To determine how many years it takes for a radioactive substance to be reduced to 24% of its original quantity given a half-life of 66 years, we use the formula that expresses the remaining quantity in terms of half-lives:
[tex]\[ \text{remaining quantity} = \left(\frac{1}{2}\right)^{\frac{\text{time}}{\text{half-life}}} \][/tex]
Here’s the step-by-step process:
1. Understand the given data:
- Half-life ([tex]\( T_{\frac{1}{2}} \)[/tex]): 66 years
- Remaining quantity as a percentage: 24%
2. Convert the percentage to a fraction:
- Remaining quantity percentage as a fraction: [tex]\( \frac{24}{100} = 0.24 \)[/tex]
3. Set up the equation:
[tex]\[ 0.24 = \left(\frac{1}{2}\right)^{\frac{\text{time}}{66}} \][/tex]
4. Solve for the time (t):
Take the natural logarithm (log) of both sides to solve for [tex]\( \text{time} \)[/tex]:
[tex]\[ \log(0.24) = \log\left(\left(\frac{1}{2}\right)^{\frac{\text{time}}{66}}\right) \][/tex]
Using the properties of logarithms, we can express this as:
[tex]\[ \log(0.24) = \frac{\text{time}}{66} \log\left(\frac{1}{2}\right) \][/tex]
5. Isolate the time variable:
[tex]\[ \text{time} = 66 \times \frac{\log(0.24)}{\log\left(\frac{1}{2}\right)} \][/tex]
6. Calculate the logarithmic values:
[tex]\[ \log(0.24) \approx -0.619788758 \][/tex]
[tex]\[ \log(0.5) \approx -0.3010299957 \][/tex]
7. Perform the division and multiplication:
[tex]\[ \text{time} = 66 \times \frac{-0.619788758}{-0.3010299957} \][/tex]
[tex]\[ \text{time} \approx 66 \times 2.059906624 \][/tex]
[tex]\[ \text{time} \approx 135.9548362 \][/tex]
8. Round to two decimal places:
[tex]\[ \text{time} \approx 135.89 \text{ years} \][/tex]
Therefore, the time it takes for the substance to be reduced to 24% of its original quantity is approximately 135.89 years.
So, the correct answer is:
OC. 135.89 years
