Curved surface area of a cylinder is `308 cm^2` , and height is 14 cm. What will be the volume of the cylinder?

Options

A. `439 cm^3`
B. `385 cm^3`
C. `539 cm^3`
D. `529 cm^3`

Answer

To find the volume of the cylinder given its curved surface area and height, we can follow these steps:

### Step 1: Understand the formula for the curved surface area of a cylinder.
The formula for the curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2\pi r h \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

### Step 2: Substitute the given values into the formula.
We know the curved surface area (CSA) is \( 308 \, \text{cm}^2 \) and the height \( h \) is \( 14 \, \text{cm} \). Therefore, we can set up the equation:
\[ 2\pi r \cdot 14 = 308 \]

### Step 3: Solve for the radius \( r \).
First, simplify the equation:
\[ 28\pi r = 308 \]
Now, divide both sides by \( 28\pi \):
\[ r = \frac{308}{28\pi} \]
Using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{308}{28 \times \frac{22}{7}} \]
\[ r = \frac{308 \times 7}{28 \times 22} \]
\[ r = \frac{2156}{616} \]
Now simplify:
\[ r = \frac{539}{154} \]
\[ r = 3.5 \, \text{cm} \]

### Step 4: Use the radius to find the volume of the cylinder.
The formula for the volume \( V \) of a cylinder is:
\[ V = \pi r^2 h \]
Substituting the values we have:
\[ V = \pi (3.5)^2 (14) \]
Calculating \( (3.5)^2 \):
\[ (3.5)^2 = 12.25 \]
Now substitute this back into the volume formula:
\[ V = \pi \cdot 12.25 \cdot 14 \]
Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{22}{7} \cdot 12.25 \cdot 14 \]
Calculating \( 12.25 \cdot 14 \):
\[ 12.25 \cdot 14 = 171.5 \]
Now substitute:
\[ V = \frac{22 \cdot 171.5}{7} \]
Calculating \( 22 \cdot 171.5 \):
\[ 22 \cdot 171.5 = 3773 \]
Now divide by \( 7 \):
\[ V = \frac{3773}{7} \approx 539 \, \text{cm}^3 \]

### Final Answer:
The volume of the cylinder is \( 539 \, \text{cm}^3 \).