Carbon-14 is a radioactive substance with a half-life of 5730 years. It is used to
determine the age of ancient objects. A Babylonian cloth fragment now has 40% of the
carbon-14 that it contained originally. How old is the fragment of cloth?

To determine the age of the Babylonian cloth fragment using the remaining percentage of carbon-14, we can use the formula for radioactive decay:

\[ N(t) = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]

where:
- \( N(t) \) is the remaining quantity of the substance after time \( t \),
- \( N_0 \) is the initial quantity of the substance,
- \( T_{1/2} \) is the half-life of the substance,
- \( t \) is the time that has passed.

Given:
- The half-life \( T_{1/2} \) of carbon-14 is 5730 years,
- The remaining carbon-14 is 40% of the original amount.

Let's set up the equation with the known values:
\[ 0.40 = \left( \frac{1}{2} \right)^{\frac{t}{5730}} \]

To solve for \( t \), we take the natural logarithm of both sides:

\[ \ln(0.40) = \ln \left( \left( \frac{1}{2} \right)^{\frac{t}{5730}} \right) \]

Using the logarithm power rule \(\ln(a^b) = b \ln(a)\), we get:

\[ \ln(0.40) = \frac{t}{5730} \ln \left( \frac{1}{2} \right) \]

Now, solve for \( t \):

\[ t = \frac{\ln(0.40)}{\ln(0.5)} \times 5730 \]

Calculate the natural logarithms:

\[ \ln(0.40) \approx -0.91629 \]
\[ \ln(0.5) \approx -0.69315 \]

Now, substitute these values into the equation:

\[ t = \frac{-0.91629}{-0.69315} \times 5730 \]
\[ t \approx 1.32193 \times 5730 \]
\[ t \approx 7576.82 \]

Therefore, the age of the Babylonian cloth fragment is approximately 7577 years.

Would you like more details or have any questions?

Here are 5 related questions you might want to ask:
1. How does the half-life of a substance affect its decay rate?
2. What is the formula for radioactive decay and how is it derived?
3. How accurate is carbon-14 dating for determining the age of ancient objects?
4. What are some other methods used for dating ancient artifacts?
5. Can carbon-14 dating be used for objects of any age?

**Tip:** When using logarithms to solve exponential decay problems, remember to apply the properties of logarithms carefully to simplify the equation correctly.

Learn how to calculate the age of a Babylonian cloth fragment using carbon-14 decay. Understand the radioactive decay formula and logarithmic calculations involved. Suitable for advanced high school students.

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