Let's go through the step-by-step solution:
1. Determine the area of the base:
- The pyramid has a square base, and each edge of the square is [tex]\(5 \: \textrm{cm}\)[/tex].
- To find the area of a square, we use the formula for the area of a square [tex]\( \text{Area} = \text{edge length} \times \text{edge length} \)[/tex].
- Substituting the given edge length:
[tex]\[
\text{Area of the base} = 5 \: \textrm{cm} \times 5 \: \textrm{cm} = 25 \: \textrm{cm}^2
\][/tex]
2. Determine the volume of the pyramid:
- The volume [tex]\(V\)[/tex] of a pyramid is given by the formula [tex]\( V = \frac{1}{3} \times \text{base area} \times \text{height} \)[/tex].
- We already calculated the base area, which is [tex]\( 25 \: \textrm{cm}^2 \)[/tex], and the height of the pyramid is given as [tex]\(7 \: \textrm{cm}\)[/tex].
- Substituting the base area and height into the volume formula:
[tex]\[
V = \frac{1}{3} \times 25 \: \textrm{cm}^2 \times 7 \: \textrm{cm}
\][/tex]
- Performing the multiplication:
[tex]\[
V = \frac{1}{3} \times 175 \: \textrm{cm}^3 = 58.33333333333333 \: \textrm{cm}^3
\][/tex]
3. Compare with the given options:
- The calculated volume of [tex]\( 58.33333333333333 \: \textrm{cm}^3 \)[/tex] can be written as a mixed number.
- Converting [tex]\( 58.33333333333333 \: \textrm{cm}^3 \)[/tex] to a mixed number, it is approximately [tex]\( 58 \frac{1}{3} \: \textrm{cm}^3 \)[/tex].
None of the given options precisely matches [tex]\( 58 \frac{1}{3} \: \textrm{cm}^3 \)[/tex], so it seems that the given options are not correct. If there is any typographical mistake in the options, that might be the reason for the discrepancy. But according to our calculations, the volume should be [tex]\( 58 \frac{1}{3} \: \textrm{cm}^3 \)[/tex].
