To solve this problem, we need to determine the ratio of the volumes of the cylinder and the cone based on the given information. We will use the formulas for the volumes of the cylinder and the cone and incorporate the relationships provided in the problem.

Given:

1. The circumference of the base of the cone is twice the circumference of the base of the cylinder.
2. The ratio of the height of the cylinder to the height of the cone is r.

Step 1: Use the formulas for volume

The formula for the volume of a cylinder is: $V_{\text{cylinder}} = \pi r_{\text{cylinder}}^2 h_{\text{cylinder}}$ where $r_{\text{cylinder}}$ is the radius of the cylinder's base, and $h_{\text{cylinder}}$ is its height.

The formula for the volume of a cone is: $V_{\text{cone}} = \frac{1}{3} \pi r_{\text{cone}}^2 h_{\text{cone}}$ where $r_{\text{cone}}$ is the radius of the cone's base, and $h_{\text{cone}}$ is its height.

Step 2: Relate the circumferences of the bases

The circumference $C$ of a circle is given by: $C = 2 \pi r$ Let $r_{\text{cylinder}}$ be the radius of the cylinder's base, and $r_{\text{cone}}$ be the radius of the cone's base.

According to the problem, the circumference of the base of the cone is twice that of the base of the cylinder. Therefore, we can write: $2 \pi r_{\text{cone}} = 2 \times (2 \pi r_{\text{cylinder}})$ Simplifying this: $r_{\text{cone}} = 2 r_{\text{cylinder}}$

Step 3: Relate the heights of the cylinder and the cone

Let $h_{\text{cylinder}}$ and $h_{\text{cone}}$ be the heights of the cylinder and cone, respectively. The ratio of the heights of the cylinder to the cone is $r$, so: $\frac{h_{\text{cylinder}}}{h_{\text{cone}}} = r$ or equivalently: $h_{\text{cylinder}} = r \times h_{\text{cone}}$

Step 4: Substitute the relationships into the volume formulas

Now, let's substitute the expressions for $r_{\text{cone}}$ and $h_{\text{cylinder}}$ into the formulas for the volumes.

Volume of the cylinder:

$V_{\text{cylinder}} = \pi r_{\text{cylinder}}^2 h_{\text{cylinder}} = \pi r_{\text{cylinder}}^2 (r \times h_{\text{cone}})$ $V_{\text{cylinder}} = \pi r_{\text{cylinder}}^2 r h_{\text{cone}}$

Volume of the cone:

$V_{\text{cone}} = \frac{1}{3} \pi r_{\text{cone}}^2 h_{\text{cone}} = \frac{1}{3} \pi (2 r_{\text{cylinder}})^2 h_{\text{cone}}$ $V_{\text{cone}} = \frac{1}{3} \pi 4 r_{\text{cylinder}}^2 h_{\text{cone}}$ $V_{\text{cone}} = \frac{4}{3} \pi r_{\text{cylinder}}^2 h_{\text{cone}}$

Step 5: Find the ratio of the volumes

Now, we find the ratio of the volume of the cylinder to the volume of the cone: $\frac{V_{\text{cylinder}}}{V_{\text{cone}}} = \frac{\pi r_{\text{cylinder}}^2 r h_{\text{cone}}}{\frac{4}{3} \pi r_{\text{cylinder}}^2 h_{\text{cone}}}$ Simplifying: $\frac{V_{\text{cylinder}}}{V_{\text{cone}}} = \frac{3r}{4}$

Final Answer:

The ratio of the volume of the cylinder to the volume of the cone is $\frac{3r}{4}$.

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Would you like further details or clarifications on any step? Here are a few related questions to explore:

1. How does changing the ratio of the heights affect the volume ratio?
2. What happens if the circumference of the cone is equal to that of the cylinder?
3. How does the volume of a cone compare to a cylinder if both have the same height and radius?
4. What is the volume ratio if the ratio of the radii is different?
5. How can we derive a formula for the volume of a truncated cone?

Tip: To simplify volume ratio problems, focus on relating key properties like radius and height, and use algebraic substitution to avoid getting overwhelmed with complex expressions!