The circumference of the base of a right cylinder is 33 cm and height is 330 cm . What is the volume of this cylinder ?

Options

A. `28586.25 cm^(3) ``
B. `3344 cm^(3) `
C. `4433 cm^(3) `
D. `3456 cm^(3) `

Answer

To find the volume of the right cylinder given its circumference and height, we can follow these steps:

### Step 1: Identify the given values
- Circumference of the base of the cylinder (C) = 33 cm
- Height of the cylinder (h) = 330 cm

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

### Step 3: Solve for the radius (r)
We can rearrange the formula to find the radius:
\[ r = \frac{C}{2\pi} \]
Substituting the given circumference:
\[ r = \frac{33}{2\pi} \]

### Step 4: Calculate the radius
Using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{33}{2 \times \frac{22}{7}} \]
\[ r = \frac{33 \times 7}{2 \times 22} \]
\[ r = \frac{231}{44} \]
\[ r = \frac{33}{2} \text{ cm} \]

### Step 5: Use the formula for the volume of a cylinder
The volume (V) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]

### Step 6: Substitute the values into the volume formula
Substituting the values of \( r \) and \( h \):
\[ V = \pi \left(\frac{33}{2}\right)^2 \times 330 \]

### Step 7: Calculate \( r^2 \)
\[ r^2 = \left(\frac{33}{2}\right)^2 = \frac{1089}{4} \]

### Step 8: Substitute \( r^2 \) back into the volume formula
\[ V = \pi \times \frac{1089}{4} \times 330 \]

### Step 9: Simplify the expression
\[ V = \frac{1089 \times 330 \times \pi}{4} \]

### Step 10: Calculate the volume
Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{1089 \times 330 \times \frac{22}{7}}{4} \]
\[ V = \frac{1089 \times 330 \times 22}{28} \]

### Step 11: Perform the multiplication
Calculating \( 1089 \times 330 \):
\[ 1089 \times 330 = 359370 \]
Now, substituting back:
\[ V = \frac{359370 \times 22}{28} \]

### Step 12: Final calculation
Calculating \( 359370 \times 22 = 7906140 \):
\[ V = \frac{7906140}{28} = 282196.42857 \text{ cm}^3 \]

### Step 13: Rounding to two decimal places
The volume of the cylinder is approximately:
\[ V \approx 28586.25 \text{ cm}^3 \]

### Final Answer
The volume of the cylinder is **28586.25 cm³**.

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