To find the product of \( \sqrt{12} \) and \( \frac{5}{6} \), we can first simplify \( \sqrt{12} \):
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]
Now, we multiply this by \( \frac{5}{6} \):
\[ \frac{5}{6} \times 2\sqrt{3} = \frac{10\sqrt{3}}{6} = \frac{5\sqrt{3}}{3} \]
Next, we can approximate \( \sqrt{3} \) (which is an irrational number) with its decimal value. The approximate value of \( \sqrt{3} \) is about 1.732.
Calculating the product:
\[ \frac{5\sqrt{3}}{3} \approx \frac{5 \times 1.732}{3} \approx \frac{8.66}{3} \approx 2.8867 \]
Now, looking at the options provided:
- A. 4.2974...; an irrational number
- B. 2.8867...; an irrational number
- C. 4.1569...; an irrational number
- D. 2.6307...; an irrational number
The closest match to our calculated value \( \approx 2.8867 \) is option B. 2.8867...; an irrational number. Thus, the answer is:
B. 2.8867...; an irrational number.