A right circular cylinder is circumscribed about a hemisphere so that they share the same base. The ratio of the volumes of cylinder and hemisphere is

Options

A. `4:3`
B. `3:1`
C. `3:4`
D. `3:2`

Answer

To find the ratio of the volumes of a right circular cylinder and a hemisphere that share the same base, we can follow these steps:

### Step 1: Define the radius and height
Let the radius of the base of both the cylinder and the hemisphere be \( R \). Since the cylinder is circumscribed about the hemisphere, the height of the cylinder will also be equal to the radius of the hemisphere, which is \( R \).

### Step 2: Calculate the volume of the cylinder
The volume \( V_c \) of a cylinder is given by the formula:
\[ V_c = \pi R^2 H \]
Substituting \( H = R \):
\[ V_c = \pi R^2 \cdot R = \pi R^3 \]

### Step 3: Calculate the volume of the hemisphere
The volume \( V_h \) of a hemisphere is given by the formula:
\[ V_h = \frac{2}{3} \pi R^3 \]

### Step 4: Find the ratio of the volumes
Now, we need to find the ratio of the volume of the cylinder to the volume of the hemisphere:
\[ \text{Ratio} = \frac{V_c}{V_h} = \frac{\pi R^3}{\frac{2}{3} \pi R^3} \]

### Step 5: Simplify the ratio
The \( \pi R^3 \) terms cancel out:
\[ \text{Ratio} = \frac{1}{\frac{2}{3}} = \frac{3}{2} \]

### Conclusion
The ratio of the volumes of the cylinder to the hemisphere is:
\[ \frac{3}{2} \]