Question

Assume that the rate of decay of a radioactive substance is proportional to the amount of the substance present. The​ half-life of a radioactive substance is the time it takes for​ one-half of the substance to disintegrate. Carbon dating is often used to determine the age of a fossil. For​ example, a humanoid skull was found in a cave in South Africa along with the remains of a campfire. Archaeologists believe the age of the skull to be the same age as the campfire. It is determined that only 2​% of the original amount of​ carbon-14 remains in the burnt wood of the campfire. Complete parts​ (a) through​ (e) below.a. Estimate the age of the skull ifthe half-life of carbon-14 is about 5720 years. b. Estimate the age of the skull if the half-life of carbon-14 is about 5620 years. The skull is about_______ years old.

Answer 1

• 32,283 years old
• 31,718 years old

Explanation:

If h is the half-life of the radioactive substance, then the proportion remaining after t years is ...

k = (1/2)^(t/h)

We can find the value of t using logarithms:

log(k) = (t/h)·log(1/2)

t = h·log(k)/log(1/2)

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a) For k = 2% and h = 5720 years, we have ...

t = 5720·log(.02)/log(.5) ≈ 5720·5.64386

t ≈ 32,283 . . . . years old

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b) For k = 2% and h = 5620 years, we have ...

t ≈ 5620·5.64386 ≈ 31,718 . . . . years old

_____

Note the proportionality of age to half-life. Once we found the multiplier corresponding to 2% remaining, we can use that for any estimate of the half-life.