To solve the problem of finding the remaining volume in the cylinder after a cone is placed inside it, we'll follow these steps:
1. Calculate the volume of the cylinder:
- The formula for the volume of a cylinder is given by:
[tex]\[
V_{\text{cylinder}} = \pi r^2 h
\][/tex]
where [tex]\( r \)[/tex] is the radius of the cylinder and [tex]\( h \)[/tex] is the height.
2. Calculate the volume of the cone:
- The radius of the cone is half the radius of the cylinder, i.e., [tex]\( \frac{r}{2} \)[/tex].
- The height of the cone is the same as the height of the cylinder, which is [tex]\( h \)[/tex].
- The formula for the volume of a cone is given by:
[tex]\[
V_{\text{cone}} = \frac{1}{3} \pi \left(\frac{r}{2}\right)^2 h
\][/tex]
- Simplifying the expression for the volume of the cone:
[tex]\[
V_{\text{cone}} = \frac{1}{3} \pi \left(\frac{r^2}{4}\right) h
\][/tex]
[tex]\[
V_{\text{cone}} = \frac{1}{3} \pi \frac{r^2}{4} h
\][/tex]
[tex]\[
V_{\text{cone}} = \frac{1}{12} \pi r^2 h
\][/tex]
3. Calculate the remaining volume in the cylinder:
- Subtract the volume of the cone from the volume of the cylinder:
[tex]\[
V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}}
\][/tex]
[tex]\[
V_{\text{remaining}} = \pi r^2 h - \frac{1}{12} \pi r^2 h
\][/tex]
- Factor out [tex]\( \pi r^2 h \)[/tex]:
[tex]\[
V_{\text{remaining}} = \left(1 - \frac{1}{12}\right) \pi r^2 h
\][/tex]
[tex]\[
V_{\text{remaining}} = \left(\frac{12}{12} - \frac{1}{12}\right) \pi r^2 h
\][/tex]
[tex]\[
V_{\text{remaining}} = \frac{11}{12} \pi r^2 h
\][/tex]
Thus, the volume of the space remaining in the cylinder after the cone is placed inside is:
[tex]\[
\boxed{\frac{11}{12} \pi r^2 h}
\][/tex]
This matches option (C).
