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A cone is placed inside a cylinder. The cone has half the radius of the cylinder, but the height of each figure is the same. The cone is tilted at an angle so its peak touches the edge of the cylinder's base. What is the volume of the space remaining in the cylinder after the cone is placed inside?

A. [tex]\frac{3}{4} \pi r^2 h[/tex]
B. [tex]\frac{2}{3} \pi r^2 h[/tex]
C. [tex]\frac{11}{12} \pi r^2 h[/tex]
D. [tex]\frac{5}{6} \pi r^2 h[/tex]
E. [tex]\frac{1}{2} \pi r^2 h[/tex]


Sagot :

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).
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