Let's break down the problem step by step to find out how many foam cubes with edge lengths of [tex]\(\frac{1}{4}\)[/tex] inches it would take to fill a given container with dimensions [tex]\( \frac{3}{4} \times 3 \times 4\frac{1}{2} \)[/tex].
### Step 1: Compute the Volume of the Container
The dimensions of the container are given as:
- Length: [tex]\( \frac{3}{4} \)[/tex] inches
- Width: [tex]\( 3 \)[/tex] inches
- Height: [tex]\( 4\frac{1}{2} \)[/tex] inches (which is equivalent to 4.5 inches)
To find the volume of the container, we multiply these three dimensions together:
[tex]\[
\text{Volume of the container} = \left(\frac{3}{4}\right) \times 3 \times 4.5
\][/tex]
### Step 2: Compute the Volume of One Foam Cube
The edge length of each foam cube is given as [tex]\(\frac{1}{4}\)[/tex] inches. The volume of a cube is calculated by raising the edge length to the third power:
[tex]\[
\text{Volume of one foam cube} = \left(\frac{1}{4}\right)^3 = \frac{1}{64} \text{ cubic inches}
\][/tex]
### Step 3: Calculate the Number of Foam Cubes Needed
To determine the number of foam cubes required to fill the container, we divide the volume of the container by the volume of one foam cube:
[tex]\[
\text{Number of foam cubes} = \frac{\text{Volume of the container}}{\text{Volume of one foam cube}}
\][/tex]
Plugging in the numbers:
[tex]\[
\text{Volume of the container} = 10.125 \text{ cubic inches}
\][/tex]
[tex]\[
\text{Volume of one foam cube} = 0.015625 \text{ cubic inches}
\][/tex]
[tex]\[
\text{Number of foam cubes} = \frac{10.125}{0.015625} = 648
\][/tex]
Thus, it would take 648 foam cubes with edge lengths of [tex]\( \frac{1}{4} \)[/tex] inches to fill the container.
### Final Answer
The correct choice is:
[tex]\[
\boxed{648}
\][/tex]
However, this answer isn't one of the provided options. Given this discrepancy, it seems there might be a mistake in the provided options themselves. Based on the computations, the true correct answer remains [tex]\( 648 \)[/tex] foam cubes, despite not matching the provided choices directly.
