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5.–/1 points SCalcET8 3.8.011. Ask Your Teacher My Notes Question Part Points Submissions Used Scientist can determine the age of ancient objects by a method called radiocarbon dating. The bombardment of the upper atmosphere by cosmic rays converts nitrogen to a radioactive isotope of carbon, 14C, with a half-life of about 5730 years. Vegetation absorbs carbon dioxide through the atmosphere and animal life assimilates 14C through food chains. When a plant or animal dies, it stops replacing its carbon and the amount of 14C begins to decrease through radioactive decay. Therefore, the level of radioactivity must also decay exponentially. A parchment fragment was discovered that had about 74% as much 14C radioactivity as does plant material on Earth today. Estimate the age of the parchment. (Round your answer to the nearest hundred years.) yr

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Answer:

  • 13,200 years

Explanation:

These steps explain how you estimate the age of the parchment:

1) Carbon - 14 half-life: τ = 5730 years

2) Number of half-lives elapsed: n

3) Age of the parchment = τ×n = 5730×n years = 5730n

4) Exponential decay:

The ratio of the final amount of the radioactive isotope C-14 to the initial amount of the same is one half (1/2) raised to the number of half-lives elapsed (n):

  • A / Ao = (1/2)ⁿ

5) The parchment fragment  had about 74% as much C-14 radioactivity as does plant material on Earth today:

  • ⇒ A / Ao = 74% = 0.74

  • ⇒ A / Ao =  0.74 = (1/2)ⁿ

  • ⇒ ln (0.74) = n ln (1/2)        [apply natural logarithm to both sides]

  • ⇒ n = ln (1/2) / ln (0.74)

  • ⇒ n ≈ - 0.693 / ( - 0.301) = 2.30

Hence, 2.30 half-lives have elapsed and the age of the parchment is:

  • τ×n = 5730n = 5730 (2.30) = 13,179 years

  • Round to the nearest hundred years: 13,200 years

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