To find the area of the base of each pyramid given that a soundproofing tile is made of 8 identical solid right pyramids with square bases, and the length of the whole tile is [tex]\(x\)[/tex] inches, let's break down the problem step-by-step.
1. Understand the Structure of the Tile:
- The tile is composed of 8 identical pyramids.
- The pyramids have square bases.
2. Determine the Layout of the Pyramids within the Tile:
- Since the pyramids are identical and each has a square base, the arrangement of pyramids should exploit symmetry.
- For simplicity, let's consider that the overall dimension [tex]\(x\)[/tex] forms a larger square composed of these smaller square-based pyramids arranged in a specific pattern.
3. Side Length Calculation:
- Each pyramid has a square base, and there are 8 of these squares forming the whole tile.
- To simplify calculations, let's assume the most logical arrangement is a 2x4 pattern or 4x2 pattern (both have area 8 smaller squares).
4. Side Length of One Pyramid's Base:
- Therefore, if [tex]\(x\)[/tex] is the entire length of the tile along one dimension containing two pyramids' bases and along the other dimension containing four pyramids' bases, the length of side of the base of each pyramid will be the smallest segment forming the length [tex]\(x\)[/tex].
- Let's set the tile dimension [tex]\(x\)[/tex] along the first pattern (2 pyramids) with equal bases:
- [tex]\(L = x / 2\)[/tex]
- Because there are four pyramids along the remaining side: [tex]\(W = x / 4\)[/tex].
- Now, the area of the base of each pyramid will be [tex]\(L \cdot W\)[/tex].
5. Determining Area:
- Since [tex]\(L = \frac{x}{4}\)[/tex],
- The area for the base of each pyramid can be achieved by squaring the side length of one of the bases:
[tex]\[L = \frac{x}{4}\][/tex]
[tex]\[L^2 = \left(\frac{x}{4} \right)^2\][/tex]
6. Conclusion:
- The area of the base of each square-based pyramid is [tex]\( \left(\frac{1}{4}x \right)^2 \)[/tex] inches.
Hence, the correct expression for the area of the base of each pyramid is:
[tex]\[
\left(\frac{1}{4} x \right)^2\text{ in}^2
\][/tex]
