of orchid is being studied. The population \(p\) at time \(t\) years after the study started is assumed to be $$p = \frac { 2800 a \mathrm { e } ^ { 0.2 t } } { 1 + a \mathrm { e } ^ { 0.2 t } } , \text { where } a \text { is a constant. }$$ Given that there were 300 orchids when the study started,

1. show that \(a = 0.12\),
2. use the equation with \(a = 0.12\) to predict the number of years before the population of orchids reaches 1850 .
3. Show that \(p = \frac { 336 } { 0.12 + \mathrm { e } ^ { - 0.2 t } }\).
4. Hence show that the population cannot exceed 2800.

1. A particular species of orchid is being studied. The population \(p\) at time \(t\) years after the study started is assumed to be

$$p = \frac { 2800 a \mathrm { e } ^ { 0.2 t } } { 1 + a \mathrm { e } ^ { 0.2 t } } , \text { where } a \text { is a constant. }$$ Given that there were 300 orchids when the study started,

1. show that \(a = 0.12\),
2. use the equation with \(a = 0.12\) to predict the number of years before the population of orchids reaches 1850.
3. Show that \(p = \frac { 336 } { 0.12 + \mathrm { e } ^ { - 0.2 t } }\).
4. Hence show that the population cannot exceed 2800.
   \end{table} Turn over
   1. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-014_689_766_276_594}
   \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x ) , - 5 \leqslant x \leqslant 5\).
   The point \(M ( 2,4 )\) is the maximum turning point of the graph.
   Sketch, on separate diagrams, the graphs of
5. \(y = \mathrm { f } ( x ) + 3\),
6. \(y = | \mathrm { f } ( x ) |\),
7. \(y = \mathrm { f } ( | x | )\). Show on each graph the coordinates of any maximum turning points.
   1. Express
   $$\frac { 2 x ^ { 2 } + 3 x } { ( 2 x + 3 ) ( x - 2 ) } - \frac { 6 } { x ^ { 2 } - x - 2 }$$ as a single fraction in its simplest form.
   3. The point \(P\) lies on the curve with equation \(y = \ln \left( \frac { 1 } { 3 } x \right)\). The \(x\)-coordinate of \(P\) is 3 . Find an equation of the normal to the curve at the point \(P\) in the form \(y = a x + b\), where \(a\) and \(b\) are constants.
   (5)
   4. (a) Differentiate with respect to \(x\) (i) \(x ^ { 2 } \mathrm { e } ^ { 3 x + 2 }\),
   (ii) \(\frac { \cos \left( 2 x ^ { 3 } \right) } { 3 x }\).
8. Given that \(x = 4 \sin ( 2 y + 6 )\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\).
   5. $$f ( x ) = 2 x ^ { 3 } - x - 4$$
9. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as $$x = \sqrt { \left( \frac { 2 } { x } + \frac { 1 } { 2 } \right) }$$ The equation \(2 x ^ { 3 } - x - 4 = 0\) has a root between 1.35 and 1.4.
10. Use the iteration formula $$x _ { n + 1 } = \sqrt { } \left( \frac { 2 } { x _ { n } } + \frac { 1 } { 2 } \right)$$ with \(x _ { 0 } = 1.35\), to find, to 2 decimal places, the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\). The only real root of \(\mathrm { f } ( x ) = 0\) is \(\alpha\).
11. By choosing a suitable interval, prove that \(\alpha = 1.392\), to 3 decimal places.
    6. $$f ( x ) = 12 \cos x - 4 \sin x$$ Given that \(\mathrm { f } ( x ) = R \cos ( x + \alpha )\), where \(R \geqslant 0\) and \(0 \leqslant \alpha \leqslant 90 ^ { \circ }\),
12. find the value of \(R\) and the value of \(\alpha\).
13. Hence solve the equation $$12 \cos x - 4 \sin x = 7$$ for \(0 \leqslant x < 360 ^ { \circ }\), giving your answers to one decimal place.
14. 1. Write down the minimum value of \(12 \cos x - 4 \sin x\).
    2. Find, to 2 decimal places, the smallest positive value of \(x\) for which this minimum value occurs. \includegraphics[max width=\textwidth, alt={}, center]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-021_60_35_2669_1853}
       7. (a) Show that
    3. \(\frac { \cos 2 x } { \cos x + \sin x } \equiv \cos x - \sin x , \quad x \neq \left( n - \frac { 1 } { 4 } \right) \pi , n \in \mathbb { Z }\),
    4. \(\frac { 1 } { 2 } ( \cos 2 x - \sin 2 x ) \equiv \cos ^ { 2 } x - \cos x \sin x - \frac { 1 } { 2 }\).
15. Hence, or otherwise, show that the equation $$\cos \theta \left( \frac { \cos 2 \theta } { \cos \theta + \sin \theta } \right) = \frac { 1 } { 2 }$$ can be written as $$\sin 2 \theta = \cos 2 \theta$$
16. Solve, for \(0 \leqslant \theta < 2 \pi\), $$\sin 2 \theta = \cos 2 \theta$$ giving your answers in terms of \(\pi\).

2 A zoologist is investigating the growth of a population of red squirrels in a forest.
She uses the equation \(N = \frac { 200 } { 1 + 9 \mathrm { e } ^ { - \frac { t } { 5 } } }\) as a model to predict the number of squirrels, \(N\), in the population \(t\) weeks after the start of the investigation. What is the size of the squirrel population at the start of the investigation?
Circle your answer.
5
20
40
200