A reactangular tin sheet is 12 cm long and 5 cm wide. It is rolled such that the ends are joined together to form a cylinder, then what will be the volume of cylinder ?

Options

A. `(180)/(pi) cm ^(3)`
B. `(120)/(pi) cm ^(3)`
C. `(100)/(pi) cm ^(3)`
D. `(60)/(pi) cm ^(3)`

Answer

To find the volume of the cylinder formed by rolling a rectangular tin sheet that is 12 cm long and 5 cm wide, we can follow these steps:

### Step 1: Identify the dimensions of the cylinder
When the rectangular sheet is rolled, the length of the sheet becomes the circumference of the base of the cylinder, and the width of the sheet becomes the height of the cylinder.

- Length of the rectangular sheet (L) = 12 cm (this will be the circumference of the cylinder)
- Width of the rectangular sheet (W) = 5 cm (this will be the height of the cylinder)

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

Since the circumference is equal to the length of the rectangular sheet:
\[ 2\pi r = 12 \]

### Step 3: Solve for the radius (r)
To find the radius, we can rearrange the equation:
\[ r = \frac{12}{2\pi} \]
\[ r = \frac{6}{\pi} \, \text{cm} \]

### Step 4: Use the formula for the volume of a cylinder
The volume (V) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( h \) is the height of the cylinder.

Substituting the values we have:
- \( r = \frac{6}{\pi} \)
- \( h = 5 \)

### Step 5: Calculate the volume
Now substituting the values into the volume formula:
\[ V = \pi \left(\frac{6}{\pi}\right)^2 \times 5 \]
\[ V = \pi \times \frac{36}{\pi^2} \times 5 \]
\[ V = \frac{180}{\pi} \, \text{cm}^3 \]

### Final Answer
The volume of the cylinder is:
\[ V = \frac{180}{\pi} \, \text{cm}^3 \]

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