To address this question, let's delve into the fundamental formulas for the volumes of a cylinder and a cone.
1. Volume of a Cylinder:
[tex]\[
V_{\text{cylinder}} = \pi r^2 h
\][/tex]
where [tex]\( r \)[/tex] is the radius of the base, and [tex]\( h \)[/tex] is the height of the cylinder.
2. Volume of a Cone:
[tex]\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\][/tex]
where [tex]\( r \)[/tex] is the radius of the base, and [tex]\( h \)[/tex] is the height of the cone.
For the cylinder and the cone to have the same volume, we need to find a relationship between their heights, given that their radii are the same.
Let the volumes of the cylinder and the cone be equal:
[tex]\[
\pi r^2 h_{\text{cylinder}} = \frac{1}{3} \pi r^2 h_{\text{cone}}
\][/tex]
Next, we can simplify and solve for the heights:
[tex]\[
h_{\text{cylinder}} = \frac{1}{3} h_{\text{cone}}
\][/tex]
This equation shows that if the height of the cylinder ([tex]\(h_{\text{cylinder}}\)[/tex]) is three times the height of the cone ([tex]\(h_{\text{cone}}\)[/tex]), both shapes can possess the same volume because the [tex]\( \pi r^2 \)[/tex] components cancel out.
Based on this analysis, we can conclude that:
[tex]\[
\text{A cylinder and a cone with the same radius could have the same volume.}
\][/tex]
Thus, the correct choice is:
[tex]\[
\boxed{\text{a. could have the same volume.}}
\][/tex]
