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Advanced Math / Nonlinear functions Difficulty: Hard

A model estimates that at the end of each year from $2015$ to $2020$ , the number of squirrels in a population was $150 percent sign$ 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 $t less than or equals 5$ ?

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Explanation

Choice B is correct. Since the model estimates that the number of squirrels in the population increased by a fixed percentage, $150 percent sign$ , each year, the model can be represented by an exponential equation of the form $n = a {(1 + \frac{p}{100})}^{t}$ , 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 $150 percent sign$ more than the number at the end of the previous year, $p equals 150$ . Substituting $150$ for $p$ in the equation $n = a {(1 + \frac{p}{100})}^{t}$ yields $n = a {(1 + \frac{150}{100})}^{t}$ , which is equivalent to $n = a {(1 + 1.5)}^{t}$ , or $n equals a left parenthesis 2.5 right parenthesis Superscript t$ . It’s given that the estimated number of squirrels at the end of $2016$ was $180$ . This means that when $t equals 1$ , $n equals 180$ . Substituting $1$ for $t$ and $180$ for $n$ in the equation $n equals a left parenthesis 2.5 right parenthesis Superscript t$ yields $180 equals a left parenthesis 2.5 right parenthesis Superscript 1$ , or $180 equals 2.5 a$ . Dividing each side of this equation by $2.5$ yields $72 equals a$ . Substituting $72$ for $a$ in the equation $n equals a left parenthesis 2.5 right parenthesis Superscript t$ yields $n equals 72 left parenthesis 2.5 right parenthesis Superscript t$ .

Choice A is incorrect. This equation represents a model where at the end of each year, the estimated number of squirrels was $150 percent sign$ of, not $150 percent sign$ 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 $150 percent sign$ of, not $150 percent sign$ 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 $180$ .