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Advanced Math / Nonlinear functions Difficulty: Hard

$p left parenthesis t right parenthesis equals 90,000 left parenthesis 1.06 right parenthesis Superscript t$

The given function $p$ models the population of Lowell $t$ years after a census. Which of the following functions best models the population of Lowell $m$ months after the census?

$r (m) = \frac{90,000}{12} {(1.06)}^{m}$

$r (m) = 90,000 {(\frac{1.06}{12})}^{m}$

$r (m) = 90,000 {(\frac{1.06}{12})}^{\frac{m}{12}}$

$r (m) = 90,000 {(1.06)}^{\frac{m}{12}}$

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Explanation

Choice D is correct. It’s given that the function $p$ models the population of Lowell $t$ years after a census. Since there are $12$ months in a year, $m$ months after the census is equivalent to $StartFraction m Over 12 EndFraction$ years after the census. Substituting $StartFraction m Over 12 EndFraction$ for $t$ in the equation $p left parenthesis t right parenthesis equals 90,000 left parenthesis 1.06 right parenthesis Superscript t$ yields $p (\frac{m}{12}) = 90,000 {(1.06)}^{\frac{m}{12}}$ . Therefore, the function $r$ that best models the population of Lowell $m$ months after the census is $r (m) = 90,000 {(1.06)}^{\frac{m}{12}}$ .

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.