A cone, a hemisphere and a cylinder stand on equal bases and have the same height. What is the ratio of their volumes?

Options

A. `2 : 1 : 3`
B. `2.5 : 1 : 3`
C. `1 : 2 : 3`
D. `1.5 : 2 : 3`

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: Identify the formulas for the volumes
- **Volume of a cone (V_cone)**:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]
- **Volume of a hemisphere (V_hemisphere)**:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \]
- **Volume of a cylinder (V_cylinder)**:
\[ V_{\text{cylinder}} = \pi r^2 h \]

### Step 2: Substitute the height in terms of radius
Since the cone, hemisphere, and cylinder have the same height and the radius is equal to the height (h = r), we can substitute h with r in the formulas.

- For the cone:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 r = \frac{1}{3} \pi r^3 \]

- For the hemisphere:
\[ V_{\text{hemisphere}} = \frac{2}{3} \pi r^3 \]

- For the cylinder:
\[ V_{\text{cylinder}} = \pi r^2 r = \pi r^3 \]

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

### Step 4: Set up the ratio of the volumes
Now we can set up the ratio of the volumes:
\[ V_{\text{cone}} : V_{\text{hemisphere}} : V_{\text{cylinder}} = \frac{1}{3} \pi r^3 : \frac{2}{3} \pi r^3 : \pi r^3 \]

### Step 5: Simplify the ratio
To simplify, we can cancel out \( \pi r^3 \) from each term:
\[ \frac{1}{3} : \frac{2}{3} : 1 \]

Now, to eliminate the fractions, we can multiply each term by 3:
\[ 1 : 2 : 3 \]

### Conclusion
The ratio of the volumes of the cone, hemisphere, and cylinder is:
\[ \text{Ratio} = 1 : 2 : 3 \]