To find the volume of a cylinder, we use the formula:
[tex]\[ V = \pi r^2 h \][/tex]
where [tex]\( r \)[/tex] is the radius of the base of the cylinder and [tex]\( h \)[/tex] is its height.
According to the problem, the height [tex]\( h \)[/tex] is twice the radius [tex]\( r \)[/tex]. This can be written as:
[tex]\[ h = 2r \][/tex]
We can substitute this expression for [tex]\( h \)[/tex] into the volume formula to get:
[tex]\[ V = \pi r^2 (2r) \][/tex]
Simplifying the equation, we have:
[tex]\[ V = \pi r^2 \cdot 2r \][/tex]
[tex]\[ V = 2 \pi r^3 \][/tex]
Thus, the expression for the volume of the cylinder, in cubic units, is:
[tex]\[ 2 \pi r^3 \][/tex]
Given the options:
1. [tex]\( 4 \pi x^2 \)[/tex]
2. [tex]\( 2 \pi x^2 \)[/tex]
3. [tex]\( \pi x^2 + 2x \)[/tex]
4. [tex]\( 2 + \pi x^3 \)[/tex]
The correct expression is not explicitly listed among the options as [tex]\( 2 \pi r^3 \)[/tex], but if we relate the given variable [tex]\( x \)[/tex] to [tex]\( r \)[/tex], the answer should be the expression [tex]\( 2 \pi x^3 \)[/tex].
The volume is not [tex]\( 2 \)[/tex] plus [tex]\( \pi x^3 \)[/tex], but rather [tex]\( 2 \pi x^3 \)[/tex]. Should none of the given options correctly include this expression entirely, it would suggest a potential issue with the problem setup or given choices, as the correct mathematical expression derived is not fully present.
