To solve this problem, let's break it down step-by-step:
1. Define the variables:
Let the radius of the cylinder be [tex]\( r \)[/tex] and the height of the cylinder be [tex]\( h \)[/tex].
2. Determine the dimensions of the cone:
According to the problem, the cone has half the radius of the cylinder and the same height. Therefore, the radius of the cone is [tex]\( \frac{r}{2} \)[/tex] and the height of the cone is [tex]\( h \)[/tex].
3. Calculate the volume of the cylinder:
The volume [tex]\( V_{\text{cylinder}} \)[/tex] of a cylinder is given by the formula:
[tex]\[
V_{\text{cylinder}} = \pi r^2 h
\][/tex]
4. Calculate the volume of the cone:
The volume [tex]\( V_{\text{cone}} \)[/tex] of a cone is given by the formula:
[tex]\[
V_{\text{cone}} = \frac{1}{3} \pi r_{\text{cone}}^2 h_{\text{cone}}
\][/tex]
Given the radius and height of the cone, we have:
[tex]\[
V_{\text{cone}} = \frac{1}{3} \pi \left( \frac{r}{2} \right)^2 h = \frac{1}{3} \pi \left( \frac{r^2}{4} \right) h = \frac{1}{3} \cdot \frac{1}{4} \pi r^2 h = \frac{1}{12} \pi r^2 h
\][/tex]
5. Calculate the remaining volume in the cylinder:
The remaining volume [tex]\( V_{\text{remaining}} \)[/tex] is the volume of the cylinder minus the volume of the cone:
[tex]\[
V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}} = \pi r^2 h - \frac{1}{12} \pi r^2 h
\][/tex]
Simplifying this, we get:
[tex]\[
V_{\text{remaining}} = \left(1 - \frac{1}{12}\right) \pi r^2 h = \left(\frac{12}{12} - \frac{1}{12}\right) \pi r^2 h = \frac{11}{12} \pi r^2 h
\][/tex]
Therefore, the volume of the space remaining in the cylinder after the cone is placed inside it is [tex]\( \frac{11}{12} \pi r^2 h \)[/tex].
The correct option is:
[tex]\[ \text{C.} \frac{11}{12} \pi r^2 h \][/tex]
