You and a friend are examining a sample from an ancient civilization.In this point sample, 2.5g of the original 10g of carbon-14 remain. Since C-14 has a half-life of 5730 years , your friend says this sample is 22,920 years old, since only 1/4 of the original sample remains it must have gone though 4 half lives. What error is your friend making? Your friend does not recognize that half-life is exponential decay, and that two half- lives have elapsed and thus the sample is actually 11460 years old Your friend has incorrectly counted half lives, and the sample is only 5730 years old. Your friend is wrong because radioactive carbon 14 will take half as long to decay with each half life and two half-lives have passed, so the sample is 8595 years old. Your friend has not made an error because only 1/4 of the sample remains, so 4 half- lives have passed.

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You and a friend are examining a sample from an ancient civilization.In this point sample, 2.5g of the original 10g of carbon-14 remain. Since C-14 has a half-life of 5730 years , your friend says this sample is 22,920 years old, since only 1/4 of the original sample remains it must have gone though 4 half lives. What error is your friend making? Your friend does not recognize that half-life is exponential decay, and that two half- lives have elapsed and thus the sample is actually 11460 years old Your friend has incorrectly counted half lives, and the sample is only 5730 years old. Your friend is wrong because radioactive carbon 14 will take half as long to decay with each half life and two half-lives have passed, so the sample is 8595 years old. Your friend has not made an error because only 1/4 of the sample remains, so 4 half- lives have passed.

Answer 1

Your friend does not recognize that half-life is exponential decay, and that two half-lives have elapsed and thus the sample is actually 11460 years old.

Answer 2

## Step1: Understand the concept of half-life. Half-life is the time required for a quantity to reduce to half of its initial value. This is a characteristic of exponential decay. ## Step2: Identify the given values. The original amount of carbon-14 is 10g, the remaining amount is 2.5g, and the half-life of carbon-14 is 5730 years. ## Step3: Determine the number of half-lives that have passed. After one half-life, the original amount of 10g would have reduced to 5g. After another half-life, this would reduce further to 2.5g. Therefore, two half-lives have passed. ## Step4: Calculate the total time that has passed by multiplying the number of half-lives by the length of a single half-life. ### \(Total\ time = Number\ of\ half-lives \times Length\ of\ a\ single\ half-life = 2 \times 5730\ years = 11460\ years\) ## Step5: Identify the error made by your friend. Your friend incorrectly counted the number of half-lives, assuming that 1/4 of the original sample remaining meant 4 half-lives have passed. However, we have calculated that only two half-lives have passed.

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