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Advanced Math / Nonlinear functions Difficulty: Medium

Bacteria are growing in a liquid growth medium. There were $300,000$ cells per milliliter during an initial observation. The number of cells per milliliter doubles every $3$ hours. How many cells per milliliter will there be $15$ hours after the initial observation?

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Explanation

Choice D is correct. Let $y$ represent the number of cells per milliliter $x$ hours after the initial observation. Since the number of cells per milliliter doubles every $3$ hours, the relationship between $x$ and $y$ can be represented by an exponential equation of the form $y = a {(b)}^{\frac{x}{k}}$ , where $a$ is the number of cells per milliliter during the initial observation and the number of cells per milliliter increases by a factor of $b$ every $k$ hours. It’s given that there were $300,000$ cells per milliliter during the initial observation. Therefore, $a equals 300,000$ . It’s also given that the number of cells per milliliter doubles, or increases by a factor of $2$ , every $3$ hours. Therefore, $b equals 2$ and $k equals 3$ . Substituting $300,000$ for $a$ , $2$ for $b$ , and $3$ for $k$ in the equation $y = a {(b)}^{\frac{x}{k}}$ yields $y = 300,000 {(2)}^{\frac{x}{3}}$ . The number of cells per milliliter there will be $15$ hours after the initial observation is the value of $y$ in this equation when $x equals 15$ . Substituting $15$ for $x$ in the equation $y = 300,000 {(2)}^{\frac{x}{3}}$ yields $y = 300,000 {(2)}^{\frac{15}{3}}$ , or $y equals 300,000 left parenthesis 2 right parenthesis Superscript 5$ . This is equivalent to $y equals 300,000 left parenthesis 32 right parenthesis$ , or $y equals 9,600,000$ . Therefore, $15$ hours after the initial observation, there will be $9,600,000$ cells per milliliter.

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.