A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their respectively volume is

Options

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

Answer

To find the ratio of the volumes of a cone, a hemisphere, and a cylinder that stand on equal bases and have the same height, we can follow these steps:

### Step 1: Define the dimensions
Let the radius of the base of the cone, hemisphere, and cylinder be \( R \) and the height of each shape be \( H \). Since they have equal bases and heights, we can denote:
- Radius \( R \)
- Height \( H = R \) (for the cone and hemisphere)
- Height \( H = 2R \) (for the cylinder, as its height is twice the radius)

### Step 2: Calculate the volume of the cone
The formula for the volume of a cone is given by:
\[ V_{cone} = \frac{1}{3} \pi R^2 H \]
Substituting \( H = R \):
\[ V_{cone} = \frac{1}{3} \pi R^2 (R) = \frac{1}{3} \pi R^3 \]

### Step 3: Calculate the volume of the hemisphere
The formula for the volume of a hemisphere is:
\[ V_{hemisphere} = \frac{2}{3} \pi R^3 \]

### Step 4: Calculate the volume of the cylinder
The formula for the volume of a cylinder is:
\[ V_{cylinder} = \pi R^2 H \]
Substituting \( H = 2R \):
\[ V_{cylinder} = \pi R^2 (2R) = 2 \pi R^3 \]

### Step 5: Write the volumes together
Now we have the volumes:
- Volume of the cone: \( V_{cone} = \frac{1}{3} \pi R^3 \)
- Volume of the hemisphere: \( V_{hemisphere} = \frac{2}{3} \pi R^3 \)
- Volume of the cylinder: \( V_{cylinder} = 2 \pi R^3 \)

### Step 6: Form the ratio of their volumes
To find the ratio of their volumes, we can write:
\[ \text{Ratio} = V_{cone} : V_{hemisphere} : V_{cylinder} = \frac{1}{3} \pi R^3 : \frac{2}{3} \pi R^3 : 2 \pi R^3 \]

### Step 7: Simplify the ratio
We can cancel \( \pi R^3 \) from all terms:
\[ \text{Ratio} = \frac{1}{3} : \frac{2}{3} : 2 \]
To eliminate the fractions, we can multiply each term by 3:
\[ \text{Ratio} = 1 : 2 : 6 \]

### Final Answer
Thus, the ratio of the volumes of the cone, hemisphere, and cylinder is:
\[ 1 : 2 : 6 \]