The volume of the cone shown in the figure below is 1/3 the volume of the cylinder. In terms of H, what is the volume of the cone? Figure displays a cone fitting a cylinder, 8 as a diameter and H as a missing value representing the height.

Given that the volume of the cone is \(\frac{1}{3}\) the volume of the cylinder, we can start by writing down the formulas for the volumes of a cylinder and a cone.

### Step 1: Volume of the Cylinder
The volume \( V_{\text{cyl}} \) of a cylinder is given by:
\[
V_{\text{cyl}} = \pi r^2 h
\]
Where:
- \( r \) is the radius of the base
- \( h \) is the height of the cylinder

Given that the diameter of the cylinder (and the cone) is 8, the radius \( r \) is:
\[
r = \frac{8}{2} = 4
\]

The height of the cylinder is \( H \). So, the volume of the cylinder is:
\[
V_{\text{cyl}} = \pi (4)^2 H = 16\pi H
\]

### Step 2: Volume of the Cone
The volume \( V_{\text{cone}} \) of a cone is given by:
\[
V_{\text{cone}} = \frac{1}{3} \pi r^2 h
\]
Where:
- \( r \) is the radius of the base
- \( h \) is the height of the cone

The height of the cone is also \( H \), so:
\[
V_{\text{cone}} = \frac{1}{3} \pi (4)^2 H = \frac{1}{3} \pi \times 16 \times H = \frac{16\pi H}{3}
\]

### Step 3: Relationship Between the Volumes
We are given that the volume of the cone is \(\frac{1}{3}\) the volume of the cylinder. We can now verify this relationship:
\[
\frac{1}{3} V_{\text{cyl}} = \frac{1}{3} \times 16\pi H = \frac{16\pi H}{3}
\]

So the volume of the cone in terms of \( H \) is:
\[
V_{\text{cone}} = \frac{16\pi H}{3}
\]

Would you like more details or have any questions?

Here are some related questions:

1. How would the volume of the cone change if the height was doubled?
2. What is the surface area of the cone?
3. How does the ratio of the radii affect the volume of a cone and cylinder?
4. Can you derive the volume of the cone using integration?
5. What would happen to the volume if the diameter of the base was increased?

**Tip:** Remember, the volume of a cone is always \(\frac{1}{3}\) of the volume of a cylinder with the same base and height.

Learn how to find the volume of a cone in terms of height H, where the cone fits perfectly within a cylinder with a diameter of 8. This problem covers key geometry concepts, including the volume formulas for cones and cylinders, and explores the relationship between their volumes. Suitable for students in Grades 9-12.

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