Answer :
Let's solve the problem step-by-step:
1. Identify the formulas and given values:
- The base of the pyramid is a square with side length [tex]\( s \)[/tex].
- The height of the pyramid [tex]\( h \)[/tex] is given as [tex]\(\frac{2}{3}\)[/tex] of the side length [tex]\( s \)[/tex].
2. Calculate the height [tex]\( h \)[/tex]:
[tex]\[ h = \frac{2}{3}s \][/tex]
3. Formula for the volume of the pyramid:
The volume [tex]\( V \)[/tex] of a pyramid is given by:
[tex]\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \][/tex]
4. Determine the base area:
The base area of the square is:
[tex]\[ \text{Base Area} = s^2 \][/tex]
5. Substitute the base area and height into the volume formula:
[tex]\[ V = \frac{1}{3} \times s^2 \times \frac{2}{3}s \][/tex]
6. Simplify the volume expression:
[tex]\[ V = \frac{1}{3} \times s^2 \times \frac{2}{3}s = \frac{1}{3} \times \frac{2}{3} \times s^3 \][/tex]
[tex]\[ V = \frac{2}{9} s^3 \][/tex]
The correct simplified expression for the volume of the pyramid is thus:
[tex]\[ V = \frac{2}{9} s^3 \][/tex]
Upon closely comparing with the given options, none matches exactly.
However, since the closest option appears to be incorrect in representing our exact value [tex]\(\frac{2}{9} s^3\)[/tex], this problem's answer seemingly is unincluded. Nonetheless, understanding the simplicity for the right expression governs understanding steps clearly aligning correct algebra.
1. Identify the formulas and given values:
- The base of the pyramid is a square with side length [tex]\( s \)[/tex].
- The height of the pyramid [tex]\( h \)[/tex] is given as [tex]\(\frac{2}{3}\)[/tex] of the side length [tex]\( s \)[/tex].
2. Calculate the height [tex]\( h \)[/tex]:
[tex]\[ h = \frac{2}{3}s \][/tex]
3. Formula for the volume of the pyramid:
The volume [tex]\( V \)[/tex] of a pyramid is given by:
[tex]\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \][/tex]
4. Determine the base area:
The base area of the square is:
[tex]\[ \text{Base Area} = s^2 \][/tex]
5. Substitute the base area and height into the volume formula:
[tex]\[ V = \frac{1}{3} \times s^2 \times \frac{2}{3}s \][/tex]
6. Simplify the volume expression:
[tex]\[ V = \frac{1}{3} \times s^2 \times \frac{2}{3}s = \frac{1}{3} \times \frac{2}{3} \times s^3 \][/tex]
[tex]\[ V = \frac{2}{9} s^3 \][/tex]
The correct simplified expression for the volume of the pyramid is thus:
[tex]\[ V = \frac{2}{9} s^3 \][/tex]
Upon closely comparing with the given options, none matches exactly.
However, since the closest option appears to be incorrect in representing our exact value [tex]\(\frac{2}{9} s^3\)[/tex], this problem's answer seemingly is unincluded. Nonetheless, understanding the simplicity for the right expression governs understanding steps clearly aligning correct algebra.
