The radii of the base as well as the heights of a cone and a cylinder are each equal to h and the radius of a hemisphere is also equal to h . Find the ratio of the volume of cylinder , hemisphere , and cone .

Answer

To find the ratio of the volumes of a cylinder, a hemisphere, and a cone, we will use the formulas for the volumes of each shape and substitute the given values.

### Step 1: Identify the dimensions
Given:
- Radius (r) of the base of the cone and cylinder = h
- Height (h) of the cone and cylinder = h
- Radius (r) of the hemisphere = h

### Step 2: Write down the volume formulas
1. **Volume of the Cylinder (V_cylinder)**:
\[ V_{cylinder} = \pi r^2 h \]
Substituting \( r = h \):
\[ V_{cylinder} = \pi h^2 h = \pi h^3 \]

2. **Volume of the Hemisphere (V_hemisphere)**:
\[ V_{hemisphere} = \frac{2}{3} \pi r^3 \]
Substituting \( r = h \):
\[ V_{hemisphere} = \frac{2}{3} \pi h^3 \]

3. **Volume of the Cone (V_cone)**:
\[ V_{cone} = \frac{1}{3} \pi r^2 h \]
Substituting \( r = h \):
\[ V_{cone} = \frac{1}{3} \pi h^2 h = \frac{1}{3} \pi h^3 \]

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

### Step 4: Find the ratio of the volumes
To find the ratio of the volumes, we can express them as:
\[ V_{cylinder} : V_{hemisphere} : V_{cone} = \pi h^3 : \frac{2}{3} \pi h^3 : \frac{1}{3} \pi h^3 \]

### Step 5: Simplify the ratio
We can cancel \( \pi h^3 \) from all terms:
\[ 1 : \frac{2}{3} : \frac{1}{3} \]

To eliminate the fractions, multiply each term by 3:
\[ 3 : 2 : 1 \]

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