The ratio of the volume of a right circular cylinder and a right circular cone of the same base and height will be :

Options

A. `2:3 `
B. `1:3`
C. `3:1`
D. `9:1`

Answer

To find the ratio of the volume of a right circular cylinder to the volume of a right circular cone with the same base and height, we can follow these steps:

### Step 1: Write down the formulas for the volumes.
- The volume \( V_c \) of a right circular cylinder is given by the formula:
\[ V_c = \pi r^2 h \]
- The volume \( V_{cone} \) of a right circular cone is given by the formula:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]

### Step 2: Set up the ratio of the volumes.
We need to find the ratio of the volume of the cylinder to the volume of the cone:
\[ \text{Ratio} = \frac{V_c}{V_{cone}} = \frac{\pi r^2 h}{\frac{1}{3} \pi r^2 h} \]

### Step 3: Simplify the ratio.
- Notice that \( \pi r^2 h \) is common in both the numerator and the denominator, so we can cancel it out:
\[ \text{Ratio} = \frac{1}{\frac{1}{3}} = 3 \]

### Step 4: Write the final ratio.
Thus, the ratio of the volume of the right circular cylinder to the volume of the right circular cone is:
\[ \text{Ratio} = 3:1 \]

### Final Answer:
The ratio of the volume of a right circular cylinder to a right circular cone of the same base and height is \( 3:1 \).

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