A model estimates that at the end of each year from 2015 to 2020, the number of squirrels in a population was 150% more than the number of squirrels in the population at the end of the previous year. The model estimates that at the end of 2016, there were 180 squirrels in the population. Which of the following equations represents this model, where n is the estimated number of squirrels in the population t years after the end of 2015 and t ≤ 5?

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Accepted Answer

Choice B is correct. Since the model estimates that the number of squirrels in the population increased by a fixed percentage, 150%, each year, the model can be represented by an exponential equation of the form n = a(1 + p ⁄ 100 ) t , where a is the estimated number of squirrels in the population at the end of 2015, and the model estimates that at the end of each year, the number is p% more than the number at the end of the previous year. Since the model estimates that at the end of each year, the number was 150% more than the number at the end of the previous year, p = 150. Substituting 150 for p in the equation yields n = a(1 + 150 ⁄ 100 ) t , which is equivalent to n = a(1 + 1.5) t , or n = a(2.5) t . It’s given that the estimated number of squirrels at the end of 2016 was 180. This means that when t = 1, n = 180. Substituting 1 for t and 180 for n in the equation n = a(2.5) t yields 180 = a(2.5) 1 , or 180 = 2.5a. Dividing each side of this equation by 2.5 yields 72 = a. Substituting 72 for a in the equation n = a(2.5) t yields n = 72(2.5) t . Choice A is incorrect. This equation represents a model where at the end of each year, the estimated number of squirrels was 150% of, not 150% more than, the estimated number at the end of the previous year. Choice C is incorrect. This equation represents a model where at the end of each year, the estimated number of squirrels was 150% of, not 150% more than, the estimated number at the end of the previous year, and the estimated number of squirrels at the end of 2015, not the end of 2016, was 180. Choice D is incorrect. This equation represents a model where the estimated number of squirrels at the end of 2015, not the end of 2016, was 180.

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