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Let's analyze and solve this question step-by-step.

### Given Information:
1. The cone has a base radius \( r \) and height \( x \).
2. The prism has a triangular cross-section with a base \( y \) and height \( y \), and its length is \( x \).
3. The volume of the cone is four times the volume of the prism.

### Step 1: Calculate the Volume of the Cone
The volume \( V_{\text{cone}} \) of a cone is given by:
\[
V_{\text{cone}} = \frac{1}{3} \times \text{area of base} \times \text{height}
\]
The area of the base of the cone (which is a circle) is \( \pi r^2 \), and the height is \( x \).
\[
V_{\text{cone}} = \frac{1}{3} \times \pi r^2 \times x = \frac{\pi r^2 x}{3}
\]

### Step 2: Calculate the Volume of the Prism
The volume \( V_{\text{prism}} \) of a prism is given by:
\[
V_{\text{prism}} = \text{area of cross-section} \times \text{length}
\]
The cross-section of the prism is a triangle with base \( y \) and height \( y \), so the area of this triangle is:
\[
\text{Area of triangle} = \frac{1}{2} \times y \times y = \frac{y^2}{2}
\]
Thus, the volume of the prism is:
\[
V_{\text{prism}} = \frac{y^2}{2} \times x = \frac{y^2 x}{2}
\]

### Step 3: Set Up the Given Condition
According to the problem, the volume of the cone is four times the volume of the prism:
\[
V_{\text{cone}} = 4 \times V_{\text{prism}}
\]
Substitute the expressions for \( V_{\text{cone}} \) and \( V_{\text{prism}} \) we derived:
\[
\frac{\pi r^2 x}{3} = 4 \times \frac{y^2 x}{2}
\]

### Step 4: Simplify the Equation
Cancel \( x \) from both sides (assuming \( x \neq 0 \)):
\[
\frac{\pi r^2}{3} = 2 y^2
\]
Multiply both sides by 3 to eliminate the denominator:
\[
\pi r^2 = 6 y^2
\]
Now, divide both sides by \( \pi \) to solve for \( r^2 \):
\[
r^2 = \frac{6 y^2}{\pi}
\]
Finally, take the square root of both sides to express \( r \) in terms of \( y \):
\[
r = \sqrt{\frac{6 y^2}{\pi}} = \frac{y \sqrt{6}}{\sqrt{\pi}}
\]

### Final Answer:
\[
r = \frac{y \sqrt{6}}{\sqrt{\pi}}
\]

Would you like more details on any step, or do you have any questions?

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Here are some additional related questions:
1. How would the result change if the volume of the cone was three times the volume of the prism?
2. What is the effect on \( r \) if the radius of the cone’s base changes but the height remains constant?
3. Can you express \( x \) in terms of \( r \) and \( y \) given the same volume relationship?
4. How would the volume of the cone change if both \( r \) and \( x \) doubled?
5. How does the formula for the volume of a cone compare to that of a cylinder with the same base and height?

**Tip:** Always check for relationships or proportionalities in geometric shapes when solving volume-related problems. This helps simplify complex expressions.

The diagram below shows a cone and a prism. All measurements are in cm. The cone has base radius r and perpendicular height x. The prism has a triangular cross section with base y and perpendicular height y. The length of the prism is x. The volume of the cone is four times the volume of the prism. Express r in terms of y.

Solve a geometry problem involving a cone and a triangular prism where the volume of the cone is four times the volume of the prism. Express the radius of the cone in terms of the prism’s base. Suitable for high school students in Grades 10-12.

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