To find the product of \( 5 - \sqrt{5} \) and \( \frac{1}{6} \), we first calculate \( 5 - \sqrt{5} \).
Calculating \( 5 - \sqrt{5} \):
- The approximate value of \( \sqrt{5} \) is about 2.236.
- Therefore, \( 5 - \sqrt{5} \approx 5 - 2.236 = 2.764 \).
Now, we will find the product:
\[ (5 - \sqrt{5}) \cdot \frac{1}{6} = \frac{5 - \sqrt{5}}{6} \]
Using the approximate value we calculated:
\[ \frac{2.764}{6} \approx 0.4607 \]
Now, let's evaluate whether this value is rational or irrational:
- The number \( 5 - \sqrt{5} \) is irrational (since \( \sqrt{5} \) is irrational, and the subtraction of a rational number from an irrational number remains irrational).
- Multiplying an irrational number (\( 5 - \sqrt{5} \)) by a rational number (\( \frac{1}{6} \)) results in an irrational number.
Thus, approximately \( 0.4607 \) is an irrational number.
However, the response choices do not include \( 0.4607; an irrational number \). Provided choices should be evaluated based on their closeness to \( \frac{5 - \sqrt{5}}{6} \):
- The closest choice is \( 0.3726 \ldots; an irrational number \) or \( 0.37; a rational number \).
The correct classification must indicate that the product is indeed irrational, but no exact computation confirms \( 0.3726 \ldots \) from \( \frac{5 - \sqrt{5}}{6} \).
However, selecting options relative to the product signify:
- 0.3726 . . . ; an irrational number seems more suitable due to irrationality.
So the final answer is:
0.3726 . . . ; an irrational number.