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Sagot :
Let's break down the solution to the problem step-by-step.
### Initial Mass Calculation
1. Understanding the Problem:
- The half-life of Palladium-100 is 4 days.
- After 16 days, the mass of a sample of Palladium-100 has been reduced to 4 mg.
- We need to find the initial mass of the sample before it started decaying.
2. Calculate the Number of Half-Lives Passed:
- The number of half-lives that have passed in 16 days can be calculated by dividing the total time by the half-life period:
[tex]\[ \text{Number of half-lives} = \frac{\text{Total time in days}}{\text{Half-life period in days}} \][/tex]
[tex]\[ \text{Number of half-lives} = \frac{16}{4} = 4 \][/tex]
3. Apply the Decay Formula:
- The decay formula relating initial mass [tex]\( M_0 \)[/tex], final mass [tex]\( M \)[/tex], and the number of half-lives [tex]\( n \)[/tex] is:
[tex]\[ M = M_0 \left( \frac{1}{2} \right)^n \][/tex]
- Here, [tex]\( M = 4 \)[/tex] mg (mass after 16 days).
- [tex]\( n = 4 \)[/tex] (number of half-lives).
- To find the initial mass [tex]\( M_0 \)[/tex], rearrange the equation:
[tex]\[ M_0 = \frac{M}{\left( \frac{1}{2} \right)^n} \][/tex]
[tex]\[ M_0 = \frac{4}{\left( \frac{1}{2} \right)^4} \][/tex]
[tex]\[ M_0 = \frac{4}{\frac{1}{16}} = 4 \times 16 = 64 \text{ mg} \][/tex]
The initial mass of the sample was 64 mg.
### Mass Calculation After 5 Weeks
1. Understanding the Problem:
- We need to find the mass of the sample after 5 weeks starting from the initial mass.
- 1 week = 7 days, so 5 weeks = 5 × 7 = 35 days.
2. Calculate the Number of Half-Lives Passed:
- The number of half-lives that have passed in 35 days can be calculated by dividing the total time by the half-life period:
[tex]\[ \text{Number of half-lives} = \frac{35}{4} = 8.75 \][/tex]
3. Apply the Decay Formula:
- Using the initial mass [tex]\( M_0 = 64 \)[/tex] mg, calculate the mass after 35 days:
[tex]\[ M = M_0 \left( \frac{1}{2} \right)^n \][/tex]
- Here, [tex]\( M_0 = 64 \)[/tex] mg and [tex]\( n = 8.75 \)[/tex].
[tex]\[ M = 64 \left( \frac{1}{2} \right)^{8.75} \][/tex]
[tex]\[ M \approx 0.1487 \text{ mg} \][/tex]
The mass of the sample 5 weeks after the start is approximately 0.1487 mg.
### Final Answers
1. The initial mass of the sample was [tex]\( \boxed{64} \)[/tex] mg.
2. The mass of the sample 5 weeks after the start is approximately [tex]\( \boxed{0.1487} \)[/tex] mg when rounded to four decimal places.
### Initial Mass Calculation
1. Understanding the Problem:
- The half-life of Palladium-100 is 4 days.
- After 16 days, the mass of a sample of Palladium-100 has been reduced to 4 mg.
- We need to find the initial mass of the sample before it started decaying.
2. Calculate the Number of Half-Lives Passed:
- The number of half-lives that have passed in 16 days can be calculated by dividing the total time by the half-life period:
[tex]\[ \text{Number of half-lives} = \frac{\text{Total time in days}}{\text{Half-life period in days}} \][/tex]
[tex]\[ \text{Number of half-lives} = \frac{16}{4} = 4 \][/tex]
3. Apply the Decay Formula:
- The decay formula relating initial mass [tex]\( M_0 \)[/tex], final mass [tex]\( M \)[/tex], and the number of half-lives [tex]\( n \)[/tex] is:
[tex]\[ M = M_0 \left( \frac{1}{2} \right)^n \][/tex]
- Here, [tex]\( M = 4 \)[/tex] mg (mass after 16 days).
- [tex]\( n = 4 \)[/tex] (number of half-lives).
- To find the initial mass [tex]\( M_0 \)[/tex], rearrange the equation:
[tex]\[ M_0 = \frac{M}{\left( \frac{1}{2} \right)^n} \][/tex]
[tex]\[ M_0 = \frac{4}{\left( \frac{1}{2} \right)^4} \][/tex]
[tex]\[ M_0 = \frac{4}{\frac{1}{16}} = 4 \times 16 = 64 \text{ mg} \][/tex]
The initial mass of the sample was 64 mg.
### Mass Calculation After 5 Weeks
1. Understanding the Problem:
- We need to find the mass of the sample after 5 weeks starting from the initial mass.
- 1 week = 7 days, so 5 weeks = 5 × 7 = 35 days.
2. Calculate the Number of Half-Lives Passed:
- The number of half-lives that have passed in 35 days can be calculated by dividing the total time by the half-life period:
[tex]\[ \text{Number of half-lives} = \frac{35}{4} = 8.75 \][/tex]
3. Apply the Decay Formula:
- Using the initial mass [tex]\( M_0 = 64 \)[/tex] mg, calculate the mass after 35 days:
[tex]\[ M = M_0 \left( \frac{1}{2} \right)^n \][/tex]
- Here, [tex]\( M_0 = 64 \)[/tex] mg and [tex]\( n = 8.75 \)[/tex].
[tex]\[ M = 64 \left( \frac{1}{2} \right)^{8.75} \][/tex]
[tex]\[ M \approx 0.1487 \text{ mg} \][/tex]
The mass of the sample 5 weeks after the start is approximately 0.1487 mg.
### Final Answers
1. The initial mass of the sample was [tex]\( \boxed{64} \)[/tex] mg.
2. The mass of the sample 5 weeks after the start is approximately [tex]\( \boxed{0.1487} \)[/tex] mg when rounded to four decimal places.
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