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4) The charred bones of a sloth in a cave in Chile represent the earliest evidence of human presence in the
southern tip of South America. A sample of the bone has a C-14 activity of 5.22 disintegrations per minute
per gram of carbon. If the C-14 activity in living organisms has an activity of 15.3 disintegrations per minute
per gram of carbon, how old are the bones? (the half-life of C-14 is 5730 years).

Respuesta :

Edufirst

Answer:

  • about 8,880 years old

Explanation:

Half-life is the number of years it takes for a quantity of a substance to decompose in half.

You may measure the amount of the substance in mass, concentration, or activity, among others.

Thus, if you start with an amount A₀ of a substance that decays with a contstan half-life, the amount, A, remaining after passing n half-lives will be:

       [tex]A=A_0\times \bigg(\dfrac{1}{2}\bigg)^n[/tex]

Then, if you know the initial amount,A₀, and the current amount remaining, A, of a substance, you can calculate the number of half-lives elapsed.

In this case:

  • A = 5.22 disintegrations/minute·gram
  • A₀ = 15.3 disintegratons/minut ·gram (when the organism was alive)

Thus:

  • (5.22/15.3) = 0.34117647

  • 0.34117647 = (0.5)ⁿ

  • n × log(0.5) = log(0.34117647)

  • n = log(0.34117647)/log(0.5)

  • n = 1.5514

Hence, about 1.55 half-lives have elapsed since the organism died.

Since the half-life of C-14 is 5,730 years, the bones are 1.55 × 5,730 = 8,881.5 ≈ 8,880 years old ← answer.

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