A right cylinder , a right cone and a hemishpere have the same height and the same base area .Find the ratio of their (i) volumes

Answer

To find the ratio of the volumes of a right cylinder, a right cone, and a hemisphere that have the same height and base area, we can follow these steps:

### Step 1: Define the Variables
Let the radius of the base of the cylinder, cone, and hemisphere be \( r \) and the height be \( h \). Since they have the same base area and height, we can denote them as \( r \) and \( h \).

### Step 2: Calculate the Volume of Each Solid
1. **Volume of the Cylinder (V_c)**:
\[ V_c = \pi r^2 h \]

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

3. **Volume of the Hemisphere (V_h)**:
\[ V_h = \frac{2}{3} \pi r^3 \]

### Step 3: Substitute Height in Terms of Radius
Since the height \( h \) is the same for all three solids, we can express \( h \) in terms of \( r \) if needed, but we will keep it as \( h \) for now.

### Step 4: Write the Volumes in Terms of \( r \) and \( h \)
Now we can express each volume in terms of \( r \) and \( h \):
- Cylinder: \( V_c = \pi r^2 h \)
- Cone: \( V_co = \frac{1}{3} \pi r^2 h \)
- Hemisphere: \( V_h = \frac{2}{3} \pi r^3 \)

### Step 5: Calculate the Ratio of the Volumes
To find the ratio of the volumes \( V_c : V_co : V_h \):
1. Substitute the volumes:
\[ V_c : V_co : V_h = \pi r^2 h : \frac{1}{3} \pi r^2 h : \frac{2}{3} \pi r^3 \]

2. Simplify the ratio:
- Cancel \( \pi \) from all terms:
\[ r^2 h : \frac{1}{3} r^2 h : \frac{2}{3} r^3 \]

3. Multiply through by 3 to eliminate fractions:
\[ 3r^2 h : r^2 h : 2r^3 \]

4. Now, factor out \( r^2 \):
\[ 3h : h : 2r \]

### Step 6: Final Ratio
Since \( h \) is common, we can express the ratio as:
\[ 3 : 1 : \frac{2r}{h} \]
If \( h = r \), we can substitute \( h \) with \( r \):
\[ 3 : 1 : 2 \]

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