To determine the area of the base of each pyramid, we need to consider the total area of the square base of the whole tile.
Given that there are 8 identical solid right pyramids with square bases, the entire tile can be seen as a composition of these pyramids.
Let the length of the whole tile be [tex]\( x \)[/tex] inches. We're looking for the expression that accurately represents the area of the base of each pyramid given in the options:
1. [tex]\( \left(\frac{1}{4} x\right)^2 \text{ in}^2 \)[/tex]
2. [tex]\( \left(\frac{1}{3} x\right)^2 \text{ in}^2 \)[/tex]
3. [tex]\( \left(\frac{1}{2} x\right)^2 \text{ in}^2 \)[/tex]
4. [tex]\( x^2 \text{ in}^2 \)[/tex]
We need to check these expressions to determine which one matches the configuration where the whole tile is made up of 8 square bases of the pyramids.
Calculating each option:
1. For [tex]\( \left(\frac{1}{4} x\right)^2 \)[/tex]:
[tex]\[
\left(\frac{1}{4} x\right)^2 = \left(\frac{1}{4}\right)^2 \cdot x^2 = \frac{1}{16} x^2
\][/tex]
2. For [tex]\( \left(\frac{1}{3} x\right)^2 \)[/tex]:
[tex]\[
\left(\frac{1}{3} x\right)^2 = \left(\frac{1}{3}\right)^2 \cdot x^2 = \frac{1}{9} x^2
\][/tex]
3. For [tex]\( \left(\frac{1}{2} x\right)^2 \)[/tex]:
[tex]\[
\left(\frac{1}{2} x\right)^2 = \left(\frac{1}{2}\right)^2 \cdot x^2 = \frac{1}{4} x^2
\][/tex]
4. For [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 \cdot x^2 = x^2
\][/tex]
From these calculations, we see that the areas are respectively [tex]\( \frac{1}{16} x^2 \)[/tex], [tex]\( \frac{1}{9} x^2 \)[/tex], [tex]\( \frac{1}{4} x^2 \)[/tex], and [tex]\( x^2 \)[/tex].
To match our given tiles composed of identical pyramids and their bases summing to form the whole tile's base area accurately, we must choose the option where the area of the base of each individual pyramid is correct.
Thus, the correct expression that shows the area of the base of each pyramid is:
[tex]\[
\left( \frac{1}{2} x \right)^2 \text{ in}^2 = \frac{1}{4} x^2 \text{ in}^2
\][/tex]
