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The problem asks for the volume of the cone in terms of \( h \), given that the volume of the cone is \( \frac{1}{3} \) the volume of the cylinder shown in the figure. The radius of the cylinder's base is 4 (half of the diameter, which is 8).

### Step-by-step approach:

1. **Volume of the Cylinder:**
   The formula for the volume of a cylinder is:
   \[
   V_{\text{cylinder}} = \pi r^2 h
   \]
   where \( r \) is the radius and \( h \) is the height.
   - For this cylinder, \( r = 4 \), so:
     \[
     V_{\text{cylinder}} = \pi (4)^2 h = 16\pi h
     \]

2. **Volume of the Cone:**
   The problem states that the volume of the cone is \( \frac{1}{3} \) of the volume of the cylinder. So, the volume of the cone is:
   \[
   V_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}} = \frac{1}{3} \times 16\pi h = \frac{16}{3} \pi h
   \]

Therefore, the volume of the cone is \( \frac{16}{3} \pi h \).

The correct answer is **\( \frac{16}{3} \pi h \)**, which matches the first option.

Would you like more details or have any questions?

### 5 Related Questions:
1. How does the volume of a cone change if its height is doubled but the radius remains the same?
2. What is the relationship between the volume of a cone and a cylinder when they have the same base and height?
3. How do you find the surface area of a cone if its slant height is known?
4. What happens to the volume of a cone if the radius of the base is halved?
5. Can you derive the formula for the volume of a cone using integration?

### Tip:
Always remember that the volume of any cone is exactly one-third of the volume of a cylinder with the same base and height.

The volume of the cone shown in the figure is 1/3 the volume of the cylinder. In terms of h, what is the volume of the cone?

Understand how to calculate the volume of a cone in terms of h, given that the cone's volume is one-third of the cylinder's volume. Learn key concepts in geometry with a step-by-step breakdown.

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