In an experiment, the population of insects, \(P(t)\), was modelled by the logistic differential equation \(\dfrac{d P}{d t}=P(2000-P)\) where \(t\) is the time in days after the beginning of the experiment. The diagram shows a direction field for this differential equation, with the point \(S\) representing the initial population. --- 2 WORK AREA LINES (style=lined) --- --- 5 WORK AREA LINES (style=blank) --- --- 4 WORK AREA LINES (style=lined) --- i. \(\text{Any solution that passes through}\ S\ \text{will follow the}\) \(\text{slope field (not crossing any lines) and approach a horizontal}\) \(\text{asymptote at}\ P=2000\ \text{from the lower side}\ (P<2000).\) ii. iii. \(P= 1000\) i. \(\text{Any solution that passes through}\ S\ \text{will follow the}\) \(\text{slope field (not crossing any lines) and approach a horizontal}\) \(\text{asymptote at}\ P=2000\ \text{from the lower side}\ (P<2000).\). ii. iii. \(\dfrac{d P}{d t}=P(2000-P)\) \(\text {Find \(P\) where \(\dfrac{d P}{d t}\) is a maximum.}\) \(\text{Consider the graph}\ \ y=P(2000-P): \) \(\Rightarrow \ \text {Graph is a concave down quadratic cutting the axis at}\ \ P=0\ \ \text{and}\ \ P=2000\) \(\Rightarrow \ \text{Max value of}\ \ P(2000-P)\ \ \Big(\text{i.e.}\ \dfrac{dP}{dt}\Big)\ \ \text{occurs at}\ \ P=1000\ \text{(axis of symmetry).}\)
Calculus, EXT1 C3 EQ-Bank 8
A researcher estimates the number of brumbies in a National Park after `t` years can be modelled by the equation
`B(t)=(18\ 000)/(1+4e^(-t))`
- Sketch the function `B(t)` over the first four years of the research. (2 marks)
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- Calculate when the brumby population should reach 13 000, giving your answer to 2 decimal places. (1 mark)
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- Show that `B^(′)(t)=(72\ 000e^t)/(e^t+4)^2` (2 marks)
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- What is the maximum growth rate of the brumby population? (3 marks)
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Show Answers Only
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- `2.34\ text{years}`
- `text{Proof (See Worked Solutions}`
- `4500\ text{brumbies per year}`
Show Worked Solution
i.
ii. `text {Find}\ t\ text{when}\ \ B(t)=13\ 000`
| `13\ 000` | `=(18\ 000)/(1+4e^(-t))` | |
| `13\ 000(1+4e^(-t))` | `=18\ 000` | |
| `1+4e^(-t)` | `=18/13` | |
| `4e^(-t)` | `=5/13` | |
| `e^(-t)` | `=5/52` | |
| `-t` | `=ln(5/52)` | |
| `t` | `=2.34\ text{years (2 d.p.)}` |
iii. `text{Show}\ \ B^(′)(t)=(72\ 000e^t)/(e^t+4)^2`
`B(t)=18\ 000(1+4e^(-t))^(-1)`
| `B^(′)(t)` | `=-1*-1*4e^(-t)*18\ 000(1+4e^(-t))^-2` | |
| `=(72\ 000)/(e^t(1+4e^(-t))^2)` | ||
| `=(72\ 000)/(e^t(1+4/e^t)^2)` | ||
| `=(72\ 000)/(e^t((e^t+4)/e^t)^2)` | ||
| `=(72\ 000)/(e^t/(e^t)^2*(e^t+4)^2)` | ||
| `=(72\ 000e^t)/(e^t+4)^2\ \ text{… as required}` |
iv. `B^(′)(t)=72\ 000e^t(e^t+4)^(-2)`
`text{Using product rule:}`
| `B^(′′)(t)` | `=72\ 000e^t(e^t+4)^(-2)+(-2e^t)(e^t+4)^(-3)72\ 000e^t` | |
| `=72\ 000e^t(1/(e^t+4)^2-(2e^t)/(e^t+4)^3)` | ||
| `=72\ 000e^t((e^t+4-2e^t)/(e^t+4)^3)` | ||
| `=72\ 000e^t((4-e^t)/(e^t+4)^3)` |
`text{Find}\ t\ text{when}\ \ B^(′′)(t)=0:`
| `4-e^t` | `=0` | |
| `e^t` | `=4` | |
| `t` | `=ln4` | |
| `=1.386…\ text{years}` |
`text{Checking concavity changes:}`
`text{Since}\ e^t>0, (e^t+4)^3>0\ \ text{for all}\ t:`
`text{At}\ t=1, 4-e^1=1.28>0\ \ =>\ \ B^(′′)(1)>0`
`text{At}\ t=2, 4-e^2=-3.4<0\ \ =>\ \ B^(′′)(2)<0`
`B^(′)(ln4)=\ text{Max growth rate}`
| `B^(′)(ln4)` | `=(72\ 000e^(ln4))/(e^(ln4)+4)^2` | |
| `=(72\ 000xx4)/((4+4)^2)` | ||
| `=4500\ text{brumbies per year}` |
Calculus, EXT1 C3 EQ-Bank 6
The population of Myna birds in a national park is decreasing at a rate proportional to the population at that time.
- Write a differential equation that describes the situation. (1 mark)
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- If the population was originally 1300 and decreased to 1040 after 5 years, find the expected population after 10 years. (3 marks)
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Calculus, EXT1 C3 EQ-Bank 4
A varroa virus is infecting commercial beehives in a regional NSW town.
All infected hives detected so far lie within a circular region with radius 16 km and researchers believe that the increase of the radius `r` km can be modelled by a differential equation, where `(dr)/(dt)=2/5sqrtr` where `t` denotes the time in months.
What does this model predict for the radius of the region affected by the pest after `t` months? (3 marks)
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Calculus, EXT1 C3 2021 HSC 14b
In a certain country, the population of deer was estimated in 1980 to be 150 000.
The population growth is given by the logistic equation `(dP)/(dt) = 0.1P((C - P)/C)` where `t` is the number of years after 1980 and `C` is the carrying capacity.
In the year 2000, the population of deer was estimated to be 600 000.
Use the fact that `C/(P(C - P)) = 1/P + 1/(C - P)` to show that the carrying capacity is approximately 1 130 000. (4 marks)
Calculus, EXT1 C3 SM-Bank 5
Bacteria are spreading over a Petri dish at a rate modelled by the differential equation
`(dP)/(dt) = P/2 (1 - P),\ 0 < P < 1`
where `P` is the proportion of the dish covered after `t` hours.
Given `2/(P(1 - P)) = 2/P + 2/(1 - P),`
- Show by integration that `(t - c)/2= log_e(P/(1 - P))`, where `c` is a constant of integration. (2 marks)
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- If half of the Petri dish is covered by the bacteria at `t = 0`, express `P` in terms of `t`. (2 marks)
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