To determine the volume of the cone given that its height is twice the radius of its base, let's proceed step-by-step.
1. Identify the Variables
- Let the radius of the cone be [tex]\( r \)[/tex].
- The height [tex]\( h \)[/tex] is given as twice the radius, so [tex]\( h = 2r \)[/tex].
2. Volume Formula for a Cone
The volume [tex]\( V \)[/tex] of a cone is given by the formula:
[tex]\[
V = \frac{1}{3} \pi r^2 h
\][/tex]
3. Substitute the Given Height
Substitute [tex]\( h = 2r \)[/tex] into the volume formula:
[tex]\[
V = \frac{1}{3} \pi r^2 (2r)
\][/tex]
4. Simplify the Expression
Simplify the expression by combining the terms:
[tex]\[
V = \frac{1}{3} \pi r^2 \cdot 2r
\][/tex]
[tex]\[
V = \frac{1}{3} \pi \cdot 2r^3
\][/tex]
[tex]\[
V = \frac{2}{3} \pi r^3
\][/tex]
5. Interpretation
Since the radius [tex]\( r \)[/tex] might be represented as [tex]\( x \)[/tex] in certain contexts, we can rewrite the expression using [tex]\( x \)[/tex] instead of [tex]\( r \)[/tex]:
[tex]\[
V = \frac{2}{3} \pi x^3
\][/tex]
Therefore, the expression that represents the volume of the cone is:
[tex]\[
\frac{2}{3} \pi x^3
\][/tex]
Thus, the correct option is:
[tex]\[
\boxed{\frac{2}{3} \pi x^3}
\][/tex]
