To solve the problem of finding the volume of the remaining space in the cylinder after the cone is placed inside, let's follow these steps in detail:
1. Determine the volume of the cylinder:
The volume [tex]\( V_{\text{cylinder}} \)[/tex] of a cylinder with radius [tex]\( r \)[/tex] and height [tex]\( h \)[/tex] is given by the formula:
[tex]\[
V_{\text{cylinder}} = \pi r^2 h
\][/tex]
2. Determine the volume of the cone:
The volume [tex]\( V_{\text{cone}} \)[/tex] of a cone with the same radius [tex]\( r \)[/tex] and height [tex]\( h \)[/tex] is given by the formula:
[tex]\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\][/tex]
3. Calculate the volume of the remaining space:
The volume of the space remaining in the cylinder after placing the cone inside is the difference between the volume of the cylinder and the volume of the cone:
[tex]\[
V_{\text{remaining}} = V_{\text{cylinder}} - V_{\text{cone}}
\][/tex]
Plugging in the formulas for [tex]\( V_{\text{cylinder}} \)[/tex] and [tex]\( V_{\text{cone}} \)[/tex]:
[tex]\[
V_{\text{remaining}} = \pi r^2 h - \frac{1}{3} \pi r^2 h
\][/tex]
Simplify the expression:
[tex]\[
V_{\text{remaining}} = \left( \pi r^2 h \right) - \left( \frac{1}{3} \pi r^2 h \right)
\][/tex]
[tex]\[
V_{\text{remaining}} = \pi r^2 h \left( 1 - \frac{1}{3} \right)
\][/tex]
[tex]\[
V_{\text{remaining}} = \pi r^2 h \left( \frac{3}{3} - \frac{1}{3} \right)
\][/tex]
[tex]\[
V_{\text{remaining}} = \pi r^2 h \left( \frac{2}{3} \right)
\][/tex]
[tex]\[
V_{\text{remaining}} = \frac{2}{3} \pi r^2 h
\][/tex]
4. Match the result with the given choices:
The volume of the space remaining in the cylinder is:
[tex]\[
\boxed{\frac{2}{3} \pi r^2 h}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{B. \frac{2}{3} \pi r^2 h}
\][/tex]
