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Advanced Math / Nonlinear functions Difficulty: Medium

A model predicts that the population of Bergen was $15,000$ in $2005$ . The model also predicts that each year for the next $5$ years, the population $p$ increased by $4 percent sign$ of the previous year's population. Which equation best represents this model, where $x$ is the number of years after $2005$ , for $x less than or equals 5$ ?

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Explanation

Choice D is correct. It's given that a model predicts the population of Bergen in $2005$ was $15,000$ . The model also predicts that each year for the next $5$ years, the population increased by $4 percent sign$ of the previous year's population. The predicted population in one of these years can be found by multiplying the predicted population from the previous year by $1.04$ . Since the predicted population in $2005$ was $15,000$ , the predicted population $1$ year later is $15,000 left parenthesis 1.04 right parenthesis$ . The predicted population $2$ years later is this value times $1.04$ , which is $15,000 left parenthesis 1.04 right parenthesis left parenthesis 1.04 right parenthesis$ , or $15,000 left parenthesis 1.04 right parenthesis squared$ . The predicted population $3$ years later is this value times $1.04$ , or $15,000 left parenthesis 1.04 right parenthesis cubed$ . More generally, the predicted population, $p$ , $x$ years after $2005$ is represented by the equation $p equals 15,000 left parenthesis 1.04 right parenthesis Superscript x$ .

Choice A is incorrect. Substituting $0$ for $x$ in this equation indicates the predicted population in $2005$ was $0.96$ rather than $15,000$ .

Choice B is incorrect. Substituting $0$ for $x$ in this equation indicates the predicted population in $2005$ was $1.04$ rather than $15,000$ .

Choice C is incorrect. This equation indicates the predicted population is decreasing, rather than increasing, by $4 percent sign$ each year.