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The problem is asking for an expression that represents the remaining amount of carbon-14, \( A \), after \( n \) years, given that carbon-14 has a half-life of 5730 years, and the fossil originally contained 572 grams.

### Step-by-Step Process:
The formula to calculate the amount of a substance left after a given time, based on its half-life, is given by the exponential decay formula:

\[
A(n) = A_0 \left( \frac{1}{2} \right)^{\frac{n}{T}}
\]

Where:
- \( A(n) \) is the amount of carbon-14 remaining after \( n \) years.
- \( A_0 \) is the initial amount of carbon-14 (572 grams in this case).
- \( T \) is the half-life of carbon-14 (5730 years in this case).
- \( n \) is the number of years.

### Plugging in the values:
- \( A_0 = 572 \) grams.
- \( T = 5730 \) years.

Thus, the expression for \( A \) becomes:

\[
A(n) = 572 \left( \frac{1}{2} \right)^{\frac{n}{5730}}
\]

This is the expression that represents the amount of carbon-14 remaining in the fossil after \( n \) years.

Would you like further details or have any questions?

Here are some related questions:
1. How do we derive the exponential decay formula for half-life problems?
2. Can this method be applied to other isotopes with different half-lives?
3. How much carbon-14 remains after 10,000 years?
4. What does it mean when the amount of carbon-14 becomes negligible?
5. How does this decay model help in determining the age of fossils?

**Tip**: When solving half-life problems, always ensure you're using the correct time period for the specific substance's half-life.

Write an expression for A to represent the amount of carbon-14 remaining in the fossil after n years.

Explore how to calculate the remaining amount of carbon-14 in a fossil after a given number of years. This problem uses the exponential decay formula and the half-life of 5730 years to find the amount of carbon-14. Suitable for Grades 9-12 students.

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