A copper rod of diameter 2 cm and length 10 cm is drawn into a wire of uniform thickness and length 10 m. Find the thickness of the wire.

Answer

To find the thickness of the wire formed from the copper rod, we can follow these steps:

### Step 1: Calculate the Volume of the Copper Rod
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height (or length) of the cylinder.

Given:
- Diameter of the copper rod = 2 cm, so the radius \( r_1 = \frac{2}{2} = 1 \) cm.
- Length of the copper rod \( h_1 = 10 \) cm.

Now, substituting the values into the volume formula:
\[ V_1 = \pi (1)^2 (10) = 10\pi \text{ cm}^3 \]

### Step 2: Calculate the Volume of the Wire
The wire is also a cylinder, and its volume \( V_2 \) can be expressed as:
\[ V_2 = \pi r_2^2 h_2 \]
where \( r_2 \) is the radius of the wire and \( h_2 \) is its length.

Given:
- Length of the wire \( h_2 = 10 \) m = 1000 cm (since 1 m = 100 cm).

### Step 3: Set the Volumes Equal
Since the volume of the copper rod is equal to the volume of the wire (as the material is the same), we have:
\[ V_1 = V_2 \]
\[ 10\pi = \pi r_2^2 (1000) \]

### Step 4: Simplify the Equation
We can cancel \( \pi \) from both sides:
\[ 10 = r_2^2 (1000) \]
Now, divide both sides by 1000:
\[ r_2^2 = \frac{10}{1000} = \frac{1}{100} \]

### Step 5: Calculate \( r_2 \)
Taking the square root of both sides:
\[ r_2 = \sqrt{\frac{1}{100}} = \frac{1}{10} \text{ cm} = 0.1 \text{ cm} \]

### Step 6: Calculate the Thickness of the Wire
The thickness of the wire is twice the radius:
\[ \text{Thickness} = 2r_2 = 2 \times 0.1 \text{ cm} = 0.2 \text{ cm} \]
To convert this into millimeters (since 1 cm = 10 mm):
\[ 0.2 \text{ cm} = 2 \text{ mm} \]

### Final Answer
The thickness of the wire is **2 mm**.

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