Answer:
72,300 years.
Explanation:
- Initial mass of this sample: 504 grams;
- Current mass of this sample: 63 grams.
What's the ratio between the current and the initial mass of this sample? In other words, what fraction of the initial sample hasn't yet decayed?
[tex]\displaystyle \frac{\text{Current Mass}}{\text{Initial Mass}} = \rm \frac{63\; g}{504\; g} = \frac{1}{8}[/tex].
The value of this fraction starts at 1 decreases to 1/2 of its initial value after every half-life. How many times shall 1/2 be multiplied to 1 before reaching 1/8? [tex]2^{3} = 8[/tex]. It takes three half-lives or [tex]3\times 24100 = 72300[/tex] years to reach that value.
In certain questions the denominator of the fraction is large. It might not even be an integer power of 2. The base-x logarithm function on calculators could help. Evaluate
[tex]\displaystyle \log_{\frac{1}{2}}{\frac{1}{8}} = 3[/tex] to find the number of half-lives required. In case the base-x logarithm function isn't available, but the natural logarithm function [tex]\ln()[/tex] is, apply the following expression (derived from the base-changing formula) to get the same result:
[tex]\displaystyle \frac{\displaystyle\ln{\left(\frac{1}{8}\right)}}{\displaystyle \ln{\left(\frac{1}{2}\right)}}[/tex].
