## Assume that the rate of decay of a radioactive substance is proportional to the amount of the substance present. The half-life of a radioac

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.

## Answers ( )

Answer:Step-by-step 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__

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.