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
(a) 9 m
(b) 18 m
(c) 27 m
(d) 36 m

Options

A. 9 m
B. 18 m
C. 27 m
D. 36 m

Answer

To find the length of the wire drawn from a copper sphere, we need to equate the volumes of the sphere and the cylinder (wire). Here’s a step-by-step solution:

### Step 1: Find 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 of the sphere is 3 cm, we first convert it to meters:
\[ r = 3 \text{ cm} = 0.03 \text{ m} \]

Now, substituting the radius into the volume formula:
\[ V = \frac{4}{3} \pi (0.03)^3 \]

Calculating \( (0.03)^3 \):
\[ (0.03)^3 = 0.000027 \text{ m}^3 \]

Now substituting this value into the volume formula:
\[ V = \frac{4}{3} \pi (0.000027) \approx 0.000113097 \text{ m}^3 \]

### Step 2: Find the volume of the wire (cylinder).
The formula for 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 (length) of the cylinder.

The diameter of the wire is given as 0.2 cm, so the radius \( r \) is:
\[ r = \frac{0.2 \text{ cm}}{2} = 0.1 \text{ cm} = 0.001 \text{ m} \]

Now substituting the radius into the volume formula:
\[ V = \pi (0.001)^2 h \]

Calculating \( (0.001)^2 \):
\[ (0.001)^2 = 0.000001 \text{ m}^2 \]

So the volume of the wire becomes:
\[ V = \pi (0.000001) h = \pi \times 0.000001 \times h \]

### Step 3: Equate the volumes of the sphere and the wire.
Since the volume of the copper remains constant, we can set the two volumes equal to each other:
\[ \frac{4}{3} \pi (0.03)^3 = \pi (0.001)^2 h \]

Cancelling \( \pi \) from both sides:
\[ \frac{4}{3} (0.03)^3 = (0.001)^2 h \]

Substituting the calculated volume of the sphere:
\[ \frac{4}{3} (0.000027) = (0.000001) h \]

### Step 4: Solve for \( h \).
First, calculate \( \frac{4}{3} (0.000027) \):
\[ \frac{4 \times 0.000027}{3} = 0.000036 \]

Now we have:
\[ 0.000036 = 0.000001 h \]

Dividing both sides by \( 0.000001 \):
\[ h = \frac{0.000036}{0.000001} = 36 \text{ m} \]

### Conclusion
The length of the wire is \( 36 \text{ m} \).