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Foxes and rabbits greenfoot download1/17/2024 When there are many rabbits present, the fox population grows. The foxes depend on the rabbits as a food source. This makes sense in the context of a predator-prey relationship. While the rabbit population decreases during the winter and recovers during the summer, the fox population seems to lag behind by three months. Graphing both functions on the same coordinate axes, we observe the mentioned âchasingâ of the two functions. ![]() Graphing $f(t)$ together with our data points confirms that our function is a good model for the given data. This suggests the use of a negative sine function $r(t) = -A \sin(B(t - D)) + C$, where $A$ is the amplitude, $B = \frac t) + 100$. ![]() Looking at the graph, we observe that the rabbit population has a vertical intercept at its midline and then decreases. Now we just have to decide if we want to use sine or cosine to model the function. We already found the amplitude, midline and period for $r(t)$ in part (a). Looking at the graph of the given points, we again see the general shape of a sine or cosine function with amplitude 50 foxes, midline 100 foxes and period 12 months. The number of foxes shows a similar pattern starting at a maximum of 150, reaching a minimum of 50 after 6 months and returning close to the maximum at the end of the year. Even though we donât have any additional data, it is reasonable to assume that this pattern will repeat itself in the next year, and we are looking at a periodic function with amplitude 500 rabbits, midline 1000 rabbits and period 12 months. Graphing the available points, we see the general shape of a sine or cosine function. The number of rabbits starts at 1000, decreases to a minimum of 500 after 3 months, increases back to 1000 in the next 3 months, reaches a maximum of 1500 after another 3 months and almost decreases to its starting value of 1000 at the end of the year. Looking at the table, we notice a pattern in the number of rabbits and foxes. Different groups might come up with different function formulas, since it is possible to use positive or negative sine or cosine functions with different horizontal shifts for both populations. It lends itself well for students working together in groups and comparing their modeling functions. In that case, students have to decide which data values to use for their model and the modeling function will not be a perfect fit for the data. Such a scenario is implemented in the task F-TF Foxes and Rabbits 3. A more realistic situation, and a good follow-up task to the current one, would be to present students with data points that do not perfectly fit on a trigonometric graph. Indeed, for the given data a sine function can be found which perfectly fits the data, a highly unusual state of affairs. ![]() ![]() The data in the current task has been simplified (it is much too regular to be completely realistic) so that students can come up with a clear workable model. Note that the verb "model" in the task means to find a function which well approximates the data presented in the table. We are now in a position to actually model the data given previously with trigonometric functions and investigate the behavior of this predator-prey situation. The example of rabbits and foxes was introduced in the task ( 8-F Foxes and Rabbits) to illustrate two functions of time given in a table.
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