To determine the correct equation representing the volume of each cone inscribed in a cylinder, let's break down the problem step-by-step.
1. Understand the Problem Statement:
- There are two identical cones inscribed in a cylinder.
- We need to find the equation for the volume of each cone using the given answer choices.
2. Volume of a Cone Formula:
The general formula for the volume [tex]\( V \)[/tex] of a cone is:
[tex]\[
V = \frac{1}{3}\pi r^2 h
\][/tex]
where:
- [tex]\( r \)[/tex] is the radius of the base of the cone.
- [tex]\( h \)[/tex] is the height of the cone.
3. Cones Inscribed in a Cylinder:
- Let's denote the radius of the cylinder (which is also the radius of each cone) as [tex]\( m \)[/tex].
- The height of the cylinder is [tex]\( H \)[/tex].
- Since two identical cones are inscribed in this cylinder, the height of each cone will be half the height of the cylinder, so [tex]\( h = \frac{H}{2} \)[/tex].
4. Substitute Values into the Volume Formula:
Substituting [tex]\( r = m \)[/tex] and [tex]\( h = \frac{H}{2} \)[/tex] into the cone volume formula:
[tex]\[
V = \frac{1}{3}\pi m^2 \left(\frac{H}{2}\right)
\][/tex]
5. Simplify the Formula:
Now, simplify the expression:
[tex]\[
V = \frac{1}{3}\pi m^2 \cdot \frac{H}{2}
\][/tex]
[tex]\[
V = \frac{1}{3} \cdot \frac{1}{2} \pi m^2 H
\][/tex]
[tex]\[
V = \frac{1}{6} \pi m^2 H
\][/tex]
6. Compare with Answer Choices:
The expression we obtained is [tex]\( V = \frac{1}{6} \pi m^2 H \)[/tex].
Among the given answer choices, the equation corresponding to this result in simplified form is:
[tex]\[
\boxed{C. \quad V=\frac{m^2 H}{6}}
\][/tex]
