The radius of a copper shpere is 12 cm. The sphere is melted and drawn into a long wire of uniform circular cross-section. If the length of the wire is 144 cm , the radius of wire is :

Options

A. 1 cm
B. 2 cm
C. 4 cm
D. none of these

Answer

To solve the problem step by step, we need to find the radius of the wire formed from the melted copper sphere.

### Step 1: Calculate the volume of the copper sphere.
The formula for the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Where \( r \) is the radius of the sphere. Here, the radius \( r \) of the copper sphere is 12 cm.

Substituting the value into the formula:
\[ V = \frac{4}{3} \pi (12)^3 \]

### Step 2: Calculate \( 12^3 \).
Calculating \( 12^3 \):
\[ 12^3 = 12 \times 12 \times 12 = 144 \times 12 = 1728 \]

### Step 3: Substitute back into the volume formula.
Now substituting \( 12^3 \) back into the volume formula:
\[ V = \frac{4}{3} \pi (1728) \]
\[ V = \frac{4 \times 1728}{3} \pi \]
\[ V = 2304 \pi \text{ cm}^3 \]

### Step 4: Set the volume of the sphere equal to the volume of the wire.
The volume of the wire (which is in the shape of a cylinder) is given by the formula:
\[ V = \pi r_w^2 h \]
Where \( r_w \) is the radius of the wire and \( h \) is the height (or length) of the wire. The height of the wire is given as 144 cm.

Setting the volumes equal to each other:
\[ 2304 \pi = \pi r_w^2 (144) \]

### Step 5: Cancel \( \pi \) from both sides.
\[ 2304 = r_w^2 \times 144 \]

### Step 6: Solve for \( r_w^2 \).
Dividing both sides by 144:
\[ r_w^2 = \frac{2304}{144} \]

### Step 7: Calculate \( \frac{2304}{144} \).
Calculating the division:
\[ r_w^2 = 16 \]

### Step 8: Find \( r_w \) by taking the square root.
Taking the square root of both sides:
\[ r_w = \sqrt{16} = 4 \text{ cm} \]

### Conclusion:
The radius of the wire is \( 4 \) cm.

---