To determine which expression shows the area of the base of each pyramid, let's consider each of the given options. The pyramids' bases are squares, meaning we calculate the area by squaring the side length of these bases.
1. The first option is [tex]\(\left(\frac{1}{4} x\right)^2\)[/tex].
- If the side length of the base of each pyramid is [tex]\(\frac{1}{4} x\)[/tex], then the area of the base is:
[tex]\[
\left(\frac{1}{4} x\right)^2 = \left(\frac{1}{4}\right)^2 \times x^2 = \frac{1}{16} x^2 \approx 0.0625 x^2
\][/tex]
2. The second option is [tex]\(\left(\frac{1}{3} x\right)^2\)[/tex].
- If the side length of the base of each pyramid is [tex]\(\frac{1}{3} x\)[/tex], then the area of the base is:
[tex]\[
\left(\frac{1}{3} x\right)^2 = \left(\frac{1}{3}\right)^2 \times x^2 = \frac{1}{9} x^2 \approx 0.111111 x^2
\][/tex]
3. The third option is [tex]\(\left(\frac{1}{2} x\right)^2\)[/tex].
- If the side length of the base of each pyramid is [tex]\(\frac{1}{2} x\)[/tex], then the area of the base is:
[tex]\[
\left(\frac{1}{2} x\right)^2 = \left(\frac{1}{2}\right)^2 \times x^2 = \frac{1}{4} x^2 = 0.25 x^2
\][/tex]
4. The fourth option is [tex]\(x^2\)[/tex].
- If the side length of the base of each pyramid is [tex]\(x\)[/tex], then the area of the base is:
[tex]\[
x^2 = 1 \times x^2 = x^2
\][/tex]
After comparing the computed areas with the expressions given in each option, we see:
- [tex]\(\left(\frac{1}{4} x\right)^2\)[/tex] results in [tex]\(0.0625 x^2\)[/tex]
- [tex]\(\left(\frac{1}{3} x\right)^2\)[/tex] results in [tex]\(0.111111 x^2\)[/tex]
- [tex]\(\left(\frac{1}{2} x\right)^2\)[/tex] results in [tex]\(0.25 x^2\)[/tex]
- [tex]\(x^2\)[/tex] remains as [tex]\(x^2\)[/tex]
Thus, the expressions for the area of the base of each pyramid match the given numerical outcomes precisely.
