A sample of an isotope has an activity of 660.5 mCi. If the isotope has a half-life of 110.3 minutes, what was the activity of the sample 408 minutes earlier? Express your answer in **units of curie (Ci)** using at least three significant figures.

To calculate the activity of the isotope 408 minutes earlier, we can use the formula for radioactive decay:

\[
A(t) = A_0 e^{-\lambda t}
\]

Where:
- \(A(t)\) is the current activity of the sample.
- \(A_0\) is the activity at the earlier time.
- \(\lambda\) is the decay constant.
- \(t\) is the time elapsed (in this case, 408 minutes).
- \(e\) is the base of the natural logarithm.

However, instead of the decay constant \(\lambda\), we can express the activity in terms of half-life (\(T_{1/2}\)):

\[
\lambda = \frac{\ln(2)}{T_{1/2}}
\]

Thus, the formula becomes:

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

### Given:
- \(A(t) = 660.5 \, \text{mCi} = 0.6605 \, \text{Ci}\) (since 1000 mCi = 1 Ci)
- \(T_{1/2} = 110.3 \, \text{minutes}\)
- \(t = 408 \, \text{minutes}\)

We need to solve for \(A_0\), the original activity:

\[
A_0 = A(t) \left( \frac{1}{2} \right)^{-\frac{t}{T_{1/2}}}
\]

Let me calculate this.The activity of the isotope 408 minutes earlier was approximately **8.578 Ci**.

Would you like more details on the calculation or further explanation?

Here are 5 related questions:
1. How does the half-life of an isotope affect its decay rate?
2. What other factors can influence the activity of a radioactive sample?
3. How is the decay constant related to half-life?
4. What practical applications rely on knowing the activity of an isotope over time?
5. How can we convert between different units of radioactive activity, such as curie and becquerel?

**Tip:** For long time intervals, it’s often easier to work with multiples of the half-life to estimate activity more quickly.

A sample of an isotope has an activity of 660.5 mCi. If the isotope has a half-life of 110.3 minutes, what was the activity of the sample 408 minutes earlier? Express your answer in units of curie (Ci) using at least three significant figures.

Learn how to calculate the activity of a radioactive isotope with an initial activity of 660.5 mCi, 408 minutes earlier. This problem involves key concepts of exponential decay, half-life, and radioactive decay calculations. Suitable for advanced high school physics or college-level students.

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