The radius of a sphere and right circular cylinder is 'r' units. Their volumes are equal. The ratio of the height and radius of the cyclinder is :

Options

A. a) `3:1`
B. b) `2:1`
C. c) `3:2`
D. d) `4:3`

Answer

To find the ratio of the height and radius of a right circular cylinder when the volumes of the sphere and cylinder are equal, we can follow these steps:

### Step-by-Step Solution:

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

2. **Write the formula for the volume of the cylinder:**
The volume \( V \) of a right circular cylinder is given by the formula:
\[ V_{cylinder} = \pi r^2 h \]
where \( r \) is the radius of the cylinder and \( h \) is the height of the cylinder.

3. **Set the volumes equal to each other:**
Since the volumes of the sphere and the cylinder are equal, we can set the two volume formulas equal:
\[ \frac{4}{3} \pi r^3 = \pi r^2 h \]

4. **Cancel out \( \pi \) from both sides:**
Dividing both sides by \( \pi \) gives:
\[ \frac{4}{3} r^3 = r^2 h \]

5. **Rearrange the equation to solve for \( h \):**
To isolate \( h \), divide both sides by \( r^2 \) (assuming \( r \neq 0 \)):
\[ h = \frac{4}{3} r \]

6. **Find the ratio of height to radius:**
The ratio of height \( h \) to radius \( r \) is:
\[ \frac{h}{r} = \frac{\frac{4}{3} r}{r} = \frac{4}{3} \]

7. **Express the ratio in the required form:**
This means that the ratio of height to radius is:
\[ h : r = 4 : 3 \]

### Final Answer:
The ratio of the height to the radius of the cylinder is \( 4 : 3 \).