The half-life of Palladium-100 is 4 days. After 24 days a sample of Palladium-100 has been reduced to a mass of 2 mg. What was the initial mass (in mg) of the sample? 

What is the mass 5 weeks after the start? 


To solve these problems, we use the formula for exponential decay, which is commonly used for half-life calculations:

\[
m(t) = m_0 \cdot \left(\frac{1}{2}\right)^{\frac{t}{T_{\text{half}}}}
\]

Where:
- \( m(t) \) is the remaining mass after time \( t \),
- \( m_0 \) is the initial mass,
- \( T_{\text{half}} \) is the half-life of the substance,
- \( t \) is the elapsed time.

### Problem 1: Finding the initial mass

Given:
- The half-life \( T_{\text{half}} = 4 \) days,
- After \( t = 24 \) days, the mass is \( m(t) = 2 \) mg.

Using the decay formula:

\[
2 = m_0 \cdot \left(\frac{1}{2}\right)^{\frac{24}{4}}
\]

Simplifying the exponent:

\[
2 = m_0 \cdot \left(\frac{1}{2}\right)^6
\]
\[
2 = m_0 \cdot \frac{1}{64}
\]

Solving for \( m_0 \):

\[
m_0 = 2 \cdot 64 = 128 \, \text{mg}
\]

So, the initial mass of the sample was **128 mg**.

### Problem 2: Finding the mass 5 weeks after the start

Convert 5 weeks to days:
- \( 5 \, \text{weeks} = 5 \times 7 = 35 \, \text{days} \).

Now, apply the decay formula with \( t = 35 \) days and \( m_0 = 128 \) mg:

\[
m(35) = 128 \cdot \left(\frac{1}{2}\right)^{\frac{35}{4}}
\]

Let's calculate the exact mass.The mass of the Palladium-100 sample after 5 weeks (35 days) will be approximately **0.297 mg**.

### Summary:
1. The initial mass of the sample was **128 mg**.
2. After 5 weeks, the remaining mass is about **0.297 mg**.

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How would the result change if the half-life were 6 days instead of 4?
2. What is the remaining mass after 10 days?
3. How long would it take for the mass to reduce to 1 mg?
4. How is the formula for exponential decay derived from basic principles?
5. Can this decay model be applied to biological half-lives?

**Tip:** Half-life decay problems often use logarithms to solve for time or the initial quantity when dealing with exponential decay.

The half-life of Palladium-100 is 4 days. After 24 days a sample of Palladium-100 has been reduced to a mass of 2 mg. What was the initial mass (in mg) of the sample? What is the mass 5 weeks after the start?

Learn how to calculate the initial mass and remaining mass of Palladium-100 using its half-life of 4 days. After 24 days, the sample has reduced to 2 mg. This detailed solution applies the exponential decay formula to determine that the initial mass was 128 mg, and after 5 weeks, the mass is approximately 0.297 mg. Suitable for students in Grades 9-12.

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