standard_description: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
problem_activity: Task
Given below is a table that gives the populations of foxes and rabbits in a national park over a 12 month period. Note that each value of $t$ corresponds to the beginning of the month and $t = 0$ corresponds to the beginning of January.
{"table": [{"tbody": [{"tr": [{"th": [{"_value": "$t$, month"}], "td": [{"_value": "0"}, {"_value": "1"}, {"_value": "2"}, {"_value": "3"}, {"_value": "4"}, {"_value": "5"}, {"_value": "6"}, {"_value": "7"}, {"_value": "8"}, {"_value": "9"}, {"_value": "10"}, {"_value": "11"}]}, {"th": [{"_value": "$r$, number of rabbits"}], "td": [{"_value": "1000"}, {"_value": "750"}, {"_value": "567"}, {"_value": "500"}, {"_value": "567"}, {"_value": "750"}, {"_value": "1000"}, {"_value": "1250"}, {"_value": "1433"}, {"_value": "1500"}, {"_value": "1433"}, {"_value": "1250"}]}, {"th": [{"_value": "$f$, number of foxes"}], "td": [{"_value": "150"}, {"_value": "143"}, {"_value": "125"}, {"_value": "100"}, {"_value": "75"}, {"_value": "57"}, {"_value": "50"}, {"_value": "57"}, {"_value": "75"}, {"_value": "100"}, {"_value": "125"}, {"_value": "143"}]}]}]}]}
Note that the number of rabbits and the number of foxes are both functions of time.
Explain why it is appropriate to model the number of rabbits and foxes as trigonometric functions of time.
Find an appropriate trigonometric function that models the number of rabbits, $r(t)$, as a function of time, $t$, in months.
Find an appropriate trigonometric function that models the number of foxes, $f(t)$, as a function of time, $t$, in months.
Graph both functions and give one possible explanation why one function seems to “chase” the other function.


aligns: yes