The lateral surface area of a cylinder is ` 1056 cm^(2)` and its height is 16 cm. What is its volume ?

Options

A. `5566 cm^(3) `
B. `4455 cm^(3) `
C. `5544 cm^(3) `
D. none of these

Answer

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

### Step 1: Understand the formula for lateral surface area
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: Substitute the known values into the formula
We know the lateral surface area is \( 1056 \, \text{cm}^2 \) and the height \( h \) is \( 16 \, \text{cm} \). Substituting these values into the formula gives:
\[ 1056 = 2 \pi r \times 16 \]

### Step 3: Simplify the equation
We can simplify the equation:
\[ 1056 = 32 \pi r \]
Now, substituting \( \pi \) with \( \frac{22}{7} \):
\[ 1056 = 32 \times \frac{22}{7} \times r \]

### Step 4: Solve for \( r \)
To isolate \( r \), we first multiply both sides by \( 7 \):
\[ 1056 \times 7 = 32 \times 22 \times r \]
Calculating \( 1056 \times 7 \):
\[ 7392 = 704 r \]
Now, divide both sides by \( 704 \):
\[ r = \frac{7392}{704} = 10.5 \, \text{cm} \]

### 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 of \( r \) and \( h \):
\[ V = \frac{22}{7} \times (10.5)^2 \times 16 \]

### Step 6: Calculate \( (10.5)^2 \)
Calculating \( (10.5)^2 \):
\[ (10.5)^2 = 110.25 \]

### Step 7: Substitute back into the volume formula
Now substituting back into the volume formula:
\[ V = \frac{22}{7} \times 110.25 \times 16 \]

### Step 8: Simplify the calculation
Calculating \( \frac{22 \times 110.25 \times 16}{7} \):
First, calculate \( 110.25 \times 16 = 1764 \):
\[ V = \frac{22 \times 1764}{7} \]
Calculating \( 22 \times 1764 = 38808 \):
\[ V = \frac{38808}{7} = 5544 \, \text{cm}^3 \]

### Final Answer
The volume of the cylinder is:
\[ \boxed{5544 \, \text{cm}^3} \]