A copper sphere of diameter 36 cm is drawn into a wire of diameter 4 mm. Find the length of the wire.

Options

A. 2000 m
B. 1500 m
C. 1855 m
D. 1944 m

Answer

To solve the problem of finding the length of the wire drawn from a copper sphere, we will follow these steps:

### Step 1: Find the volume of the copper sphere.
The formula for the volume \( V \) of a sphere is given by:

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

Given the diameter of the sphere is 36 cm, we can find the radius \( r \):

\[ r = \frac{\text{diameter}}{2} = \frac{36 \text{ cm}}{2} = 18 \text{ cm} \]

Now, substituting the radius into the volume formula:

\[ V = \frac{4}{3} \pi (18)^3 \]

Calculating \( (18)^3 \):

\[ (18)^3 = 5832 \]

Thus, the volume of the sphere becomes:

\[ V = \frac{4}{3} \pi (5832) = \frac{23328}{3} \pi = 7776 \pi \text{ cm}^3 \]

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

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

The diameter of the wire is 4 mm, which we need to convert to cm:

\[ \text{Diameter in cm} = \frac{4 \text{ mm}}{10} = 0.4 \text{ cm} \]

Now, we find the radius \( r \) of the wire:

\[ r = \frac{\text{diameter}}{2} = \frac{0.4 \text{ cm}}{2} = 0.2 \text{ cm} \]

Substituting the radius into the volume formula:

\[ V = \pi (0.2)^2 h = \pi (0.04) h = 0.04 \pi h \text{ cm}^3 \]

### Step 3: Set the volumes equal to each other.
Since the volume of the sphere is equal to the volume of the wire, we have:

\[ 7776 \pi = 0.04 \pi h \]

We can cancel \( \pi \) from both sides:

\[ 7776 = 0.04 h \]

### Step 4: Solve for \( h \).
To find \( h \), we rearrange the equation:

\[ h = \frac{7776}{0.04} \]

Calculating \( h \):

\[ h = 7776 \div 0.04 = 7776 \times 25 = 194400 \text{ cm} \]

### Step 5: Convert the height from cm to meters.
Since 1 meter = 100 cm, we convert \( h \) to meters:

\[ h = \frac{194400 \text{ cm}}{100} = 1944 \text{ m} \]

### Final Answer:
The length of the wire is \( 1944 \text{ m} \).

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