using the half life formula, answer this question: The half-life of Palladium-100 is 4 days. After 12 
days a sample of Pd-100 has been reduced to a mass of 4.00 mg. 
(a) Determine the starting mass. (b) What is the mass 8 weeks after 
the start?

The half-life formula can be written as:

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

where:
- \( A(t) \) is the remaining amount of the substance at time \( t \),
- \( A_0 \) is the initial amount (starting mass),
- \( T_{\frac{1}{2}} \) is the half-life of the substance,
- \( t \) is the time that has passed.

### (a) Determine the starting mass:

Given:
- The half-life of Palladium-100 \( T_{\frac{1}{2}} = 4 \) days,
- After 12 days, the remaining mass \( A(12) = 4.00 \) mg.

First, we can solve for the initial mass \( A_0 \) using the formula:

\[
A(12) = A_0 \times \left(\frac{1}{2}\right)^{\frac{12}{4}}
\]

\[
4.00 = A_0 \times \left(\frac{1}{2}\right)^3
\]

\[
4.00 = A_0 \times \frac{1}{8}
\]

Multiplying both sides by 8:

\[
A_0 = 4.00 \times 8 = 32.00 \, \text{mg}
\]

Thus, the starting mass of the sample is **32.00 mg**.

### (b) What is the mass 8 weeks after the start?

First, convert 8 weeks into days:

\[
8 \, \text{weeks} = 8 \times 7 = 56 \, \text{days}
\]

Now, use the half-life formula again to calculate the remaining mass \( A(56) \):

\[
A(56) = A_0 \times \left(\frac{1}{2}\right)^{\frac{56}{4}}
\]

\[
A(56) = 32.00 \times \left(\frac{1}{2}\right)^{14}
\]

Since \( \left(\frac{1}{2}\right)^{14} = \frac{1}{16384} \), we can calculate:

\[
A(56) = 32.00 \times \frac{1}{16384} = 0.001953125 \, \text{mg}
\]

Thus, after 8 weeks, the mass of Palladium-100 will be approximately **0.00195 mg**.

---

Would you like further details or clarifications on any of the steps?

Here are 5 related questions:
1. How is the half-life formula derived?
2. How would the calculations change if the half-life were in hours?
3. What would be the mass after 20 days instead of 56 days?
4. How do half-life calculations apply to radioactive decay in real-world scenarios?
5. Can the half-life formula be used to model population decay?

**Tip:** In half-life problems, always ensure the time is converted into the same units as the half-life value.

Using the half-life formula, answer this question: The half-life of Palladium-100 is 4 days. After 12 days a sample of Pd-100 has been reduced to a mass of 4.00 mg. (a) Determine the starting mass. (b) What is the mass 8 weeks after the start?

Learn how to solve a Palladium-100 half-life problem using the exponential decay formula. This detailed solution covers calculating the starting mass after 12 days and predicting the mass 8 weeks later. Key concepts include half-life and exponential decay, suitable for Grades 10-12.

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