A right circular cylinder is formed by rolling a rectanglular paper 12 cm long and 3 cm wide along its length. The radius of the base of the cylinder will be

Options

A. `(3)/(2pi) cm`
B. `(6)/(pi) cm`
C. `(9)/(2pi) cm`
D. `2pi cm`

Answer

To find the radius of the base of the cylinder formed by rolling a rectangular paper, we can follow these steps:

### Step 1: Identify the dimensions of the rectangular paper
The rectangular paper has a length of 12 cm and a width of 3 cm.

### Step 2: Understand how the cylinder is formed
When the rectangular paper is rolled along its length, the length of the paper becomes the circumference of the base of the cylinder. Therefore, we have:
- Circumference of the base of the cylinder = Length of the rectangular paper = 12 cm

### Step 3: Use the formula for the circumference of a circle
The formula for the circumference (C) of a circle is given by:
\[ C = 2\pi r \]
where \( r \) is the radius of the base of the cylinder.

### Step 4: Set up the equation
From the information given, we can set up the equation:
\[ 2\pi r = 12 \]

### Step 5: Solve for the radius \( r \)
To find \( r \), we can rearrange the equation:
\[ r = \frac{12}{2\pi} \]
\[ r = \frac{12}{2 \times 3.14} \] (using \( \pi \approx 3.14 \))
\[ r = \frac{12}{6.28} \]
\[ r \approx 1.91 \, \text{cm} \]

### Step 6: Simplify the expression
We can also express this in a fraction:
\[ r = \frac{12}{2\pi} = \frac{6}{\pi} \]

### Step 7: Final value of the radius
Using the approximate value of \( \pi \):
\[ r \approx \frac{6}{3.14} = \frac{600}{314} \approx \frac{300}{157} \]
This gives us a radius of approximately \( 1.91 \, \text{cm} \).

### Conclusion
Thus, the radius of the base of the cylinder is:
\[ r = \frac{6}{\pi} \, \text{cm} \text{ or approximately } 1.91 \, \text{cm} \]