If perimeter of the base of a cylinder is 66 cm. Then find volume of cylinder if height of cylinder is 0.04 m

Options

A. A)1111 `cm^(3)`
B. B)1386 `cm^(3)`
C. C)2046 `cm^(3)`
D. D)1186 `cm^(3)`

Answer

To solve the problem, we need to find the volume of a cylinder given the perimeter of its base and its height. Here are the steps to solve the problem:

### Step 1: Understand the relationship between perimeter and radius
The perimeter (circumference) of the base of a cylinder is given by the formula:
\[ C = 2\pi r \]
where \( r \) is the radius of the base.

### Step 2: Set up the equation using the given perimeter
We know the perimeter of the base is 66 cm, so we can set up the equation:
\[ 2\pi r = 66 \]

### Step 3: Solve for the radius \( r \)
To find the radius, we can rearrange the equation:
\[ r = \frac{66}{2\pi} \]
Substituting the value of \( \pi \) (approximately \( \frac{22}{7} \)):
\[ r = \frac{66}{2 \times \frac{22}{7}} = \frac{66 \times 7}{44} = \frac{462}{44} = \frac{21}{2} \text{ cm} \]

### Step 4: Convert the height from meters to centimeters
The height of the cylinder is given as 0.04 m. To convert this to centimeters:
\[ 0.04 \text{ m} = 0.04 \times 100 = 4 \text{ cm} \]

### Step 5: Use the formula for the volume of the cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Substituting the values we have:
\[ V = \pi \left(\frac{21}{2}\right)^2 \times 4 \]

### Step 6: Calculate \( r^2 \)
Calculating \( r^2 \):
\[ r^2 = \left(\frac{21}{2}\right)^2 = \frac{441}{4} \]

### Step 7: Substitute and calculate the volume
Now substituting \( r^2 \) into the volume formula:
\[ V = \pi \times \frac{441}{4} \times 4 \]
The \( 4 \) in the numerator and denominator cancels out:
\[ V = \pi \times 441 \]
Substituting \( \pi \) with \( \frac{22}{7} \):
\[ V = \frac{22}{7} \times 441 \]

### Step 8: Calculate the final volume
Calculating:
\[ V = \frac{22 \times 441}{7} = \frac{9702}{7} = 1386 \text{ cm}^3 \]

Thus, the volume of the cylinder is:
\[ \boxed{1386 \text{ cm}^3} \]