Let's solve this problem step-by-step.
1. Understanding the given data:
- The half-life of Palladium-100 is 4 days.
- After 20 days, the mass of the Palladium-100 sample is 7 mg.
- We need to find the initial mass of the sample.
- We also need to determine the mass 6 weeks after the start. (Note: 6 weeks equals 42 days.)
2. Finding the initial mass:
The decay of a substance over time can be described by the half-life formula:
[tex]\[
\text{final mass} = \text{initial mass} \times \left(0.5\right)^{\frac{\text{time elapsed}}{\text{half-life}}}
\][/tex]
Rearranging this formula to solve for the initial mass:
[tex]\[
\text{initial mass} = \frac{\text{final mass}}{\left(0.5\right)^{\frac{\text{time elapsed}}{\text{half-life}}}}
\][/tex]
Substituting in the known values:
[tex]\[
\text{initial mass} = \frac{7 \text{ mg}}{\left(0.5\right)^{\frac{20 \text{ days}}{4 \text{ days}}}}
\][/tex]
Simplifying the exponent first:
[tex]\[
\frac{20 \text{ days}}{4 \text{ days}} = 5
\][/tex]
So:
[tex]\[
\left(0.5\right)^5 = 0.03125
\][/tex]
Now, calculate the initial mass:
[tex]\[
\text{initial mass} = \frac{7 \text{ mg}}{0.03125}
\][/tex]
[tex]\[
\text{initial mass} = 224 \text{ mg}
\][/tex]
Therefore, the initial mass of the sample was 224 mg.
3. Finding the mass 6 weeks after the start:
Using the half-life decay formula again, but now for 42 days (6 weeks):
[tex]\[
\text{mass after 42 days} = \text{initial mass} \times \left(0.5\right)^{\frac{\text{time elapsed}}{\text{half-life}}}
\][/tex]
Substituting in the values:
[tex]\[
\text{mass after 42 days} = 224 \text{ mg} \times \left(0.5\right)^{\frac{42 \text{ days}}{4 \text{ days}}}
\][/tex]
Simplifying the exponent first:
[tex]\[
\frac{42 \text{ days}}{4 \text{ days}} = 10.5
\][/tex]
So:
[tex]\[
\text{mass after 42 days} = 224 \text{ mg} \times \left(0.5\right)^{10.5}
\][/tex]
Now, calculating [tex]\(\left(0.5\right)^{10.5}\)[/tex]:
[tex]\[
\left(0.5\right)^{10.5} \approx 0.0006903
\][/tex]
Finally, calculating the remaining mass:
[tex]\[
\text{mass after 42 days} \approx 224 \text{ mg} \times 0.0006903
\][/tex]
[tex]\[
\text{mass after 42 days} \approx 0.1547 \text{ mg}
\][/tex]
Therefore, the mass of the sample 6 weeks after the start is 0.1547 mg, rounded to four decimal places.
So, the initial mass of the sample was 224 mg, and the mass 6 weeks after the start was 0.1547 mg.
