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Carbon dating involves the measurement of the amount of carbon- $\newline$ $14$ in a sample. Archaeologists sometimes use carbon dating to estimate the number of years since an organism died. The function shows the ratio, $R(t)$ , of carbon- $\newline$ $14$ remaining in a sample to the original amount of carbon- $\newline$ $14$ after $t$ years. If one sample was $3$ times as old as another sample of the same material, how would the carbon- $\newline$ $14$ ratios of the samples relate? Choose $1$ answer: $\newline$ (Choice A) The ratio of the older sample would be $3$ times the ratio of the newer sample. $\newline$ (Choice B) The ratio of the newer sample would be $3$ times the ratio of the older sample. $\newline$ (Choice C) The ratio of the older sample would be the cube of the ratio of the newer sample. $\newline$ (Choice D) The ratio of the newer sample would be the cube of the ratio of the older sample.

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Write and solve equations for proportional relationships

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Q. Carbon dating involves the measurement of the amount of carbon-

\newline

14

in a sample. Archaeologists sometimes use carbon dating to estimate the number of years since an organism died. The function shows the ratio,

R(t)

, of carbon-

\newline

14

remaining in a sample to the original amount of carbon-

\newline

14

after

t

years. If one sample was

3

times as old as another sample of the same material, how would the carbon-

\newline

14

ratios of the samples relate? Choose

1

answer:

\newline

(Choice A) The ratio of the older sample would be

3

times the ratio of the newer sample.

\newline

(Choice B) The ratio of the newer sample would be

3

times the ratio of the older sample.

\newline

(Choice C) The ratio of the older sample would be the cube of the ratio of the newer sample.

\newline

(Choice D) The ratio of the newer sample would be the cube of the ratio of the older sample.

Understand Carbon Dating Concept: Understand the concept of carbon dating and the decay of carbon $-14$ . Carbon $-14$ decays over time, and the rate of decay is exponential. This means that the amount of carbon $-14$ in a sample decreases to half its original amount over a period known as the half-life. The function $R(t)$ represents the ratio of carbon $-14$ remaining after $t$ years. Since the decay is exponential, the ratio of carbon $-14$ in the older sample will not be a simple multiple of the ratio in the newer sample.
Analyze Age-Carbon $-14$ Ratio Relationship: Analyze the relationship between the age of the samples and the carbon $-14$ ratio. If one sample is $3$ times as old as another, the amount of carbon $-14$ in the older sample will have gone through more decay cycles than the younger sample. This means that the ratio $R(t)$ for the older sample will be smaller because it has had more time to decay.
Determine Correct Ratio Relationship: Determine the correct relationship between the ratios. $\newline$ Since the decay is exponential, the relationship between the ages of the samples and their respective carbon $-14$ ratios is also exponential. Therefore, the ratio of the older sample will not be $3$ times the ratio of the newer sample, nor will the ratio of the newer sample be $3$ times the ratio of the older sample. Instead, the ratio of the older sample will be a power of the ratio of the newer sample, based on the exponential decay function.
Choose Based on Exponential Decay: Choose the correct answer based on the exponential decay relationship. $\newline$ Given the exponential nature of decay, if the older sample is $3$ times as old as the younger sample, the ratio of the older sample will be a power of the ratio of the newer sample. The correct relationship is that the ratio of the newer sample will be a power of the ratio of the older sample. The power will be related to the number of half-lives that have passed, which is not necessarily a cube relationship unless the half-life specifically corresponds to the age difference being a factor of $3$ . Therefore, the correct answer is not a simple cube relationship, but rather an exponential one that depends on the specific half-life of carbon- $14$ .

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