https://file.aiquickdraw.com/mathbot/file/17271912390087eq2oesf.jpg

The question you have uploaded is about modeling the decay of carbon-14. The problem states that:

- The half-life of carbon-14 is 4,500 years.
- The initial amount of carbon-14 is 20 grams.
- You need to determine the amount of carbon-14 remaining as a function of time \( t \), using an appropriate model.

### Key concept:
The decay of carbon-14 can be modeled using an exponential decay function. The general form of an exponential decay function is:

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

Where:
- \( A \) is the initial amount (in this case, 20 grams),
- \( T \) is the half-life (in this case, 4,500 years),
- \( t \) is the time in years,
- \( f(t) \) is the amount remaining at time \( t \).

So, the correct model for the carbon-14 decay is:

\[
f(t) = 20 \left( \frac{1}{2} \right)^{\frac{t}{4500}}
\]

Looking at the options in your image, the correct answer is the one already selected:

\[
f(t) = 20 \left( \frac{1}{2} \right)^{\frac{t}{4500}}
\]

Would you like further explanation or details on exponential decay?

### Relative questions:
1. What is the difference between exponential decay and exponential growth?
2. How is the half-life related to the exponential decay formula?
3. How would the formula change if the half-life was different?
4. How can we use this model to calculate how much carbon-14 remains after a certain number of years?
5. What would happen to the function if the initial amount of carbon-14 was changed?

### Tip:
Always verify units and make sure the half-life in the problem matches the base of the exponential function used in the decay model.

The half-life of carbon-14 is 4,500 years. Given an initial 20 gram sample, express the amount of carbon-14 remaining as a function of time, t. Determine an appropriate model for this scenario.

Explore how to model the decay of a 20 gram carbon-14 sample over time, using the half-life of 4,500 years. This detailed explanation includes the exponential decay formula and its application, suitable for high school students in Grades 9-12.

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