The half-life of Palladium-100 is 4 days. After 12 days a sample of Palladium-100 has been reduced to a mass of 4 mg. If the initial mass is 32, what is the 5 weeks after the start?

To solve this, we can use the formula for exponential decay, which is based on the half-life:

\[
M(t) = M_0 \left(\frac{1}{2}\right)^{\frac{t}{T}}
\]

Where:
- \( M(t) \) is the mass after time \( t \),
- \( M_0 \) is the initial mass,
- \( T \) is the half-life,
- \( t \) is the time passed.

Given:
- The half-life \( T = 4 \) days,
- Initial mass \( M_0 = 32 \) mg,
- We want to find the mass after \( t = 5 \) weeks (which is 35 days).

### Step 1: Apply the decay formula
Substitute the values into the formula:

\[
M(35) = 32 \left(\frac{1}{2}\right)^{\frac{35}{4}}
\]

### Step 2: Simplify the exponent
\[
\frac{35}{4} = 8.75
\]

Now the equation becomes:

\[
M(35) = 32 \left(\frac{1}{2}\right)^{8.75}
\]

### Step 3: Evaluate the power
We can calculate:

\[
\left(\frac{1}{2}\right)^{8.75} \approx 0.0027
\]

### Step 4: Calculate the final mass
Now, multiply:

\[
M(35) = 32 \times 0.0027 \approx 0.0864 \, \text{mg}
\]

### Conclusion:
After 5 weeks (or 35 days), the mass of Palladium-100 will be approximately **0.0864 mg**.

Would you like more details on any part of the solution?

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Here are 5 related questions to expand your understanding:
1. How does the concept of half-life apply in radioactive decay?
2. How would the calculation change if the half-life were different?
3. What is the significance of exponential decay in real-world applications?
4. How can you determine the time it takes for a substance to decay to a specific mass?
5. How does the half-life relate to the rate of decay in percentages?

**Tip:** Always double-check if you're working with the right units for time, especially when converting between days, weeks, or other periods.

The half-life of Palladium-100 is 4 days. After 12 days a sample of Palladium-100 has been reduced to a mass of 4 mg. If the initial mass is 32 mg, what is the mass 5 weeks after the start?

Learn how to calculate the mass of Palladium-100 after 5 weeks using the half-life formula. The problem demonstrates exponential decay in radioactive substances, starting with an initial mass of 32 mg and a half-life of 4 days. Suitable for students in Grades 9-12.

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