The length of the wire of 0.2 cm radius that can be drawn after melting a solid copper sphere of diameter 18 cm, is----

Options

A. 24.3 m
B. 243 m
C. 2430 m
D. 24300 m

Answer

To find the length of the wire that can be drawn from a solid copper sphere after melting it, we need to follow these steps:

### Step 1: Calculate the volume of the solid 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.

Given the diameter of the sphere is 18 cm, the radius \( r \) is:
\[ r = \frac{diameter}{2} = \frac{18}{2} = 9 \text{ cm} \]

Now, substituting the value of \( r \) into the volume formula:
\[ V = \frac{4}{3} \pi (9)^3 \]
Calculating \( 9^3 \):
\[ 9^3 = 729 \]
Now substituting back:
\[ V = \frac{4}{3} \pi (729) = \frac{2916}{3} \pi = 972 \pi \text{ cm}^3 \]

### Step 2: Calculate the volume of the wire.
The volume \( V \) of a cylinder (which is the shape of the wire) 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.

Given the radius of the wire is 0.2 cm, we can express the volume of the wire as:
\[ V = \pi (0.2)^2 h \]
Calculating \( (0.2)^2 \):
\[ (0.2)^2 = 0.04 \]
So, the volume of the wire becomes:
\[ V = \pi (0.04) h = 0.04 \pi h \]

### Step 3: Set the volume of the sphere equal to the volume of the wire.
Since the volume of the melted sphere will equal the volume of the wire, we can set them equal to each other:
\[ 972 \pi = 0.04 \pi h \]

### Step 4: Solve for \( h \).
We can cancel \( \pi \) from both sides:
\[ 972 = 0.04 h \]
Now, solving for \( h \):
\[ h = \frac{972}{0.04} \]
Calculating \( \frac{972}{0.04} \):
\[ h = 972 \times 25 = 24300 \text{ cm} \]

### Final Answer:
The length of the wire that can be drawn after melting the solid copper sphere is **24300 cm**.

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