The volume of right circular cylinder is the same numerical as its total surface area .Find the smallest integral value for the radius of the cylinder .

Options

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

Explanation

To solve the problem, we need to equate the volume of a right circular cylinder to its total surface area and find the smallest integral value for the radius \( r \).

### Step-by-Step Solution:

1. **Write the formulas for volume and surface area**:
- The volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
- The total surface area \( A \) of a cylinder is given by:
\[ A = 2\pi rh + 2\pi r^2 \]

2. **Set the volume equal to the surface area**:
Since the problem states that the volume is numerically equal to the total surface area, we can set up the equation:
\[ \pi r^2 h = 2\pi rh + 2\pi r^2 \]

3. **Simplify the equation**:
We can divide both sides by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ r^2 h = 2rh + 2r^2 \]
Rearranging gives:
\[ r^2 h - 2rh - 2r^2 = 0 \]

4. **Factor out common terms**:
We can factor out \( r \) from the equation:
\[ r(h - 2r) = 2h \]

5. **Solve for \( h \)**:
Rearranging gives:
\[ h = \frac{2r}{r - 2} \]

6. **Determine conditions for \( h \)**:
For \( h \) to be positive, the denominator \( r - 2 \) must also be positive:
\[ r - 2 > 0 \implies r > 2 \]

7. **Find the smallest integral value for \( r \)**:
The smallest integer greater than 2 is 3. Therefore, the smallest integral value for the radius \( r \) is:
\[ r = 3 \]

### Final Answer:
The smallest integral value for the radius of the cylinder is \( \boxed{3} \).