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Advanced Math / Nonlinear functions Difficulty: Medium

A scientist initially measures $12,000$ bacteria in a growth medium. $4$ hours later, the scientist measures $24,000$ bacteria. Assuming exponential growth, the formula $upper P equals upper C left parenthesis 2 right parenthesis Superscript r t$ gives the number of bacteria in the growth medium, where $r$ and $C$ are constants and $P$ is the number of bacteria $t$ hours after the initial measurement. What is the value of $r$ ?

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Explanation

Choice B is correct. It’s given that the formula $upper P equals upper C left parenthesis 2 right parenthesis Superscript r t$ gives the number of bacteria in a growth medium, where $r$ and $C$ are constants and $P$ is the number of bacteria $t$ hours after the initial measurement. It’s also given that a scientist initially measures $12,000$ bacteria in the growth medium. Since the initial measurement is $0$ hours after the initial measurement, it follows that when $t equals 0$ , $upper P equals 12,000$ . Substituting $0$ for $t$ and $12,000$ for $P$ in the given equation yields $12,000 equals upper C left parenthesis 2 right parenthesis Superscript r left parenthesis 0 right parenthesis$ , or $12,000 equals upper C left parenthesis 2 right parenthesis Superscript 0$ , which is equivalent to $12,000 equals upper C$ . It’s given that $4$ hours later, the scientist measures $24,000$ bacteria, or when $t equals 4$ , $upper P equals 24,000$ . Substituting $4$ for $t$ , $24,000$ for $P$ , and $12,000$ for $C$ in the given equation yields $24,000 equals 12,000 left parenthesis 2 right parenthesis Superscript 4 r$ . Dividing each side of this equation by $12,000$ yields $2 equals 2 Superscript 4 r$ , or $2 Superscript 1 Baseline equals 2 Superscript 4 r$ , which is equivalent to $1 equals 4 r$ . Dividing both sides of this equation by $4$ yields $\frac{1}{4} = r$ . Therefore, the value of $r$ is $\frac{1}{4}$ .

Choice A is incorrect. This is the value of the reciprocal of $C$ .

Choice C is incorrect. This is the value of the reciprocal of $r$ .

Choice D is incorrect. This is the value of $C$ .