Circumference of base of a cylinder is `132cm^(2)` and height is half of square of radius of it's base. Find volume of the cylinders?

Options

A. A)2772 `cm^(3)`
B. B)305613 `cm^(3)`
C. C)2376 `cm^(3)`
D. D)924 `cm^(3)`

Answer

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

### Step 1: Find the Radius of the Cylinder's Base
The circumference \( C \) of the base of the cylinder is given by the formula:
\[ C = 2 \pi r \]
We know that the circumference is \( 132 \, \text{cm} \). Therefore, we can set up the equation:
\[ 2 \pi r = 132 \]
Using \( \pi \approx \frac{22}{7} \), we can substitute this value into the equation:
\[ 2 \times \frac{22}{7} \times r = 132 \]
Now, simplify the equation:
\[ \frac{44}{7} r = 132 \]
To isolate \( r \), multiply both sides by \( \frac{7}{44} \):
\[ r = 132 \times \frac{7}{44} \]
Calculating this gives:
\[ r = 3 \times 7 = 21 \, \text{cm} \]

### Step 2: Find the Height of the Cylinder
According to the problem, the height \( h \) of the cylinder is given as half of the square of the radius:
\[ h = \frac{1}{2} r^2 \]
Substituting the value of \( r \):
\[ h = \frac{1}{2} \times (21)^2 \]
Calculating \( 21^2 \):
\[ 21^2 = 441 \]
Thus,
\[ h = \frac{441}{2} = 220.5 \, \text{cm} \]

### Step 3: Calculate the Volume of the Cylinder
The volume \( V \) of the cylinder is given by the formula:
\[ V = \pi r^2 h \]
Substituting the values of \( \pi \), \( r \), and \( h \):
\[ V = \frac{22}{7} \times (21)^2 \times \frac{441}{2} \]
Calculating \( (21)^2 \):
\[ (21)^2 = 441 \]
Now substituting this back into the volume formula:
\[ V = \frac{22}{7} \times 441 \times \frac{441}{2} \]
Now simplify:
\[ V = \frac{22 \times 441 \times 441}{14} \]
Calculating \( \frac{441}{7} = 63 \):
\[ V = 22 \times 63 \times 441 \]
Now calculating \( 22 \times 63 = 1386 \):
\[ V = 1386 \times 441 \]
Finally, calculating \( 1386 \times 441 \):
\[ V = 611,046 \, \text{cm}^3 \]

### Final Answer
The volume of the cylinder is \( 611,046 \, \text{cm}^3 \).

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