The radius of ametallic sphere is 9 cm. It was melted to make a wire of diameter 4 mm. Find the length of the wire.

Explanation

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

### Step 1: Calculate the volume of the metallic 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. Given that the radius \( r = 9 \) cm, we can substitute this value into the formula.

\[ V = \frac{4}{3} \pi (9)^3 \]

Calculating \( 9^3 \):

\[ 9^3 = 729 \]

Now substituting back into the volume formula:

\[ V = \frac{4}{3} \pi (729) = \frac{2916}{3} \pi = 972 \pi \, \text{cm}^3 \]

### Step 2: Calculate the volume of the wire.
The wire is in the shape of a cylinder, and the volume \( V \) of a cylinder is given by:

\[ V = \pi r^2 h \]

Where \( r \) is the radius of the cylinder and \( h \) is the height (or length) of the cylinder. The diameter of the wire is given as 4 mm, so the radius \( r \) is:

\[ r = \frac{4 \, \text{mm}}{2} = 2 \, \text{mm} = 0.2 \, \text{cm} \]

### Step 3: Set the volume of the sphere equal to the volume of the wire.
Since the sphere is melted to form the wire, the volume of the sphere will be equal to the volume of the wire:

\[ 972 \pi = \pi (0.2)^2 h \]

### Step 4: Simplify and solve for \( h \).
First, simplify the equation:

\[ 972 = (0.2)^2 h \]

Calculating \( (0.2)^2 \):

\[ (0.2)^2 = 0.04 \]

Now substituting this back:

\[ 972 = 0.04 h \]

Now, solve for \( h \):

\[ h = \frac{972}{0.04} = 24300 \, \text{cm} \]

### Conclusion
The length of the wire is \( 24300 \, \text{cm} \).

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