To determine which choice is equivalent to the product of [tex]\(\sqrt{\frac{3}{16}} \cdot \sqrt{\frac{3}{9}}\)[/tex], let's break down the problem step by step.
1. Calculate [tex]\(\sqrt{\frac{3}{16}}\)[/tex]:
[tex]\[
\sqrt{\frac{3}{16}} \approx 0.433
\][/tex]
2. Calculate [tex]\(\sqrt{\frac{3}{9}}\)[/tex]:
[tex]\[
\sqrt{\frac{3}{9}} \approx 0.577
\][/tex]
3. Multiply the two square roots together:
[tex]\[
\sqrt{\frac{3}{16}} \cdot \sqrt{\frac{3}{9}} \approx 0.433 \cdot 0.577 \approx 0.25
\][/tex]
4. Convert the result to a fraction:
The decimal [tex]\(0.25\)[/tex] can be written as the fraction [tex]\(\frac{1}{4}\)[/tex].
5. Compare [tex]\(\frac{1}{4}\)[/tex] with the given choices:
[tex]\[
\begin{aligned}
&A. \quad \frac{3}{12} = \frac{1}{4} \\
&B. \quad \frac{3}{4} \\
&C. \quad \frac{6}{12} = \frac{1}{2} \\
&D. \quad \frac{1}{2} \\
&E. \quad \frac{7}{12}
\end{aligned}
\][/tex]
The fraction [tex]\(\frac{3}{12}\)[/tex] simplifies to [tex]\(\frac{1}{4}\)[/tex], which matches our calculated result.
Thus, the correct choice is:
[tex]\[
\boxed{\text{A}}
\][/tex]
