A copper rod of radius 2cm and length 7 cm is drawn into a wire of length 112 cm of uniform thickness. The thickness of the wire is `:`

Options

A. 0.5cm
B. 0.7 cm
C. 2 cm
D. 1 cm

Answer

To solve the problem of finding the thickness of the wire formed from a copper rod, we can follow these steps:

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

Given:
- Radius of the rod \( r = 2 \, \text{cm} \)
- Length of the rod \( h = 7 \, \text{cm} \)

Substituting the values:
\[ V = \pi (2)^2 (7) = \pi (4)(7) = 28\pi \, \text{cm}^3 \]

### Step 2: Calculate the Volume of the Wire
The volume of the wire can also be expressed in terms of its thickness \( t \) and length \( L \). The wire is also cylindrical, and its volume is given by:
\[ V = \pi \left( \frac{t}{2} \right)^2 L \]
where \( t \) is the thickness (which is the diameter of the wire), and \( L \) is the length of the wire.

Given:
- Length of the wire \( L = 112 \, \text{cm} \)

Substituting the values:
\[ V = \pi \left( \frac{t}{2} \right)^2 (112) = \pi \left( \frac{t^2}{4} \right) (112) = 28\pi t^2 \, \text{cm}^3 \]

### Step 3: Set the Volumes Equal
Since the mass of the copper does not change when it is drawn into a wire, the volume of the rod must equal the volume of the wire:
\[ 28\pi = 28\pi t^2 \]

### Step 4: Simplify the Equation
Dividing both sides by \( 28\pi \):
\[ 1 = t^2 \]

### Step 5: Solve for Thickness \( t \)
Taking the square root of both sides:
\[ t = 1 \, \text{cm} \]

Thus, the thickness of the wire is \( 1 \, \text{cm} \).

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