A right circular cylinder and a right circular cone have the same radius and the same volume. The ratio of the height of the cylinder to that of the cone is

Options

A. `3 : 5`
B. `2 : 5`
C. `3 : 1`
D. `1 : 3`

Explanation

To find the ratio of the height of a right circular cylinder to that of a right circular cone when both have the same radius and volume, we can follow these steps:

### Step 1: Write the formula for the volume of the cylinder and the cone.

The volume \( V \) of a right circular cylinder is given by the formula:
\[ V_{cylinder} = \pi r^2 h_{cylinder} \]
where \( r \) is the radius and \( h_{cylinder} \) is the height of the cylinder.

The volume \( V \) of a right circular cone is given by the formula:
\[ V_{cone} = \frac{1}{3} \pi r^2 h_{cone} \]
where \( h_{cone} \) is the height of the cone.

### Step 2: Set the volumes equal to each other.

Since the cylinder and the cone have the same volume, we can set the two volume equations equal to each other:
\[ \pi r^2 h_{cylinder} = \frac{1}{3} \pi r^2 h_{cone} \]

### Step 3: Simplify the equation.

We can simplify the equation by dividing both sides by \( \pi r^2 \) (assuming \( r \neq 0 \)):
\[ h_{cylinder} = \frac{1}{3} h_{cone} \]

### Step 4: Find the ratio of the heights.

To find the ratio of the height of the cylinder to the height of the cone, we can express it as:
\[ \frac{h_{cylinder}}{h_{cone}} = \frac{1}{3} \]

### Conclusion

Thus, the ratio of the height of the cylinder to the height of the cone is:
\[ \frac{h_{cylinder}}{h_{cone}} = \frac{1}{3} \]

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