The diameter of a copper sphere is 18 cm. It is melted and drawn into a long wire of uniform cross section. If the length of the wire is 108 m, find its diameter.

Answer

To solve the problem, we need to find the diameter of a wire formed by melting a copper sphere. The steps to solve this problem are as follows:

### Step 1: Find the radius of the copper sphere
The diameter of the sphere is given as 18 cm. Therefore, the radius \( r \) can be calculated as:
\[ r = \frac{\text{Diameter}}{2} = \frac{18 \, \text{cm}}{2} = 9 \, \text{cm} \]

**Hint:** Remember that the radius is half of the diameter.

### Step 2: Calculate the volume of the copper sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
Substituting \( r = 9 \, \text{cm} \):
\[ V = \frac{4}{3} \pi (9)^3 = \frac{4}{3} \pi (729) = \frac{2916}{3} \pi = 972 \pi \, \text{cm}^3 \]

**Hint:** Use the formula for the volume of a sphere and remember to cube the radius.

### Step 3: Convert the volume to cubic meters
Since the length of the wire is given in meters, we need to convert the volume from cubic centimeters to cubic meters. We know that:
\[ 1 \, \text{m}^3 = 10^6 \, \text{cm}^3 \]
Thus,
\[ 972 \pi \, \text{cm}^3 = \frac{972 \pi}{10^6} \, \text{m}^3 \]

**Hint:** Be careful with unit conversions, especially when dealing with cubic measurements.

### Step 4: Calculate the volume of the wire
The wire has a cylindrical shape, and its volume \( V \) can also be expressed as:
\[ V = \pi r^2 h \]
where \( r \) is the radius of the wire and \( h \) is the height (or length) of the wire. The length of the wire is given as 108 m. Thus:
\[ V = \pi r^2 (108) \]

**Hint:** The volume of a cylinder is calculated using the formula \( \pi r^2 h \).

### Step 5: Set the volumes equal
Since the volume of the melted sphere is equal to the volume of the wire:
\[ 972 \pi \, \text{cm}^3 = \pi r^2 (108) \]
Dividing both sides by \( \pi \):
\[ 972 = r^2 (108) \]

**Hint:** You can simplify the equation by canceling out \( \pi \).

### Step 6: Solve for \( r^2 \)
Now, solve for \( r^2 \):
\[ r^2 = \frac{972}{108} = 9 \]

**Hint:** Perform the division carefully to find \( r^2 \).

### Step 7: Find the radius \( r \)
Taking the square root of both sides:
\[ r = \sqrt{9} = 3 \, \text{cm} \]

**Hint:** Remember that the radius is the square root of \( r^2 \).

### Step 8: Calculate the diameter of the wire
The diameter \( d \) of the wire is given by:
\[ d = 2r = 2 \times 3 \, \text{cm} = 6 \, \text{cm} \]

**Hint:** The diameter is twice the radius.

### Final Answer
The diameter of the wire is **6 cm**.