The volume of a right circular cylinder whose height is 40 cm , and circumference of its base is 66 cm, is

Options

A. `55440 cm^(3)`
B. `3465 cm^(3)`
C. `7720 cm^(3)`
D. `13860 cm^(3)`

Answer

To find the volume of a right circular cylinder, we will follow these steps:

### Step 1: Identify the given values
- Height (h) = 40 cm
- Circumference of the base (C) = 66 cm

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

### Step 3: Solve for the radius (r)
Given \( C = 66 \) cm, we can substitute this into the formula:
\[ 66 = 2 \pi r \]
Using \( \pi \approx \frac{22}{7} \):
\[ 66 = 2 \times \frac{22}{7} \times r \]
Now, simplify:
\[ 66 = \frac{44}{7} r \]
To isolate \( r \), multiply both sides by \( \frac{7}{44} \):
\[ r = \frac{66 \times 7}{44} \]
\[ r = \frac{462}{44} \]
\[ r = \frac{21}{2} \text{ cm} \]

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

### Step 5: Substitute the values into the volume formula
Substituting \( r = \frac{21}{2} \) cm and \( h = 40 \) cm:
\[ V = \pi \left(\frac{21}{2}\right)^2 \times 40 \]
\[ V = \frac{22}{7} \times \left(\frac{441}{4}\right) \times 40 \]

### Step 6: Simplify the expression
Calculating \( \left(\frac{21}{2}\right)^2 \):
\[ \left(\frac{21}{2}\right)^2 = \frac{441}{4} \]
Now substituting back:
\[ V = \frac{22}{7} \times \frac{441}{4} \times 40 \]

### Step 7: Calculate the volume
First, simplify \( \frac{441 \times 40}{4} \):
\[ \frac{441 \times 40}{4} = 441 \times 10 = 4410 \]
Now substitute this into the volume formula:
\[ V = \frac{22}{7} \times 4410 \]
Now calculate:
\[ V = \frac{22 \times 4410}{7} \]
\[ V = \frac{97020}{7} \]
\[ V = 13860 \text{ cm}^3 \]

### Final Answer
The volume of the right circular cylinder is **13860 cm³**.

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