To determine which number produces a rational number when multiplied by \(\frac{1}{3}\), we need to investigate each given option:
1. Option A: \(2.236067978 \ldots\)
- Multiplying this number by \(\frac{1}{3}\) results in:
[tex]\[
2.236067978 \times \frac{1}{3} = 0.7453559926666667 \ldots
\][/tex]
Checking if this product is a rational number, we see that \(0.7453559926666667 \ldots\) is a non-terminating, non-repeating decimal, thus it is irrational.
2. Option B: \(\pi \)
- Multiplying \(\pi \) by \(\frac{1}{3}\) results in:
[tex]\[
\pi \times \frac{1}{3} = \frac{\pi}{3}
\][/tex]
Since \(\pi\) is an irrational number, \(\frac{\pi}{3}\) remains irrational as the multiplication of an irrational number by a non-zero rational number (here \(\frac{1}{3}\)) is still irrational.
3. Option C: \(\frac{3}{7} \)
- Multiplying \(\frac{3}{7}\) by \(\frac{1}{3}\) results in:
[tex]\[
\frac{3}{7} \times \frac{1}{3} = \frac{3 \cdot 1}{7 \cdot 3} = \frac{1}{7}
\][/tex]
Here, \(\frac{1}{7}\) is clearly a rational number as it can be expressed as the ratio of two integers \(1\) and \(7\).
4. Option D: \(\sqrt{12}\)
- Multiplying \(\sqrt{12}\) by \(\frac{1}{3}\) results in:
[tex]\[
\sqrt{12} \times \frac{1}{3} = \frac{\sqrt{12}}{3} = \frac{2\sqrt{3}}{3} \approx 1.154700538 \ldots
\][/tex]
\(\frac{2\sqrt{3}}{3}\) remains irrational because \(\sqrt{3}\) is irrational, and thus any non-integer multiple of \(\sqrt{3}\) is also irrational.
From these evaluations, it is clear that only Option C: \(\frac{3}{7}\) produces a rational number when multiplied by \(\frac{1}{3}\). So, the correct answer is:
[tex]\[ \boxed{\frac{3}{7}} \][/tex]
