Two right circular cones x and y are made. x having three times the radius of y and y having half the volume of x. Calculate the ratio between the heights of x and y.

Answer

To solve the problem step by step, we will use the properties of cones and the relationships given in the question.

### Step 1: Define the Variables
Let:
- \( r_y \) = radius of cone y
- \( r_x \) = radius of cone x

According to the problem, we have:
\[ r_x = 3r_y \]

### Step 2: Volume of the Cones
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Where \( r \) is the radius and \( h \) is the height of the cone.

For cone x:
\[ V_x = \frac{1}{3} \pi r_x^2 h_x \]

For cone y:
\[ V_y = \frac{1}{3} \pi r_y^2 h_y \]

### Step 3: Relationship Between Volumes
According to the problem, the volume of cone y is half the volume of cone x:
\[ V_y = \frac{1}{2} V_x \]

Substituting the volume formulas:
\[ \frac{1}{3} \pi r_y^2 h_y = \frac{1}{2} \left( \frac{1}{3} \pi r_x^2 h_x \right) \]

### Step 4: Simplifying the Equation
We can cancel \( \frac{1}{3} \pi \) from both sides:
\[ r_y^2 h_y = \frac{1}{2} r_x^2 h_x \]

### Step 5: Substitute \( r_x \)
Now substitute \( r_x = 3r_y \) into the equation:
\[ r_y^2 h_y = \frac{1}{2} (3r_y)^2 h_x \]

This simplifies to:
\[ r_y^2 h_y = \frac{1}{2} \cdot 9 r_y^2 h_x \]

### Step 6: Cancel \( r_y^2 \)
Assuming \( r_y \neq 0 \), we can divide both sides by \( r_y^2 \):
\[ h_y = \frac{9}{2} h_x \]

### Step 7: Finding the Ratio of Heights
To find the ratio of the heights \( h_x \) and \( h_y \):
\[ \frac{h_x}{h_y} = \frac{h_x}{\frac{9}{2} h_x} = \frac{2}{9} \]

Thus, the ratio of the heights of cone x to cone y is:
\[ \frac{h_x}{h_y} = \frac{2}{9} \]

### Final Answer
The required ratio of the heights of cones x and y is:
\[ 2 : 9 \]

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