The lateral surfac area of a cylinder of length 14 m is `220m^(2)`. Find the volume of the cylinder.

Answer

To solve the problem of finding the volume of a cylinder given its lateral surface area and height, we can follow these steps:

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

### Step 2: Plug in the known values
From the question, we know:
- Lateral surface area (LSA) = 220 m²
- Height (h) = 14 m

Substituting these values into the formula:
\[ 220 = 2 \pi r \times 14 \]

### Step 3: Simplify the equation
We can simplify the equation:
\[ 220 = 28 \pi r \]

### Step 4: Solve for the radius (r)
To find \( r \), we rearrange the equation:
\[ r = \frac{220}{28 \pi} \]

Using \( \pi \approx \frac{22}{7} \):
\[ r = \frac{220}{28 \times \frac{22}{7}} \]
\[ r = \frac{220 \times 7}{28 \times 22} \]

Now we can simplify:
- \( 220 \div 22 = 10 \)
- \( 28 \div 7 = 4 \)

Thus:
\[ r = \frac{10 \times 7}{4} = \frac{70}{4} = 17.5 \text{ m} \]

### Step 5: Calculate 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:
\[ V = \pi \times (17.5)^2 \times 14 \]

Calculating \( (17.5)^2 \):
\[ (17.5)^2 = 306.25 \]

Now substituting back into the volume formula:
\[ V = \frac{22}{7} \times 306.25 \times 14 \]

Calculating:
\[ V = \frac{22 \times 306.25 \times 14}{7} \]

Now, simplifying:
- \( 14 \div 7 = 2 \)
Thus:
\[ V = 22 \times 306.25 \times 2 \]
\[ V = 44 \times 306.25 = 13475 \text{ m}^3 \]

### Final Answer
The volume of the cylinder is \( 13475 \text{ m}^3 \).

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