If A denotes the volume of a right circular cylinder of same height as its diameter and B is the volume of a sphere of same radius then `(A)/(B)` is :

Options

A. `(4)/(3)`
B. `(3)/(2)`
C. `(2)/(3)`
D. `(3)/(4)`

Answer

To solve the problem, we need to find the ratio \( \frac{A}{B} \), where \( A \) is the volume of a right circular cylinder and \( B \) is the volume of a sphere. The height of the cylinder is equal to its diameter.

### Step-by-Step Solution:

1. **Define the radius**:
Let the radius of the cylinder and sphere be \( r \).

2. **Determine the height of the cylinder**:
Since the height of the cylinder is equal to its diameter, we have:
\[ \text{Height of the cylinder} = \text{Diameter} = 2r \]

3. **Calculate the volume of the cylinder (A)**:
The formula for the volume of a right circular cylinder is given by:
\[ A = \pi r^2 h \]
Substituting the height \( h = 2r \):
\[ A = \pi r^2 (2r) = 2\pi r^3 \]

4. **Calculate the volume of the sphere (B)**:
The formula for the volume of a sphere is given by:
\[ B = \frac{4}{3} \pi r^3 \]

5. **Find the ratio \( \frac{A}{B} \)**:
Now we can find the ratio of the volumes:
\[ \frac{A}{B} = \frac{2\pi r^3}{\frac{4}{3} \pi r^3} \]
The \( \pi r^3 \) terms cancel out:
\[ \frac{A}{B} = \frac{2}{\frac{4}{3}} = 2 \times \frac{3}{4} = \frac{6}{4} = \frac{3}{2} \]

6. **Conclusion**:
Therefore, the value of \( \frac{A}{B} \) is:
\[ \frac{A}{B} = \frac{3}{2} \]