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Advanced Math / Nonlinear functions Difficulty: Hard

$upper P left parenthesis t right parenthesis equals 260 left parenthesis 1.04 right parenthesis Superscript left parenthesis six fourths right parenthesis t$

The function $P$ models the population, in thousands, of a certain city $t$ years after $2003$ . According to the model, the population is predicted to increase by $4 percent sign$ every $n$ months. What is the value of $n$ ?

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Explanation

Choice A is correct. It’s given that the function $P$ models the population, in thousands, of a certain city $t$ years after $2003$ . The value of the base of the given exponential function, $1.04$ , corresponds to an increase of $4 percent sign$ for every increase of $1$ in the exponent, $(\frac{6}{4}) t$ . If the exponent is equal to $0$ , then $left parenthesis six fourths right parenthesis t equals 0$ . Multiplying both sides of this equation by $(\frac{4}{6})$ yields $t equals 0$ . If the exponent is equal to $1$ , then $left parenthesis six fourths right parenthesis t equals 1$ . Multiplying both sides of this equation by $(\frac{4}{6})$ yields $t = \frac{4}{6}$ , or $t = \frac{2}{3}$ . Therefore, the population is predicted to increase by $4 percent sign$ every $\frac{2}{3}$ of a year. It’s given that the population is predicted to increase by $4 percent sign$ every $n$ months. Since there are $12$ months in a year, $\frac{2}{3}$ of a year is equivalent to $left parenthesis two thirds right parenthesis left parenthesis 12 right parenthesis$ , or $8$ , months. Therefore, the value of $n$ is $8$ .

Choice B is incorrect. This is the number of months in which the population is predicted to increase by $4 percent sign$ according to the model $upper P left parenthesis t right parenthesis equals 260 left parenthesis 1.04 right parenthesis Superscript t$ , not $upper P left parenthesis t right parenthesis equals 260 left parenthesis 1.04 right parenthesis Superscript left parenthesis six fourths right parenthesis t$ .

Choice C is incorrect. This is the number of months in which the population is predicted to increase by $4 percent sign$ according to the model $upper P left parenthesis t right parenthesis equals 260 left parenthesis 1.04 right parenthesis Superscript left parenthesis four sixths right parenthesis t$ , not $upper P left parenthesis t right parenthesis equals 260 left parenthesis 1.04 right parenthesis Superscript left parenthesis six fourths right parenthesis t$ .

Choice D is incorrect. This is the number of months in which the population is predicted to increase by $4 percent sign$ according to the model $upper P left parenthesis t right parenthesis equals 260 left parenthesis 1.04 right parenthesis Superscript left parenthesis one sixth right parenthesis t$ , not $upper P left parenthesis t right parenthesis equals 260 left parenthesis 1.04 right parenthesis Superscript left parenthesis six fourths right parenthesis t$ .