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Sagot :
To solve this problem, we need to determine how many days have passed for each isotope based on their initial weight, ending measured weight, and half-life. Let's break this down step-by-step for each isotope.
### Isotope A
Isotope A has:
- Initial weight: 95 units
- Ending measured weight: 5.9 units
- Half-life: 6 days
The decay formula based on half-life is:
[tex]\[ \text{measured weight} = \text{initial weight} \times (0.5)^{\frac{t}{T}} \][/tex]
where [tex]\( t \)[/tex] is the time passed, and [tex]\( T \)[/tex] is the half-life.
Rewriting the formula to solve for [tex]\( t \)[/tex]:
[tex]\[ \frac{\text{measured weight}}{\text{initial weight}} = (0.5)^{\frac{t}{T}} \][/tex]
[tex]\[ \ln\left( \frac{\text{measured weight}}{\text{initial weight}} \right) = \frac{t}{T} \ln(0.5) \][/tex]
[tex]\[ t = \frac{\ln\left( \frac{\text{measured weight}}{\text{initial weight}} \right)}{\ln(0.5)} \times T \][/tex]
Plugging in the values for Isotope A:
[tex]\[ t_A = \frac{\ln\left( \frac{5.9}{95} \right)}{\ln(0.5)} \times 6 \][/tex]
After calculating, we find:
[tex]\[ t_A \approx 24 \text{ days} \][/tex]
### Isotope B
Isotope B has:
- Initial weight: 20 units
- Ending measured weight: 2.5 units
- Half-life: 2 days
Using the same decay formula and solving for [tex]\( t \)[/tex]:
[tex]\[ t_B = \frac{\ln\left( \frac{2.5}{20} \right)}{\ln(0.5)} \times 2 \][/tex]
After calculating, we find:
[tex]\[ t_B \approx 6 \text{ days} \][/tex]
### Isotope C
Isotope C has:
- Initial weight: 45 units
- Ending measured weight: 22.5 units
- Half-life: 10 days
Using the same decay formula and solving for [tex]\( t \)[/tex]:
[tex]\[ t_C = \frac{\ln\left( \frac{22.5}{45} \right)}{\ln(0.5)} \times 10 \][/tex]
After calculating, we find:
[tex]\[ t_C \approx 10 \text{ days} \][/tex]
### Conclusion
Based on the calculations:
- Isotope A was measured at day 24.
- Isotope B was measured at day 6.
- Isotope C was measured at day 10.
Therefore, the correct statement is:
Isotope A was measured at day 24, Isotope B was measured at day 6, and Isotope C was measured at day 10.
### Isotope A
Isotope A has:
- Initial weight: 95 units
- Ending measured weight: 5.9 units
- Half-life: 6 days
The decay formula based on half-life is:
[tex]\[ \text{measured weight} = \text{initial weight} \times (0.5)^{\frac{t}{T}} \][/tex]
where [tex]\( t \)[/tex] is the time passed, and [tex]\( T \)[/tex] is the half-life.
Rewriting the formula to solve for [tex]\( t \)[/tex]:
[tex]\[ \frac{\text{measured weight}}{\text{initial weight}} = (0.5)^{\frac{t}{T}} \][/tex]
[tex]\[ \ln\left( \frac{\text{measured weight}}{\text{initial weight}} \right) = \frac{t}{T} \ln(0.5) \][/tex]
[tex]\[ t = \frac{\ln\left( \frac{\text{measured weight}}{\text{initial weight}} \right)}{\ln(0.5)} \times T \][/tex]
Plugging in the values for Isotope A:
[tex]\[ t_A = \frac{\ln\left( \frac{5.9}{95} \right)}{\ln(0.5)} \times 6 \][/tex]
After calculating, we find:
[tex]\[ t_A \approx 24 \text{ days} \][/tex]
### Isotope B
Isotope B has:
- Initial weight: 20 units
- Ending measured weight: 2.5 units
- Half-life: 2 days
Using the same decay formula and solving for [tex]\( t \)[/tex]:
[tex]\[ t_B = \frac{\ln\left( \frac{2.5}{20} \right)}{\ln(0.5)} \times 2 \][/tex]
After calculating, we find:
[tex]\[ t_B \approx 6 \text{ days} \][/tex]
### Isotope C
Isotope C has:
- Initial weight: 45 units
- Ending measured weight: 22.5 units
- Half-life: 10 days
Using the same decay formula and solving for [tex]\( t \)[/tex]:
[tex]\[ t_C = \frac{\ln\left( \frac{22.5}{45} \right)}{\ln(0.5)} \times 10 \][/tex]
After calculating, we find:
[tex]\[ t_C \approx 10 \text{ days} \][/tex]
### Conclusion
Based on the calculations:
- Isotope A was measured at day 24.
- Isotope B was measured at day 6.
- Isotope C was measured at day 10.
Therefore, the correct statement is:
Isotope A was measured at day 24, Isotope B was measured at day 6, and Isotope C was measured at day 10.
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