The radius of a metallic sphere is 3 cm.it is melted and drawn into a wire of uniform circular section of 0.1 cm. the length of the wire will be (in m):

Options

A. 44
B. 24
C. 36
D. 40

Answer

To solve the problem step by step, we will calculate the volume of the metallic sphere, then determine the length of the wire formed from that volume.

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

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

Given that the radius \( r \) of the sphere is 3 cm, we can substitute this value into the formula:

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

Calculating \( (3)^3 \):

\[ (3)^3 = 27 \]

Now substituting back into the volume formula:

\[ V = \frac{4}{3} \pi \times 27 = 36 \pi \text{ cm}^3 \]

### Step 2: Calculate the Volume of the Wire
The wire is formed with a uniform circular section. 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 circular section of the wire and \( h \) is the length of the wire. The radius of the wire is given as 0.1 cm. Thus, we can express the volume of the wire as:

\[ V = \pi (0.1)^2 h \]

Calculating \( (0.1)^2 \):

\[ (0.1)^2 = 0.01 \]

So the volume of the wire becomes:

\[ V = \pi \times 0.01 \times h = 0.01 \pi h \text{ cm}^3 \]

### Step 3: Set the Volumes Equal
Since the sphere is melted to form the wire, the volumes must be equal:

\[ 36 \pi = 0.01 \pi h \]

### Step 4: Solve for \( h \)
We can cancel \( \pi \) from both sides:

\[ 36 = 0.01 h \]

Now, to find \( h \), we divide both sides by 0.01:

\[ h = \frac{36}{0.01} = 3600 \text{ cm} \]

### Step 5: Convert Length from cm to m
To convert centimeters to meters, we divide by 100:

\[ h = \frac{3600}{100} = 36 \text{ m} \]

### Final Answer
The length of the wire will be **36 meters**.

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