To find the volume of a pyramid with a square base, let's go through the solution step by step.
1. Identify the side length of the base:
The side length of the square base, [tex]\( s \)[/tex], is given as 5.
2. Calculate the height of the pyramid:
The height of the pyramid is [tex]\(\frac{2}{3}\)[/tex] that of the side length:
[tex]\[
\text{Height} = \frac{2}{3} \times 5 = \frac{10}{3}
\][/tex]
3. Find the area of the base:
Since the base is a square with side length [tex]\( s = 5 \)[/tex]:
[tex]\[
\text{Base Area} = s^2 = 5^2 = 25
\][/tex]
4. Use the formula for the volume of a pyramid:
The volume [tex]\( V \)[/tex] of a pyramid is given by [tex]\(\frac{1}{3} \times \text{Base Area} \times \text{Height}\)[/tex]:
[tex]\[
V = \frac{1}{3} \times 25 \times \frac{10}{3}
\][/tex]
5. Simplify the expression:
[tex]\[
V = \frac{1}{3} \times 25 \times \frac{10}{3} = \frac{25 \times 10}{9} = \frac{250}{9} \approx 27.77777777777777
\][/tex]
Given this true numerical result, identify the correct expression from the options provided:
- Option A: [tex]\(V = 2 s^2\)[/tex]
- Option B: [tex]\(V = 25^3\)[/tex]
- Option C: [tex]\(V = \frac{2}{3} s^2\)[/tex]
- Option D: [tex]\(V = \frac{2}{5}\)[/tex]
- Option E: [tex]\(V = \frac{1}{3} s^3\)[/tex]
The correct mathematical expression for the volume, given [tex]\( s = 5 \)[/tex], aligns with:
[tex]\[
V = \frac{1}{3} s^3
\][/tex]
This corresponds to Option E, [tex]\( V = \frac{1}{3} s^3 \)[/tex]. Thus, the expression that represents the volume of the pyramid is:
E. [tex]\( V = \frac{1}{3} s^3 \)[/tex]
