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Advanced Math / Nonlinear functions Difficulty: Medium

A function $p$ estimates that there were $2,000$ animals in a population in $1998$ . Each year from $1998$ to $2010$ , the function estimates that the number of animals in this population increased by $3 percent sign$ of the number of animals in the population the previous year. Which equation defines this function, where $p (x)$ is the estimated number of animals in the population $x$ years after $1998$ ?

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Explanation

Choice C is correct. It's given that a function $p$ estimates that there were $2,000$ animals in a population in $1998$ and that each year from $1998$ to $2010$ , the number of animals in this population increased by $3 percent sign$ of the number of animals in the population the previous year. It follows that this situation can be represented by the function $p (x) = a {(1 + \frac{r}{100})}^{x}$ , where $p (x)$ is the estimated number of animals in the population $x$ years after $1998$ , $a$ is the estimated number of animals in the population in $1998$ , and each year the estimated number of animals increased by $r %$ . Substituting $2,000$ for $a$ and $3$ for $r$ in this function yields $p (x) = 2,000 {(1 + \frac{3}{100})}^{x}$ , or $p left parenthesis x right parenthesis equals 2,000 left parenthesis 1.03 right parenthesis Superscript x$ .

Choice A is incorrect. This function represents a population in which each year the number of animals increased by $200 percent sign$ , not $3 percent sign$ , of the number of animals in the population the previous year.

Choice B is incorrect. This function represents a population in which each year the number of animals increased by $97 percent sign$ , not $3 percent sign$ , of the number of animals in the population the previous year.

Choice D is incorrect. This function represents a population in which each year the number of animals decreased, rather than increased, by $3 percent sign$ of the number of animals in the population the previous year.