A copper rod of 1 cm diameter and 8 cm length is drawn into a wire of uniform diameter and 18 m length. The raidus (in cm) of the wire is

Options

A. `(1)/(15)`
B. `(1)/(30)`
C. `(2)/(15)`
D. 15

Answer

To find the radius of the wire drawn from the copper rod, we will follow these steps:

### Step 1: Find the volume of the copper rod
The volume \( V \) of a cylinder (which is the shape of the rod) 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 = 1 cm, so the radius \( r = \frac{1}{2} \) cm = 0.5 cm
- Length of the copper rod \( h = 8 \) cm

Now, substituting the values into the volume formula:
\[ V = \pi (0.5)^2 (8) = \pi \times 0.25 \times 8 = 2\pi \text{ cm}^3 \]

### Step 2: Find the volume of the wire
The wire is also a cylinder, and its volume will be equal to the volume of the copper rod since the material is conserved when the rod is drawn into the wire.

Let the radius of the wire be \( R \) and the length of the wire \( H = 18 \) m = 1800 cm (since we need to keep the units consistent).

The volume of the wire is given by:
\[ V = \pi R^2 H \]

### Step 3: Set the volumes equal to each other
Since the volume of the copper rod is equal to the volume of the wire:
\[ 2\pi = \pi R^2 (1800) \]

### Step 4: Simplify the equation
We can divide both sides by \( \pi \):
\[ 2 = R^2 \times 1800 \]

### Step 5: Solve for \( R^2 \)
Rearranging gives:
\[ R^2 = \frac{2}{1800} = \frac{1}{900} \]

### Step 6: Solve for \( R \)
Taking the square root of both sides:
\[ R = \sqrt{\frac{1}{900}} = \frac{1}{30} \text{ cm} \]

### Conclusion
The radius of the wire is \( \frac{1}{30} \) cm.

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