On a remote island, there are only two species of animals: foxes and rabbits. The foxes are the predators and the rabbits are their prey.
The populations of foxes and rabbits increase and decrease in a periodic pattern, with the period of both populations being the same, as shown in the graph below, for all `t \geq 0`, where time `t` is measured in weeks.
One point of minimum fox population, (20, 700), and one point of maximum fox population, (100, 2500), are also shown on the graph.
The graph has been drawn to scale.
The population of rabbits can be modelled by the rule `r(t)=1700 \sin \left(\frac{\pi t}{80}\right)+2500`.
- i. State the initial population of rabbits. (1 mark)
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- ii. State the minimum and maximum population of rabbits. (1 mark)
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- iii. State the number of weeks between maximum populations of rabbits. (1 mark)
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The population of foxes can be modelled by the rule `f(t)=a \sin (b(t-60))+1600`.
- Show that `a=900` and `b=\frac{\pi}{80}`. (2 marks)
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- Find the maximum combined population of foxes and rabbits. Give your answer correct to the nearest whole number. (1 mark)
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- What is the number of weeks between the periods when the combined population of foxes and rabbits is a maximum? (1 mark)
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The population of foxes is better modelled by the transformation of `y=\sin (t)` under `Q` given by
- Find the average population during the first 300 weeks for the combined population of foxes and rabbits, where the population of foxes is modelled by the transformation of `y=\sin(t)` under the transformation `Q`. Give your answer correct to the nearest whole number. (4 marks)
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Over a longer period of time, it is found that the increase and decrease in the population of rabbits gets smaller and smaller.
The population of rabbits over a longer period of time can be modelled by the rule
- Find the average rate of change between the first two times when the population of rabbits is at a maximum. Give your answer correct to one decimal place. (2 marks)
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- Find the time, where `t>40`, in weeks, when the rate of change of the rabbit population is at its greatest positive value. Give your answer correct to the nearest whole number. (2 marks)
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- Over time, the rabbit population approaches a particular value.
- State this value. (1 mark)
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