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Sagot :
To determine what the values in the columns labeled "Fraction remaining" and "Percentage remaining" measure, we need to understand the context given by the chart on the rate of decay.
The table shows:
- The number of half-lives that have elapsed.
- The corresponding fraction of material that has not decayed.
- The percentage of the remaining undecayed material.
Here is the detailed breakdown of the values:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{Half-lives elapsed} & \text{Fraction remaining} & \text{Percentage remaining} \\ \hline 0 & 1 & 100 \\ \hline 1 & \frac{1}{2} & 50 \\ \hline 2 & \frac{1}{4} & 25 \\ \hline 3 & \frac{1}{8} & 12.5 \\ \hline 4 & \frac{1}{16} & 6.25 \\ \hline 5 & \frac{1}{32} & 3.125 \\ \hline \end{array} \][/tex]
### Step-by-Step Analysis:
1. Half-lives Elapsed: This column indicates the number of half-lives that have been completed. A half-life is the amount of time required for half of the radioactive atoms in a sample to decay.
2. Fraction Remaining:
- At 0 half-lives, all the material is present (1).
- After 1 half-life, half the material remains ([tex]\(\frac{1}{2}\)[/tex]).
- After 2 half-lives, a quarter of the material remains ([tex]\(\frac{1}{4}\)[/tex]).
- After 3 half-lives, one-eighth of the material remains ([tex]\(\frac{1}{8}\)[/tex]).
- After 4 half-lives, one-sixteenth of the material remains ([tex]\(\frac{1}{16}\)[/tex]).
- After 5 half-lives, one-thirty-second of the material remains ([tex]\(\frac{1}{32}\)[/tex]).
3. Percentage Remaining: This is simply the fraction converted to a percentage.
- 1 (or [tex]\(\frac{1}{1}\)[/tex]) is 100%.
- [tex]\(\frac{1}{2}\)[/tex] is 50%.
- [tex]\(\frac{1}{4}\)[/tex] is 25%.
- [tex]\(\frac{1}{8}\)[/tex] is 12.5%.
- [tex]\(\frac{1}{16}\)[/tex] is 6.25%.
- [tex]\(\frac{1}{32}\)[/tex] is 3.125%.
### Conclusion:
Both the "Fraction remaining" and "Percentage remaining" columns measure the same thing: the amount of material that has not decayed after a given number of half-lives. This consistent tracking makes it clear that these columns trace the decay progression of a substance, specifically focusing on the remaining undecomposed quantity.
Therefore, the value being measured in these columns is:
amount of material that has not decayed.
The table shows:
- The number of half-lives that have elapsed.
- The corresponding fraction of material that has not decayed.
- The percentage of the remaining undecayed material.
Here is the detailed breakdown of the values:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{Half-lives elapsed} & \text{Fraction remaining} & \text{Percentage remaining} \\ \hline 0 & 1 & 100 \\ \hline 1 & \frac{1}{2} & 50 \\ \hline 2 & \frac{1}{4} & 25 \\ \hline 3 & \frac{1}{8} & 12.5 \\ \hline 4 & \frac{1}{16} & 6.25 \\ \hline 5 & \frac{1}{32} & 3.125 \\ \hline \end{array} \][/tex]
### Step-by-Step Analysis:
1. Half-lives Elapsed: This column indicates the number of half-lives that have been completed. A half-life is the amount of time required for half of the radioactive atoms in a sample to decay.
2. Fraction Remaining:
- At 0 half-lives, all the material is present (1).
- After 1 half-life, half the material remains ([tex]\(\frac{1}{2}\)[/tex]).
- After 2 half-lives, a quarter of the material remains ([tex]\(\frac{1}{4}\)[/tex]).
- After 3 half-lives, one-eighth of the material remains ([tex]\(\frac{1}{8}\)[/tex]).
- After 4 half-lives, one-sixteenth of the material remains ([tex]\(\frac{1}{16}\)[/tex]).
- After 5 half-lives, one-thirty-second of the material remains ([tex]\(\frac{1}{32}\)[/tex]).
3. Percentage Remaining: This is simply the fraction converted to a percentage.
- 1 (or [tex]\(\frac{1}{1}\)[/tex]) is 100%.
- [tex]\(\frac{1}{2}\)[/tex] is 50%.
- [tex]\(\frac{1}{4}\)[/tex] is 25%.
- [tex]\(\frac{1}{8}\)[/tex] is 12.5%.
- [tex]\(\frac{1}{16}\)[/tex] is 6.25%.
- [tex]\(\frac{1}{32}\)[/tex] is 3.125%.
### Conclusion:
Both the "Fraction remaining" and "Percentage remaining" columns measure the same thing: the amount of material that has not decayed after a given number of half-lives. This consistent tracking makes it clear that these columns trace the decay progression of a substance, specifically focusing on the remaining undecomposed quantity.
Therefore, the value being measured in these columns is:
amount of material that has not decayed.
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