A copper wire of diameter 6mm is evenly wrapped on the cylinder of length 18cm and diameter 49cm to cover the whole surface. Find:
the volume of the wire.

Answer

To find the volume of the copper wire wrapped around the cylinder, we will follow these steps:

### Step 1: Convert the diameter of the wire to centimeters
The diameter of the copper wire is given as 6 mm. To convert this to centimeters, we divide by 10.

\[ \text{Diameter of wire} = \frac{6 \text{ mm}}{10} = 0.6 \text{ cm} \]

### Step 2: Calculate the radius of the wire
The radius of the wire is half of the diameter.

\[ \text{Radius of wire} = \frac{0.6 \text{ cm}}{2} = 0.3 \text{ cm} \]

### Step 3: Identify the dimensions of the cylinder
The length of the cylinder is given as 18 cm, and the diameter of the cylinder is 49 cm. We can find the radius of the cylinder.

\[ \text{Radius of cylinder} = \frac{49 \text{ cm}}{2} = 24.5 \text{ cm} \]

### Step 4: Calculate the number of turns of the wire around the cylinder
The number of turns can be calculated by dividing the length of the cylinder by the diameter of the wire.

\[ \text{Number of turns} = \frac{\text{Length of cylinder}}{\text{Diameter of wire}} = \frac{18 \text{ cm}}{0.6 \text{ cm}} = 30 \]

### Step 5: Calculate the circumference of the cylinder
The length of wire in one turn is equal to the circumference of the base of the cylinder, which can be calculated using the formula \(C = 2\pi R\).

\[ \text{Circumference} = 2 \times \frac{22}{7} \times 24.5 \text{ cm} = 154 \text{ cm} \]

### Step 6: Calculate the total length of the wire
The total length of the wire used in 30 turns is:

\[ \text{Total length of wire} = \text{Number of turns} \times \text{Circumference} = 30 \times 154 \text{ cm} = 4620 \text{ cm} \]

### Step 7: Calculate the volume of the wire
The volume of the wire can be calculated using the formula for the volume of a cylinder, \(V = \pi R^2 H\), where \(H\) is the length of the wire.

\[ \text{Volume} = \pi \times (0.3 \text{ cm})^2 \times 4620 \text{ cm} \]

Calculating this gives:

\[ \text{Volume} = \frac{22}{7} \times 0.09 \text{ cm}^2 \times 4620 \text{ cm} = 1306.8 \text{ cm}^3 \]

### Final Answer
The volume of the copper wire is \(1306.8 \text{ cm}^3\).

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