To find the product of the square root of 12 and \( \frac{5}{6} \), we first calculate the square root of 12:
\[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]
Next, we multiply \( 2\sqrt{3} \) by \( \frac{5}{6} \):
\[ \text{Product} = 2\sqrt{3} \times \frac{5}{6} = \frac{10\sqrt{3}}{6} = \frac{5\sqrt{3}}{3} \]
Next, we can approximate \( \sqrt{3} \) which is approximately 1.732:
\[ \frac{5\sqrt{3}}{3} \approx \frac{5 \times 1.732}{3} = \frac{8.66}{3} \approx 2.8867 \]
Now, we can compare our result to the options given:
a) 2.6307
b) 2.8867
c) 4.2974
The closest match is b) 2.8867.
Thus, the product of the square root of 12 and \( \frac{5}{6} \) is approximately 2.8867. This is a rational number since it can be expressed as a fraction, and also an irrational number due to the inclusion of \( \sqrt{3} \), leading to a non-terminating decimal. However, since the required answer relates to options provided, we recognize the approximate value.