To solve this problem, we need to compare the volume of the cylinder and the volume of the cone placed inside it, and then find the volume of the remaining space.
1. Volume of the Cylinder:
The general formula for the volume of a cylinder is given by:
[tex]\[
V_{cylinder} = \pi r^2 h
\][/tex]
where [tex]\( r \)[/tex] is the radius of the cylinder and [tex]\( h \)[/tex] is the height.
2. Volume of the Cone:
The general formula for the volume of a cone is given by:
[tex]\[
V_{cone} = \frac{1}{3} \pi r^2 h
\][/tex]
Since the cone's radius is half of the cylinder's radius, let [tex]\( r \)[/tex] be the radius of the cylinder, and thus the cone's radius will be [tex]\( \frac{r}{2} \)[/tex]. The height of the cone is the same as the height of the cylinder, [tex]\( h \)[/tex].
Substituting the radius of the cone:
[tex]\[
V_{cone} = \frac{1}{3} \pi \left(\frac{r}{2}\right)^2 h = \frac{1}{3} \pi \frac{r^2}{4} h = \frac{1}{12} \pi r^2 h
\][/tex]
3. Volume of the Remaining Space:
The volume of the remaining space inside the cylinder can be found by subtracting the volume of the cone from the volume of the cylinder:
[tex]\[
V_{remaining} = V_{cylinder} - V_{cone} = \pi r^2 h - \frac{1}{12} \pi r^2 h
\][/tex]
Simplify the expression:
[tex]\[
V_{remaining} = \pi r^2 h \left(1 - \frac{1}{12}\right) = \pi r^2 h \left(\frac{12}{12} - \frac{1}{12}\right) = \pi r^2 h \left(\frac{11}{12}\right) = \frac{11}{12} \pi r^2 h
\][/tex]
Thus, the volume of the space remaining in the cylinder after the cone is placed inside is:
[tex]\(\frac{11}{12} \pi r^2 h\)[/tex].
The correct answer is:
C. [tex]\(\frac{11}{12} \pi r^2 h\)[/tex]
