A biology class at Central High School predicted that a local population of animals will double in size every 12 years. The population at the beginning of 2014 was estimated to be 50 animals. If P represents the population n years after 2014, then which of the following equations represents the class’s model of the population over time?
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Choice D is correct. The student first recognizes that a population that doubles in size over equal time periods is increasing at an exponential rate. An exponential growth model can be written in the form where is the population at time 0, since 2 raised to the zeroth power is 1 and is the doubling time, since when 2 is raised to the first power, and the population will be From the way the variables were defined, the population at time is 50 and the doubling time is 12.
Choice A is not the correct answer. This answer may result from a misunderstanding of exponential equations or of the context. A student who tries to model the scenario with a linear equation or who misunderstands that the y-intercept of a linear model should represent the initial number of animals may produce this equation.
Choice B is not the correct answer. This answer may result from a misunderstanding of exponential equations or of the scenario. A student who tries to model the scenario with a linear equation may produce this equation.
Choice C is not the correct answer. A student who tries to produce an exponential model, but does not understand how the 12 years affects the model, may incorrectly write the exponent.
Students must identify the correct mathematical notation for an exponential relationship that represents a real-world situation.