To determine how many foam cubes with edge lengths of [tex]\(\frac{1}{4}\)[/tex] inch are needed to fill the given container with dimensions [tex]\(\frac{3}{4} \times 3 \times x\)[/tex], we need to follow a series of steps involving volume calculation and division.
### Step-by-Step Solution:
1. Calculate the Volume of the Container:
- The dimensions of the container are given as [tex]\(\frac{3}{4}\)[/tex] inch, 3 inches, and some unspecified length [tex]\(x\)[/tex]. However, it appears from the numerical result that the specified dimensions might actually equal an equivalent product resulting in a container volume.
- Given that the volume of the container calculated is [tex]\(2.25\)[/tex] cubic inches, we use that as the total volume we need to fill.
2. Calculate the Volume of One Foam Cube:
- Each foam cube has an edge length of [tex]\(\frac{1}{4}\)[/tex] inch.
- Volume of one foam cube [tex]\( = (\frac{1}{4} \text{ inch})^3 = \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} = \frac{1}{64}\)[/tex] cubic inches or [tex]\(0.015625\)[/tex] cubic inches.
3. Determine the Number of Foam Cubes Needed:
- The number of foam cubes required is the total volume of the container divided by the volume of one foam cube.
- Using the provided volumes:
[tex]\[
\text{Number of cubes} = \frac{2.25 \text{ cubic inches}}{0.015625 \text{ cubic inches per cube}} = 144 \text{ cubes}
\][/tex]
### Conclusion:
The number of foam cubes with edge lengths of [tex]\(\frac{1}{4}\)[/tex] inch required to fill the given container is [tex]\(144\)[/tex] cubes.
Thus, none of the provided answer choices (a, b, c, d) match the calculated number directly. There might be a potential error in the listed choices or misunderstanding in interpreting the problem dimensions. The correct number of foam cubes, based on the provided calculations, is:
[tex]\[
\boxed{144} \text{ cubes}
\][/tex]
