The curved surface area of a cylinder is `4400cm^(2)` and the circumference of its base is 110 cm. Find the volume of the cylinder.

Answer

To solve the problem step by step, we will use the given information about the curved surface area and the circumference of the cylinder to find its volume.

### Step 1: Identify the given data
- Curved Surface Area (CSA) of the cylinder = 4400 cm²
- Circumference of the base of the cylinder = 110 cm

### Step 2: Use the formula for the circumference of a cylinder to find the radius
The formula for the circumference (C) of a cylinder is:
\[ C = 2\pi r \]
Given that the circumference is 110 cm, we can set up the equation:
\[ 110 = 2 \times \frac{22}{7} \times r \]

### Step 3: Solve for the radius (r)
Rearranging the equation to solve for r:
\[ r = \frac{110 \times 7}{2 \times 22} \]
Calculating this gives:
\[ r = \frac{770}{44} = \frac{35}{2} = 17.5 \, \text{cm} \]

### Step 4: Use the formula for the curved surface area to find the height (h)
The formula for the curved surface area (CSA) of a cylinder is:
\[ CSA = 2\pi rh \]
Substituting the known values:
\[ 4400 = 2 \times \frac{22}{7} \times \frac{35}{2} \times h \]

### Step 5: Simplify and solve for height (h)
The 2s cancel out:
\[ 4400 = \frac{22 \times 35}{7} \times h \]
Calculating \( \frac{22 \times 35}{7} \):
\[ \frac{770}{7} = 110 \]
Thus, we have:
\[ 4400 = 110h \]
Now, solving for h:
\[ h = \frac{4400}{110} = 40 \, \text{cm} \]

### Step 6: Calculate the volume of the cylinder
The formula for the volume (V) of a cylinder is:
\[ V = \pi r^2 h \]
Substituting the values we found:
\[ V = \frac{22}{7} \times \left(\frac{35}{2}\right)^2 \times 40 \]

### Step 7: Simplify the volume calculation
Calculating \( \left(\frac{35}{2}\right)^2 = \frac{1225}{4} \):
\[ V = \frac{22}{7} \times \frac{1225}{4} \times 40 \]
Now, simplifying this:
\[ V = \frac{22 \times 1225 \times 40}{28} \]
\[ V = \frac{22 \times 1225 \times 10}{7} \]
Calculating \( 22 \times 10 = 220 \):
\[ V = \frac{220 \times 1225}{7} \]
Calculating \( \frac{220 \times 1225}{7} = 3850 \]

### Final Answer
The volume of the cylinder is **3850 cm³**.

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