The radius of a metallic sphere is 60 mm .It is melted and recast in to wire of diameter 0.8 mm .Find the length of the wire.

Explanation

To find the length of the wire made from melting a metallic sphere, we will follow these steps:

### Step 1: Calculate the Volume of the Sphere
The formula for the volume of a sphere is given by:

\[ V = \frac{4}{3} \pi r^3 \]

Given that the radius \( r \) of the sphere is 60 mm, we can convert this to centimeters:

\[ r = 60 \text{ mm} = 6 \text{ cm} \]

Now, substituting the value of \( r \) into the volume formula:

\[ V = \frac{4}{3} \pi (6)^3 \]

Calculating \( (6)^3 \):

\[ (6)^3 = 216 \]

Now substituting this back into the volume formula:

\[ V = \frac{4}{3} \pi (216) = \frac{864}{3} \pi = 288 \pi \text{ cm}^3 \]

### Step 2: Calculate the Volume of the Wire
The wire has a diameter of 0.8 mm, which we need to convert to centimeters:

\[ \text{Diameter} = 0.8 \text{ mm} = 0.08 \text{ cm} \]

The radius \( r_w \) of the wire is:

\[ r_w = \frac{0.08}{2} = 0.04 \text{ cm} \]

The volume \( V_w \) of the wire is given by the formula:

\[ V_w = \pi r_w^2 L \]

Substituting \( r_w \):

\[ V_w = \pi (0.04)^2 L = \pi (0.0016) L = 0.0016 \pi L \text{ cm}^3 \]

### Step 3: Set the Volumes Equal
Since the volume of the sphere is equal to the volume of the wire (as the sphere is melted to form the wire), we have:

\[ 288 \pi = 0.0016 \pi L \]

We can cancel \( \pi \) from both sides:

\[ 288 = 0.0016 L \]

### Step 4: Solve for Length \( L \)
Now, we solve for \( L \):

\[ L = \frac{288}{0.0016} \]

Calculating \( L \):

\[ L = 288 \div 0.0016 = 180000 \text{ cm} \]

### Step 5: Convert Length to Meters
To convert the length from centimeters to meters, we divide by 100:

\[ L = \frac{180000}{100} = 1800 \text{ m} \]

### Final Answer
The length of the wire is **1800 meters**.

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