A rectangular paper 11 cm by 8 cm can be exactly wrapped to cover the curved surface of a cylinder of height 8 cm. the volume of the cylinder is

Options

A. `66" cm"^(3)`
B. `77" cm"^(3)`
C. `88" cm"^(3)`
D. `121" cm"^(3)`

Answer

To find the volume of the cylinder that can be wrapped with a rectangular paper of dimensions 11 cm by 8 cm, we will follow these steps:

### Step 1: Understand the relationship between the rectangular paper and the cylinder
The rectangular paper wraps around the curved surface area of the cylinder. The height of the cylinder is given as 8 cm.

### Step 2: Calculate the area of the rectangular paper
The area of the rectangular paper can be calculated using the formula:
\[ \text{Area} = \text{Length} \times \text{Breadth} \]
Substituting the values:
\[ \text{Area} = 11 \, \text{cm} \times 8 \, \text{cm} = 88 \, \text{cm}^2 \]

### Step 3: Set the area of the rectangular paper equal to the curved surface area of the cylinder
The curved surface area (CSA) of a cylinder is given by the formula:
\[ \text{CSA} = 2 \pi r h \]
Where \( r \) is the radius and \( h \) is the height. Since the height \( h \) is given as 8 cm, we can write:
\[ 88 \, \text{cm}^2 = 2 \pi r \times 8 \, \text{cm} \]

### Step 4: Substitute the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \):
\[ 88 = 2 \times \frac{22}{7} \times r \times 8 \]

### Step 5: Simplify the equation
\[ 88 = \frac{352}{7} r \]
Multiplying both sides by 7 to eliminate the fraction:
\[ 88 \times 7 = 352 r \]
\[ 616 = 352 r \]

### Step 6: Solve for \( r \)
\[ r = \frac{616}{352} = \frac{77}{44} = \frac{7}{4} \, \text{cm} \]

### Step 7: Calculate the volume of the cylinder
The volume \( V \) of the cylinder is given by:
\[ V = \pi r^2 h \]
Substituting the values of \( \pi \), \( r \), and \( h \):
\[ V = \frac{22}{7} \times \left(\frac{7}{4}\right)^2 \times 8 \]
Calculating \( \left(\frac{7}{4}\right)^2 \):
\[ \left(\frac{7}{4}\right)^2 = \frac{49}{16} \]
Now substituting back into the volume formula:
\[ V = \frac{22}{7} \times \frac{49}{16} \times 8 \]
\[ V = \frac{22 \times 49 \times 8}{7 \times 16} \]
Cancelling \( 8 \) with \( 16 \):
\[ V = \frac{22 \times 49 \times 1}{7 \times 2} \]
Calculating \( 22 \div 7 = 3.14 \) and \( 49 \div 7 = 7 \):
\[ V = \frac{22 \times 7}{2} = \frac{154}{2} = 77 \, \text{cm}^3 \]

### Final Answer
The volume of the cylinder is \( 77 \, \text{cm}^3 \).

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