To determine which expression represents the volume of the cone, we need to apply the formula for the volume of a cone and relate it to the given condition where both the base diameter and the height of the cone are equal to [tex]\( x \)[/tex] units.
1. 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]
where [tex]\( r \)[/tex] is the radius of the base, and [tex]\( h \)[/tex] is the height of the cone.
2. Relate the problem's conditions to the formula:
Given that the base diameter and the height of the cone are both [tex]\( x \)[/tex] units:
- The diameter of the base [tex]\( D \)[/tex] is [tex]\( x \)[/tex], so the radius [tex]\( r \)[/tex], being half of the diameter, is:
[tex]\[
r = \frac{x}{2}
\][/tex]
- The height [tex]\( h \)[/tex] of the cone is:
[tex]\[
h = x
\][/tex]
3. Substitute [tex]\( r \)[/tex] and [tex]\( h \)[/tex] into the volume formula:
Substitute [tex]\( r = \frac{x}{2} \)[/tex] and [tex]\( h = x \)[/tex] into the volume formula [tex]\( V = \frac{1}{3} \pi r^2 h \)[/tex]:
[tex]\[
V = \frac{1}{3} \pi \left( \frac{x}{2} \right)^2 x
\][/tex]
4. Simplify the expression:
- Calculate [tex]\( \left( \frac{x}{2} \right)^2 \)[/tex]:
[tex]\[
\left( \frac{x}{2} \right)^2 = \frac{x^2}{4}
\][/tex]
- Substitute [tex]\( \frac{x^2}{4} \)[/tex] into the volume formula:
[tex]\[
V = \frac{1}{3} \pi \cdot \frac{x^2}{4} \cdot x
\][/tex]
- Combine the [tex]\( x \)[/tex] terms:
[tex]\[
V = \frac{1}{3} \pi \cdot \frac{x^2}{4} \cdot x = \frac{1}{3} \cdot \frac{\pi x^3}{4} = \frac{\pi x^3}{12}
\][/tex]
5. Conclusion:
The simplified expression that represents the volume of the cone is:
[tex]\[
\frac{1}{12} \pi x^3
\][/tex]
Thus, the correct expression is:
[tex]\[
\boxed{\frac{1}{12} \pi x^3}
\][/tex]
