The problem asks to find the ratio $x : y$ for a cone and a cylinder, where both shapes have the same volume.

Information given:

1. The cone has a base radius $x$ and a height $y$.
2. The cylinder has a base radius $x$ and a height $y$.

Step 1: Volume Formulas

• The volume of a cone is given by: $V_{\text{cone}} = \frac{1}{3} \pi x^2 y$
• The volume of a cylinder is given by: $V_{\text{cylinder}} = \pi x^2 y$

Step 2: Equating the volumes

Since the cone and the cylinder have the same volume, we can set the two volume formulas equal to each other: $\frac{1}{3} \pi x^2 y_{\text{cone}} = \pi x^2 y_{\text{cylinder}}$ This simplifies to: $\frac{1}{3} y_{\text{cone}} = y_{\text{cylinder}}$ Thus, the height of the cone is three times the height of the cylinder.

Step 3: Finding the ratio

From the simplification above, we know that $y_{\text{cone}} = 3 y_{\text{cylinder}}$. Therefore, the ratio $x : y$ depends on the respective heights.

Would you like a more detailed explanation or have any other questions?

Here are some follow-up questions for further exploration:

1. What happens to the ratio if the cone's height is halved?
2. How does the surface area of the two shapes compare given the same base radius?
3. If the cylinder's height was doubled, how would the volume comparison change?
4. What is the effect on the ratio $x : y$ if the cylinder's radius changes?
5. How would the problem change if the cone and cylinder had different base radii?

Tip: Always check units and assumptions, such as "same volume" in geometric problems, to ensure the relationships are valid.