The half-life of radioactive carbon 14 is 5730 years. How much of a 16-gram sample will remain after 500 years?
*
1 point
15.2 grams
14.7 grams
14.2 grams
13.5 grams

To solve this, we use the formula for radioactive decay:

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

where:
- \(A(t)\) is the amount remaining after time \(t\),
- \(A_0\) is the initial amount,
- \(T\) is the half-life,
- \(t\) is the elapsed time.

Here:
- \(A_0 = 16\) grams,
- \(T = 5730\) years,
- \(t = 500\) years.

Now, substitute these values into the equation:

\[
A(500) = 16 \times \left( \frac{1}{2} \right)^{\frac{500}{5730}}
\]

First, calculate the fraction \(\frac{500}{5730}\):

\[
\frac{500}{5730} \approx 0.0873
\]

Now, calculate \(\left( \frac{1}{2} \right)^{0.0873}\):

\[
\left( \frac{1}{2} \right)^{0.0873} \approx 0.9433
\]

Finally, calculate \(A(500)\):

\[
A(500) = 16 \times 0.9433 \approx 15.09 \, \text{grams}
\]

This rounds to **15.1 grams**, which is closest to **15.2 grams**.

Thus, the correct answer is:
\[
\boxed{15.2 \, \text{grams}}
\]

Would you like more details on this process or have any questions?

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Here are 5 related questions for deeper understanding:
1. How do you derive the exponential decay formula for half-life?
2. What would the remaining sample be after 5730 years for a 16-gram sample?
3. How does the decay constant relate to half-life in radioactive decay?
4. What other elements undergo radioactive decay, and how do their half-lives compare to carbon-14?
5. How can you use logarithms to solve for time when the remaining amount and initial amount are known?

**Tip**: Always round intermediate values carefully to avoid significant errors in the final result.

The half-life of radioactive carbon 14 is 5730 years. How much of a 16-gram sample will remain after 500 years?

Find out how much of a 16-gram sample of carbon-14 remains after 500 years using the exponential decay formula. This step-by-step solution explains how to calculate the remaining amount of carbon-14, covering concepts like half-life and radioactive decay. Suitable for high school students in Grades 10-12.

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