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A radioactive substance has a half-life of 66 years. How many years does it take for the quantity to be reduced to 24% of its original amount? Save any rounding for the final calculation. Show your answer to two decimal places.

A. 26.13 years
B. 134.89 years
C. 135.89 years
D. 71.51 years
E. 83.46 years


Sagot :

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