In an experiment, the population of insects, , was modelled by the logistic differential equation

where is the time in days after the beginning of the experiment.

The diagram shows a direction field for this differential equation, with the point representing the initial population.
 

1. Explain why the graph of the solution that passes through the point cannot also pass through the point .   (1 mark)
   

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2. Clearly sketch the graph of the solution that passes through the point .   (1 mark)
   

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3. Find the predicted value of the population, , at which the rate of growth of the population is largest.   (2 marks)
   

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A researcher estimates the number of brumbies in a National Park after years can be modelled by the equation

1. Sketch the function over the first four years of the research.   (2 marks)
   

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2. Calculate when the brumby population should reach 13 000, giving your answer to 2 decimal places.   (1 mark)
   

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3. Show that     (2 marks)
   

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4. What is the maximum growth rate of the brumby population?   (3 marks)
   

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The population of Myna birds in a national park is decreasing at a rate proportional to the population at that time.

1. Write a differential equation that describes the situation.   (1 mark)
   

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2. 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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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 km can be modelled by a differential equation, where where denotes the time in months.

What does this model predict for the radius of the region affected by the pest after months?   (3 marks)

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    where   is the number of years after 1980 and is the carrying capacity.

In the year 2000, the population of deer was estimated to be 600 000.

Use the fact that    to show that the carrying capacity is approximately 1 130 000.  (4 marks)

Bacteria are spreading over a Petri dish at a rate modelled by the differential equation

where    is the proportion of the dish covered after    hours.

Given 

1. Show by integration that  , where    is a constant of integration.  (2 marks)
   

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2. If half of the Petri dish is covered by the bacteria at  , express    in terms of  .  (2 marks)
   

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