To determine the volume of a cone with a base diameter and height both equal to [tex]\(x\)[/tex] units, we will use the formula for the volume of a cone:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Here's the step-by-step solution:
1. Identify the given values:
- The diameter of the base of the cone is [tex]\(x\)[/tex].
- The height of the cone is [tex]\(x\)[/tex].
2. Find the radius of the base:
The radius [tex]\(r\)[/tex] is half of the diameter. So,
[tex]\[ r = \frac{x}{2} \][/tex]
3. Substitute the radius and height into the volume formula:
[tex]\[ V = \frac{1}{3} \pi \left( \frac{x}{2} \right)^2 \cdot x \][/tex]
4. Simplify the expression:
- Square the radius:
[tex]\[ \left( \frac{x}{2} \right)^2 = \frac{x^2}{4} \][/tex]
- Substitute this back into the volume formula:
[tex]\[ V = \frac{1}{3} \pi \cdot \frac{x^2}{4} \cdot x \][/tex]
5. Combine the factors:
[tex]\[ V = \frac{1}{3} \cdot \frac{1}{4} \pi x^2 \cdot x \][/tex]
[tex]\[ V = \frac{1}{12} \pi x^3 \][/tex]
6. The expression that represents the volume of the cone is:
[tex]\[ \frac{1}{12} \pi x^3 \][/tex]
Therefore, the correct answer is:
[tex]\[ \frac{1}{12} \pi x^3 \][/tex]
