The magnitude of the volume of a closed right circular cylinder of unit height divided by the magnitude of the total surface area of the cylinder (r being the radius of the cylinder) is equal to

Options

A. `1/2(1+(1)/(r))`
B. `1/2(1+(1)/(r+1))`
C. `1/2(1-(1)/(r))`
D. `1/2(1-(1)/(r+1))`

Answer

To solve the problem of finding the magnitude of the volume of a closed right circular cylinder of unit height divided by the magnitude of the total surface area of the cylinder, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the formulas**:
- The volume \( V \) of a right circular cylinder is given by the formula:
\[ V = \pi r^2 h \]
- The total surface area \( A \) of a closed right circular cylinder is given by the formula:
\[ A = 2\pi r (r + h) \]

2. **Substitute the height**:
- Since the height \( h \) of the cylinder is given as 1 unit, we can substitute \( h = 1 \) into both formulas.
- For volume:
\[ V = \pi r^2 \cdot 1 = \pi r^2 \]
- For total surface area:
\[ A = 2\pi r (r + 1) = 2\pi r^2 + 2\pi r \]

3. **Set up the division**:
- We need to find the ratio of the volume to the total surface area:
\[ \frac{V}{A} = \frac{\pi r^2}{2\pi r^2 + 2\pi r} \]

4. **Simplify the expression**:
- We can simplify the fraction:
\[ \frac{V}{A} = \frac{\pi r^2}{2\pi (r^2 + r)} \]
- Cancel out \( \pi \) from the numerator and denominator:
\[ \frac{V}{A} = \frac{r^2}{2(r^2 + r)} \]

5. **Further simplification**:
- We can factor out \( r \) from the denominator:
\[ \frac{V}{A} = \frac{r^2}{2r(r + 1)} = \frac{r}{2(r + 1)} \]

### Final Result:
Thus, the magnitude of the volume of a closed right circular cylinder of unit height divided by the magnitude of the total surface area of the cylinder is:
\[ \frac{r}{2(r + 1)} \]