Let's determine the volume of the space remaining in the cylinder after placing the cone inside it.
First, let's consider the two figures separately:
1. 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 of the cylinder.
2. Volume of the Cone:
The formula for the volume of a cone is given by [tex]\( V_{\text{cone}} = \frac{1}{3} \pi r_{\text{cone}}^2 h \)[/tex], where [tex]\( r_{\text{cone}} \)[/tex] is the radius of the base of the cone and [tex]\( h \)[/tex] is the height of the cone.
Given that the radius of the cone is half the radius of the cylinder, we have:
[tex]\[ r_{\text{cone}} = \frac{r}{2} \][/tex]
Now we can express the volume of the cone as:
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi \left(\frac{r}{2}\right)^2 h \][/tex]
[tex]\[ V_{\text{cone}} = \frac{1}{3} \pi \left( \frac{r^2}{4} \right) h \][/tex]
[tex]\[ V_{\text{cone}} = \frac{1}{3} \cdot \frac{\pi r^2 h}{4} \][/tex]
[tex]\[ V_{\text{cone}} = \frac{1}{12} \pi r^2 h \][/tex]
Next, to find the volume of the space remaining in the cylinder after placing the cone inside it, we subtract the volume of the cone from the volume of the cylinder:
[tex]\[ \text{Remaining Volume} = V_{\text{cylinder}} - V_{\text{cone}} \][/tex]
[tex]\[ \text{Remaining Volume} = \pi r^2 h - \frac{1}{12} \pi r^2 h \][/tex]
[tex]\[ \text{Remaining Volume} = \pi r^2 h \left(1 - \frac{1}{12}\right) \][/tex]
[tex]\[ \text{Remaining Volume} = \pi r^2 h \left(\frac{12}{12} - \frac{1}{12}\right) \][/tex]
[tex]\[ \text{Remaining Volume} = \pi r^2 h \cdot \frac{11}{12} \][/tex]
[tex]\[ \text{Remaining Volume} = \frac{11}{12} \pi r^2 h \][/tex]
Therefore, the volume of the space remaining in the cylinder after placing the cone is:
[tex]\[ \boxed{\frac{11}{12} \pi r^2 h} \][/tex]
