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Advanced Math Difficulty: Hard

$f (x) = {(1.84)}^{\frac{x}{4}}$

The function $f$ is defined by the given equation. The equation can be rewritten as $f (x) = {(1 + \frac{p}{100})}^{x}$ , where $p$ is a constant. Which of the following is closest to the value of $p$ ?

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Explanation

Choice A is correct. The equation $f (x) = {(1.84)}^{\frac{x}{4}}$ can be rewritten as $f left parenthesis x right parenthesis equals left parenthesis 1.84 right parenthesis Superscript left parenthesis one fourth right parenthesis left parenthesis x right parenthesis$ , which is equivalent to $f left parenthesis x right parenthesis equals left parenthesis 1.84 Superscript one fourth Baseline right parenthesis Superscript x$ , or approximately $f left parenthesis x right parenthesis equals left parenthesis 1.16467 right parenthesis Superscript x$ . Since it's given that $f (x) = {(1.84)}^{\frac{x}{4}}$ can be rewritten as $f (x) = {(1 + \frac{p}{100})}^{x}$ , where $p$ is a constant, it follows that $1 + \frac{p}{100}$ is approximately equal to $1.16467$ . Therefore, $StartFraction p Over 100 EndFraction$ is approximately equal to $0.16467$ . It follows that the value of $p$ is approximately equal to $16.467$ . Of the given choices, $16$ is closest to the value of $p$ .

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

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

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