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To solve for the decay constant \( k \) given the half-life, we use the following relationship between half-life and the decay rate constant in radioactive decay:

\[
k = \frac{\ln(2)}{t_{1/2}}
\]

Where:
- \( t_{1/2} \) is the half-life of the substance,
- \( \ln(2) \approx 0.693147 \).

From the image, the half-life is given as 18.2 days.

Now, we can plug the value of \( t_{1/2} = 18.2 \) days into the formula:

\[
k = \frac{0.693147}{18.2}
\]

Calculating this:

\[
k = 0.038078
\]

So, the decay constant \( k \) is approximately \( 0.038078 \) (rounded to six decimal places).

Would you like more details or have any questions about this? 

Here are 5 related questions:
1. How would you calculate the decay constant if the half-life was given in hours?
2. Can the decay constant change over time for a substance?
3. How can this decay constant \( k \) be used to find the remaining amount of a substance after a certain period?
4. What is the significance of the natural logarithm \( \ln(2) \) in the decay formula?
5. How does the half-life of a substance affect its practical applications in medicine or science?

**Tip**: Remember, for any radioactive decay process, the shorter the half-life, the faster the decay rate (larger \( k \)).

Complete the table shown for the half-life of a certain radioactive substance, where the half-life is 18.2 days. Calculate the decay rate k.

Learn how to calculate the decay rate constant k for a substance with a half-life of 18.2 days. This problem covers concepts like exponential decay, logarithms, and involves the decay formula k = ln(2) / t_{1/2}. Suitable for students in Grades 10-12.

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