https://file.aiquickdraw.com/mathbot/file/172806448009254g5sjn7.jpg

Let's solve the problem step by step.

### Given:
- Half-life of Palladium-100 = 4 days
- After 24 days, the sample has been reduced to a mass of 3 mg.

We are asked to find:

1. **Initial mass** of the sample.
2. The **mass** of the sample **7 weeks** after the start.

### Step 1: Exponential Decay Formula
The general formula for radioactive decay is:

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

Where:
- \( M(t) \) is the mass at time \( t \),
- \( M_0 \) is the initial mass (what we need to find),
- \( T_{1/2} \) is the half-life (4 days),
- \( t \) is the elapsed time (in days).

### Step 2: Finding Initial Mass
From the problem, after 24 days, the mass is 3 mg.

We plug into the decay formula:

\[
3 = M_0 \cdot \left(\frac{1}{2}\right)^{\frac{24}{4}}
\]

Simplifying:

\[
3 = M_0 \cdot \left(\frac{1}{2}\right)^6
\]
\[
3 = M_0 \cdot \frac{1}{64}
\]
\[
M_0 = 3 \cdot 64 = 192 \text{ mg}
\]

So, the **initial mass** of the sample is **192 mg**.

### Step 3: Finding Mass after 7 weeks
We now need to find the mass after 7 weeks. Since there are 7 days in a week, 7 weeks equals:

\[
7 \times 7 = 49 \text{ days}
\]

We use the same decay formula with \( M_0 = 192 \text{ mg} \), \( t = 49 \text{ days} \), and \( T_{1/2} = 4 \text{ days} \):

\[
M(49) = 192 \cdot \left(\frac{1}{2}\right)^{\frac{49}{4}}
\]
\[
M(49) = 192 \cdot \left(\frac{1}{2}\right)^{12.25}
\]

Let's calculate this:

\[
M(49) \approx 192 \cdot 0.0001835
\]
\[
M(49) \approx 0.0353 \text{ mg}
\]

So, after 7 weeks, the mass will be approximately **0.0353 mg**.

### Summary of Answers:
1. The initial mass of the sample is **192 mg**.
2. The mass after 7 weeks is approximately **0.0353 mg**.

Would you like further clarification or details?

### Related Questions:
1. How does the decay constant relate to the half-life in radioactive decay?
2. Can we find the decay constant for Palladium-100 based on the given half-life?
3. How would the formula change if the half-life were expressed in hours instead of days?
4. What would happen to the mass after 14 weeks instead of 7 weeks?
5. What is the fractional amount of the sample remaining after 24 days?

### Tip:
Always ensure time units are consistent when applying decay formulas—especially when converting between days, weeks, or hours.

The half-life of Palladium-100 is 4 days. After 24 days, a sample of Palladium-100 has been reduced to a mass of 3 mg. What was the initial mass of the sample? What is the mass (in mg) 7 weeks after the start?

Solve a radioactive decay problem involving Palladium-100 with a half-life of 4 days. Calculate the initial mass and the remaining mass after 7 weeks, using exponential decay laws. Suitable for high school students in grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation