A cone, a cylinder and a hemisphere stand on equal bases and have equal heights. The ratio of their volumes is

Options

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

Answer

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

### Step-by-Step Solution:

1. **Define Variables**:
Let the radius of the base of the cone, cylinder, and hemisphere be \( R \) and the height of all three shapes be \( H \).

2. **Volume of the Cone**:
The formula for the volume of a cone is given by:
\[ V_{\text{cone}} = \frac{1}{3} \pi R^2 H \]

3. **Volume of the Hemisphere**:
The formula for the volume of a hemisphere is given by:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi R^3 \]
Since the height of the hemisphere is equal to the radius \( R \), we can express the height \( H \) in terms of \( R \) (i.e., \( H = R \)). Thus, substituting \( R \) for \( H \):
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi R^3 \]

4. **Volume of the Cylinder**:
The formula for the volume of a cylinder is given by:
\[ V_{\text{cylinder}} = \pi R^2 H \]
Again, substituting \( H \) with \( R \):
\[ V_{\text{cylinder}} = \pi R^2 R = \pi R^3 \]

5. **Form the Ratios**:
Now we can write the volumes in terms of \( R \):
\[ V_{\text{cone}} : V_{\text{hemisphere}} : V_{\text{cylinder}} = \frac{1}{3} \pi R^2 H : \frac{2}{3} \pi R^3 : \pi R^3 \]
Substituting \( H = R \):
\[ = \frac{1}{3} \pi R^2 R : \frac{2}{3} \pi R^3 : \pi R^3 \]
Simplifying this gives:
\[ = \frac{1}{3} \pi R^3 : \frac{2}{3} \pi R^3 : \pi R^3 \]

6. **Cancel Out Common Terms**:
We can cancel \( \pi R^3 \) from each term:
\[ = \frac{1}{3} : \frac{2}{3} : 1 \]

7. **Convert to Whole Numbers**:
To express the ratio in whole numbers, multiply each term by 3:
\[ = 1 : 2 : 3 \]

### Final Answer:
The ratio of the volumes of the cone, hemisphere, and cylinder is:
\[ \text{Cone : Hemisphere : Cylinder} = 1 : 2 : 3 \]