To determine which expression represents the volume of a cone where the height is twice the radius of its base, let's go through the process step-by-step.
### Step 1: Recall the Formula for the Volume of 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]
Where:
- [tex]\( r \)[/tex] is the radius of the base of the cone.
- [tex]\( h \)[/tex] is the height of the cone.
### Step 2: Substitute the Given Condition
According to the problem, the height [tex]\( h \)[/tex] of the cone is twice the radius [tex]\( r \)[/tex]. Therefore, we can express the height [tex]\( h \)[/tex] as:
[tex]\[ h = 2r \][/tex]
### Step 3: Substitute [tex]\( h \)[/tex] into the Volume Formula
Now substitute [tex]\( h = 2r \)[/tex] into the volume formula:
[tex]\[ V = \frac{1}{3} \pi r^2 (2r) \][/tex]
### Step 4: Simplify the Expression
Simplify the expression by multiplying inside the parentheses:
[tex]\[ V = \frac{1}{3} \pi r^2 \cdot 2r \][/tex]
[tex]\[ V = \frac{1}{3} \pi r^2 \cdot 2r = \frac{1}{3} \pi \cdot 2 \cdot r^3 \][/tex]
[tex]\[ V = \frac{2}{3} \pi r^3 \][/tex]
### Conclusion
The expression that represents the volume of the cone, in cubic units, is:
[tex]\[ \frac{2}{3} \pi r^3 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\frac{2}{3} \pi x^3} \][/tex]
