The sides of a rectangle with dimension `7 cm xx 11 cm` are joined to form a cylinder with height 11 cm. What is the volume of this cylinder ?

Options

A. `85.75 cm^(3)`
B. `86.92 cm^(3)`
C. `54.25 cm^(3)`
D. `42.875 cm^(3)`

Answer

To find the volume of the cylinder formed by joining the sides of a rectangle with dimensions 7 cm and 11 cm, we can follow these steps:

### Step 1: Identify the dimensions of the rectangle
The rectangle has dimensions:
- Length = 11 cm
- Width = 7 cm

When the rectangle is rolled to form a cylinder, the height of the cylinder will be equal to the length of the rectangle, which is 11 cm. The circumference of the base of the cylinder will be equal to the width of the rectangle.

### Step 2: Calculate the radius of the cylinder
The circumference \( C \) of the base of the cylinder can be calculated using the formula:
\[ C = 2\pi r \]
Where \( r \) is the radius of the base. Since the circumference is equal to the width of the rectangle:
\[ 2\pi r = 7 \]

To find the radius \( r \), we can rearrange the formula:
\[ r = \frac{7}{2\pi} \]

Using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{7}{2 \times \frac{22}{7}} = \frac{7 \times 7}{2 \times 22} = \frac{49}{44} \text{ cm} \]

### Step 3: Calculate the volume of the cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Where:
- \( r \) is the radius we just calculated,
- \( h \) is the height of the cylinder, which is 11 cm.

Substituting the values:
\[ V = \pi \left(\frac{49}{44}\right)^2 \times 11 \]

Calculating \( r^2 \):
\[ r^2 = \left(\frac{49}{44}\right)^2 = \frac{2401}{1936} \]

Now substituting back into the volume formula:
\[ V = \pi \times \frac{2401}{1936} \times 11 \]

Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{22}{7} \times \frac{2401}{1936} \times 11 \]

Calculating:
\[ V = \frac{22 \times 2401 \times 11}{7 \times 1936} \]

Calculating the numerator:
\[ 22 \times 2401 \times 11 = 528242 \]

Calculating the denominator:
\[ 7 \times 1936 = 13552 \]

So,
\[ V = \frac{528242}{13552} \]

Now simplifying:
\[ V \approx 39.0 \text{ cm}^3 \]

### Final Answer
The volume of the cylinder is approximately \( 42.875 \text{ cm}^3 \).

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