In the year 2005, a picture supposedly painted by a famous artist some time after 1755 but before 1805 contains 95.5 percent of its carbon-14 (half-life 5730 years).
From this information, could this picture have been painted by this artist? 
No

Approximately how old is the painting? 
 years

To determine the approximate age of the painting based on the remaining percentage of Carbon-14, we use the concept of **radioactive decay** and the formula related to half-life decay:

The formula is:

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

Where:
- \(N(t)\) is the amount of Carbon-14 remaining after time \(t\),
- \(N_0\) is the initial amount of Carbon-14,
- \(t\) is the time elapsed (age of the painting),
- \(T_{1/2}\) is the half-life of Carbon-14 (5730 years).

In this case, \(N(t)/N_0 = 95.5\%\), or 0.955, and \(T_{1/2} = 5730\) years.

We need to solve for \(t\) (the age of the painting). Rearranging the formula and taking the natural logarithm of both sides, we get:

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

\[
\ln(0.955) = \frac{t}{T_{1/2}} \cdot \ln\left( \frac{1}{2} \right)
\]

Now, solving for \(t\). I'll calculate this for you.The approximate age of the painting is **381 years**.

Let me know if you want more details or have any other questions!

Here are 5 related questions to expand your understanding:

1. How does the half-life of carbon-14 affect the accuracy of dating ancient objects?
2. Could environmental factors alter the rate of decay of carbon-14?
3. How is radiocarbon dating used in archaeology to date organic artifacts?
4. What would happen if the painting had only 50% of its original carbon-14?
5. How do scientists ensure the accuracy of radiocarbon dating over long periods?

**Tip**: When calculating decay over long periods, small changes in percentage remaining can lead to significant differences in age estimates.

In the year 2005, a picture supposedly painted by a famous artist some time after 1755 but before 1805 contains 95.5 percent of its carbon-14 (half-life 5730 years). From this information, could this picture have been painted by this artist? Approximately how old is the painting?

Learn how to calculate the age of a painting by using the radioactive decay of Carbon-14. This problem involves applying the half-life decay formula, solving logarithmic equations, and understanding exponential decay. Suitable for students in Grades 10-12, this example calculates an age of approximately 381 years.

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