The radioactivity of a given sample of whisky due to tritium (half life `12.3` years) was found to be only `3 %` of that measured in a recently purchased bottle marked `''7` years old''. The sample must have been prepared about.

Options

A. Before 70 years
B. Before 220 years
C. Before 420 years
D. Before 300 years

Explanation

To determine when the sample of whisky was prepared, we need to analyze the information given about the radioactivity of tritium in the whisky. Here’s a step-by-step solution:

### Step 1: Understand the half-life of tritium
The half-life of tritium is given as 12.3 years. This means that every 12.3 years, the amount of tritium (and hence its radioactivity) in the sample reduces to half of its previous value.

### Step 2: Determine the percentage of radioactivity
The problem states that the radioactivity of the sample is only 3% of that measured in a recently purchased bottle marked "7 years old." This means that the sample has undergone several half-lives to reach this level of radioactivity.

### Step 3: Calculate the number of half-lives
We need to find out how many half-lives it takes for the radioactivity to drop to 3%. We can use the formula for the remaining percentage of a radioactive substance after n half-lives:
\[ \text{Remaining Activity} = \left( \frac{1}{2} \right)^n \times 100\% \]
Setting this equal to 3%, we have:
\[ \left( \frac{1}{2} \right)^n \times 100\% = 3\% \]
Dividing both sides by 100% gives:
\[ \left( \frac{1}{2} \right)^n = 0.03 \]

### Step 4: Solve for n
Taking the logarithm of both sides:
\[ n \log\left(\frac{1}{2}\right) = \log(0.03) \]
Now, we can calculate n:
\[ n = \frac{\log(0.03)}{\log(0.5)} \]
Using a calculator:
- \(\log(0.03) \approx -1.5228787\)
- \(\log(0.5) \approx -0.3010299\)

So,
\[ n \approx \frac{-1.5228787}{-0.3010299} \approx 5.06 \]
This means it takes approximately 5 half-lives for the radioactivity to drop to 3%.

### Step 5: Calculate the total time elapsed
Since each half-life is 12.3 years, the total time for 5 half-lives is:
\[ \text{Total Time} = n \times \text{half-life} = 5 \times 12.3 \text{ years} = 61.5 \text{ years} \]

### Step 6: Add the age of the recently purchased bottle
The recently purchased bottle is marked as "7 years old." Therefore, the total time since the sample was prepared is:
\[ \text{Time since preparation} = 61.5 \text{ years} + 7 \text{ years} = 68.5 \text{ years} \]

### Conclusion
The sample must have been prepared approximately 68.5 years ago.

### Final Answer
The sample must have been prepared about **69 years ago** (rounding to the nearest whole number).

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