Questions: Of the coast of Florida has been found fossil shark teeth that have belonged to tens of thousands of years extinct ancestor of the white shark. How old is a shark's tooth, when the concentration of the radioactive isotope carbon-14 has decreased to 20.7 percent of the original concentration of the radioactive isotope carbon-14 from the living organism? The half-life of the radioactive isotope carbon-14 is about 5730 years. Answer tolerance limits have been set ± 1 a.

Solution Steps

We need to determine the age of a shark's tooth based on the decay of carbon-14. The concentration of carbon-14 has decreased to 20.7% of its original value. The half-life of carbon-14 is 5730 years. We will use the exponential decay formula to find the age.

The formula for exponential decay is given by:

\[ N(t) = N_0 \cdot e^{-\lambda t} \]

where:

• \( N(t) \) is the remaining quantity of the substance after time \( t \),
• \( N_0 \) is the initial quantity of the substance,
• \( \lambda \) is the decay constant,
• \( t \) is the time elapsed.

The decay constant \( \lambda \) is related to the half-life \( T_{1/2} \) by the formula:

\[ \lambda = \frac{\ln(2)}{T_{1/2}} \]

Substituting the given half-life:

\[ \lambda = \frac{\ln(2)}{5730} \approx 0.000121 \]

We know that the concentration has decreased to 20.7% of the original, so:

\[ \frac{N(t)}{N_0} = 0.207 \]

Substitute into the decay formula:

\[ 0.207 = e^{-\lambda t} \]

Take the natural logarithm of both sides to solve for \( t \):

\[ \ln(0.207) = -\lambda t \]

\[ t = \frac{\ln(0.207)}{-\lambda} \]

Substitute the value of \( \lambda \):

\[ t = \frac{\ln(0.207)}{-0.000121} \approx 13,500 \]

Final Answer