To determine the expression that represents the volume of a cylinder given that its height is twice the radius of its base, follow these steps:
1. Understand the formula for the volume of a cylinder:
The volume [tex]\( V \)[/tex] of a cylinder is given by the formula:
[tex]\[
V = \pi r^2 h
\][/tex]
where [tex]\( r \)[/tex] is the radius of the base and [tex]\( h \)[/tex] is the height of the cylinder.
2. Identify the relationship between the height and the radius:
According to the problem, the height [tex]\( h \)[/tex] of the cylinder is twice the radius [tex]\( r \)[/tex]. Therefore, we can write:
[tex]\[
h = 2r
\][/tex]
3. Substitute the height in the volume formula:
Substitute [tex]\( h = 2r \)[/tex] into the volume formula [tex]\( V = \pi r^2 h \)[/tex]:
[tex]\[
V = \pi r^2 (2r)
\][/tex]
4. Simplify the expression:
Simplify the expression by multiplying [tex]\( \pi r^2 \)[/tex] by [tex]\( 2r \)[/tex]:
[tex]\[
V = 2\pi r^3
\][/tex]
So, the volume of the cylinder, in cubic units, is given by:
[tex]\[
2\pi r^3
\][/tex]
Given the choices:
- [tex]\( 4 \pi x^2 \)[/tex]
- [tex]\( 2 \pi x^3 \)[/tex]
- [tex]\( \pi x^2 + 2x \)[/tex]
- [tex]\( 2 + \pi x^3 \)[/tex]
The correct expression that represents the volume of the cylinder, where [tex]\( x \)[/tex] (the variable used here) is the radius [tex]\( r \)[/tex], is:
[tex]\[
2 \pi x^3
\][/tex]
Thus, the answer is:
[tex]\[
2 \pi x^3
\][/tex]
