Calculating Fossil Age Using Carbon-14 Dating

This question involves determining the age of a fossil using the principles of radioactive decay, specifically Carbon-14 ($^{14}C$) dating.

Key information provided:

• Remaining $^{14}C$ in the fossil: $3.125\%$ of the original amount.
• Half-life ($t_{1/2}$) of $^{14}C$: 5730 years.

Radioactive Decay Formula

The amount of a radioactive isotope remaining after time $t$ is given by the formula:

$ N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} $

Where:

• $N(t)$ is the amount remaining at time $t$.
• $N_0$ is the initial amount.
• $t$ is the time elapsed (age of the fossil).
• $t_{1/2}$ is the half-life of the isotope.

We can rewrite the formula in terms of the fraction remaining:

$ \frac{N(t)}{N_0} = \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} $

Determining the Number of Half-Lives

The fossil contains $3.125\%$ of the original $^{14}C$. Convert this percentage to a fraction:

$ \frac{N(t)}{N_0} = \frac{3.125}{100} = 0.03125 $

We need to find how many half-lives correspond to this fraction. Notice that $0.03125$ is a power of $\frac{1}{2}$:

• $(\frac{1}{2})^1 = 0.5$
• $(\frac{1}{2})^2 = 0.25$
• $(\frac{1}{2})^3 = 0.125$
• $(\frac{1}{2})^4 = 0.0625$
• $(\frac{1}{2})^5 = 0.03125$

Therefore, the fraction remaining is $(\frac{1}{2})^5$. Comparing this to the decay formula:

$ \left(\frac{1}{2}\right)^5 = \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} $

This implies that 5 half-lives have passed:

$ \frac{t}{t_{1/2}} = 5 $

Calculating the Fossil's Age

Now, calculate the total time ($t$) using the number of half-lives and the half-life duration:

$ t = 5 \times t_{1/2} $

$ t = 5 \times 5730 \text{ years} $

$ t = 28650 \text{ years} $

Conclusion

The calculated age of the fossil is 28650 years. This value lies between 28000 and 29000 years, consistent with the provided answer range.