The radius of a sphere is r and radius of base of a cylinder is r and height is 2r. The ratio of their volumes will be

Options

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

Answer

To find the ratio of the volumes of a sphere and a cylinder with given dimensions, we can follow these steps:

### Step 1: Identify the formulas for the volumes
- The volume \( V \) of a sphere is given by the formula:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
- The volume \( V \) of a cylinder is given by the formula:
\[ V_{\text{cylinder}} = \pi r^2 h \]

### Step 2: Substitute the given values
- For the sphere, the radius \( r \) is \( r \).
- For the cylinder, the radius \( r \) is also \( r \) and the height \( h \) is \( 2r \).

Substituting these values into the formulas:
- Volume of the sphere:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]
- Volume of the cylinder:
\[ V_{\text{cylinder}} = \pi r^2 (2r) = 2\pi r^3 \]

### Step 3: Set up the ratio of the volumes
Now, we need to find the ratio of the volume of the sphere to the volume of the cylinder:
\[ \text{Ratio} = \frac{V_{\text{sphere}}}{V_{\text{cylinder}}} = \frac{\frac{4}{3} \pi r^3}{2 \pi r^3} \]

### Step 4: Simplify the ratio
- Cancel out \( \pi \) and \( r^3 \) from the numerator and the denominator:
\[ \text{Ratio} = \frac{\frac{4}{3}}{2} \]
- This can be rewritten as:
\[ \text{Ratio} = \frac{4}{3} \times \frac{1}{2} = \frac{4}{6} = \frac{2}{3} \]

### Step 5: State the final ratio
Thus, the ratio of the volumes of the sphere to the cylinder is:
\[ \text{Ratio} = 2 : 3 \]

### Final Answer
The ratio of the volumes is \( 2 : 3 \).

---