The volume of a right circular cylinder and that of a sphere are equal and their radii are also equal. If the height of the cylinder be h and diameter of the sphere d, then which of the following relation is correct ?

Options

A. h = d
B. 2h = d
C. 2h = 3d
D. 3h = 2d

Answer

To solve the problem, we need to establish the relationship between the height of a right circular cylinder and the diameter of a sphere, given that their volumes are equal and their radii are equal.

### Step-by-Step Solution:

1. **Understand the given information**:
- The radius of the cylinder (r) is equal to the radius of the sphere (r).
- The height of the cylinder is denoted as \( h \).
- The diameter of the sphere is denoted as \( d \).

2. **Express the diameter in terms of the radius**:
- The diameter of the sphere is twice the radius:
\[ d = 2r \]

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

4. **Write the formula for the volume of the cylinder**:
- The volume \( V_c \) of a right circular cylinder is given by:
\[ V_c = \pi r^2 h \]

5. **Set the volumes equal to each other**:
- Since the volumes are equal, we have:
\[ \frac{4}{3} \pi r^3 = \pi r^2 h \]

6. **Cancel out common terms**:
- We can cancel \( \pi \) from both sides (assuming \( \pi \neq 0 \)):
\[ \frac{4}{3} r^3 = r^2 h \]

7. **Divide both sides by \( r^2 \)** (assuming \( r \neq 0 \)):
- This simplifies to:
\[ \frac{4}{3} r = h \]

8. **Express \( h \) in terms of \( r \)**:
- Rearranging gives us:
\[ h = \frac{4}{3} r \]

9. **Substitute \( r \) in terms of \( d \)**:
- From the earlier step, we know \( d = 2r \), thus:
\[ r = \frac{d}{2} \]

10. **Substitute \( r \) into the equation for \( h \)**:
- Replace \( r \) in the equation \( h = \frac{4}{3} r \):
\[ h = \frac{4}{3} \left( \frac{d}{2} \right) \]

11. **Simplify the equation**:
- This simplifies to:
\[ h = \frac{4d}{6} = \frac{2d}{3} \]

12. **Rearranging gives the final relation**:
- Multiplying both sides by 3 gives:
\[ 3h = 2d \]

### Conclusion:
The correct relationship between the height of the cylinder and the diameter of the sphere is:
\[ 3h = 2d \]