To determine the area of the base of each pyramid, we need to follow these logical steps:
1. Understand the Geometry: The soundproofing tile is made up of 8 identical solid right pyramids with square bases and it forms a square tile with a total side length of [tex]\( x \)[/tex].
2. Base Length Distribution: Since the entire tile's side length is [tex]\( x \)[/tex] and there are 8 pyramids, we need to understand how the pyramids are positioned. Consider that 4 pyramids form one layer and 4 pyramids form another layer on top of the first, keeping its total length covered by both the layers.
3. Base of Each Pyramid: If the whole tile's length is [tex]\( x \)[/tex], the base of each pyramid needs to match the distribution over the tile.
4. Length of Each Base: Since 2 pyramids need to fit along the side of the square, each of these pyramids will therefore have a base length that is half the total length, so it will be [tex]\( x/2 \)[/tex].
5. Area of Each Base: The area of the base of each pyramid is given by the square of the base's length. If each base length is [tex]\( x/2 \)[/tex],
[tex]\[
\text{Area of the base} = \left(\frac{x}{2}\right)^2 = \frac{x^2}{4}
\][/tex]
6. Conclusion: Therefore, the correct expression for the area of the base of each pyramid is:
[tex]\[
\left(\frac{1}{2} x\right)^2 \text{ in}^2
\][/tex]
Among the given choices, the correct one is:
[tex]\[
\left(\frac{1}{2} x\right)^2 \, \text{in}^2
\][/tex]
