A model estimates that at the end of each year from 2015 to 2020 , the number of squirrels in

Question

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 &#8804; 5 ?

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 n = a 1 + p 100 t yields n = a 1 + 150 100 t , which is equivalent to n = a 1 + 1.5 t , or n = a 2.5 t . It&rsquo;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 = 2.5 a . 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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