To determine how long it would take for an isotope to be reduced to 12.5% of its original amount, we need to follow these steps:
1. Understand the Concept of Half-Life: The half-life of a substance is the time it takes for half of the substance to decay or be reduced.
2. Identify the Fraction Remaining: We need to find out how many half-lives correspond to having 12.5% of the original sample remaining. Since 12.5% can be written as a fraction, it corresponds to 0.125 of the original amount.
3. Express 12.5% as a Power of 1/2: Each half-life reduces the sample by half. To connect this with our remaining amount, observe that:
[tex]\[
\left(\frac{1}{2}\right)^n = 0.125
\][/tex]
where [tex]\( n \)[/tex] is the number of half-lives required.
4. Solve for [tex]\( n \)[/tex]: Recognize that [tex]\( 0.125 \)[/tex] is the same as [tex]\( \left(\frac{1}{2}\right)^3 \)[/tex]. Thus:
[tex]\[
n = 3
\][/tex]
This indicates that it takes 3 half-lives to reduce the sample to 12.5% of its original amount.
5. Calculate the Total Time: Knowing that one half-life is 5 days, we then calculate the total time taken by multiplying the number of half-lives by the duration of one half-life:
[tex]\[
\text{Total time} = 3 \times 5 \text{ days} = 15 \text{ days}
\][/tex]
Thus, it would take 15 days for the isotope to decay to 12.5% of its original amount.
