A copper sphere of radius 3 cm is beaten and drawn into a wire of diameter 0.2 cm. The length of the wire is :

Options

A. 9 m
B. 12 m
C. 18 m
D. 36 m

Answer

To find the length of the wire drawn from a copper sphere, we need to equate the volume of the sphere to the volume of the wire. Here are the steps to solve the problem:

### 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. Given that the radius \( r = 3 \) cm, we can substitute this value into the formula:
\[ V = \frac{4}{3} \pi (3)^3 \]
Calculating \( (3)^3 \):
\[ (3)^3 = 27 \]
Now substituting this back into the volume formula:
\[ V = \frac{4}{3} \pi (27) = 36 \pi \text{ cm}^3 \]

### Step 2: Calculate the volume of the wire.
The wire is in the shape of a cylinder. The formula for the volume \( V \) of a cylinder is:
\[ V = \pi r^2 h \]
Where \( r \) is the radius of the base of the cylinder and \( h \) is the height (or length) of the cylinder. The diameter of the wire is given as 0.2 cm, so the radius \( r \) is:
\[ r = \frac{0.2}{2} = 0.1 \text{ cm} \]
Now substituting the radius into the volume formula:
\[ V = \pi (0.1)^2 h = \pi (0.01) h = 0.01 \pi h \text{ cm}^3 \]

### Step 3: Equate the volumes of the sphere and the wire.
Since the volume of the copper sphere is equal to the volume of the wire, we can set the two equations equal to each other:
\[ 36 \pi = 0.01 \pi h \]
We can divide both sides by \( \pi \) (assuming \( \pi \neq 0 \)):
\[ 36 = 0.01 h \]

### Step 4: Solve for \( h \).
To find \( h \), we can rearrange the equation:
\[ h = \frac{36}{0.01} \]
Calculating this gives:
\[ h = 3600 \text{ cm} \]

### Conclusion:
The length of the wire is \( 3600 \) cm, or \( 36 \) meters.