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

Options

A. a) 1 cm
B. b) 0.9 cm
C. c) 0.3 cm
D. d) 0.6cm

Answer

To find the diameter of the wire formed from the melted copper sphere, we will follow these steps:

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

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

### Step 3: Convert the length of the wire into centimeters.
The length of the wire is given as 108 m. We need to convert this into centimeters:
\[ 108 \, \text{m} = 108 \times 100 \, \text{cm} = 10800 \, \text{cm} \]

### Step 4: Set up the volume of the cylinder (wire).
The volume of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Where \( r \) is the radius of the cylinder and \( h \) is the height (length) of the cylinder. We know the volume of the wire is equal to the volume of the sphere:
\[ 972 \pi \, \text{cm}^3 = \pi r^2 (10800 \, \text{cm}) \]

### Step 5: Cancel out \( \pi \) and solve for \( r^2 \).
\[ 972 = r^2 (10800) \]
Now, divide both sides by 10800:
\[ r^2 = \frac{972}{10800} \]
Calculating the right side:
\[ r^2 = \frac{972 \div 108}{10800 \div 108} = \frac{9}{100} = 0.09 \]

### Step 6: Find the radius and then the diameter of the wire.
Taking the square root of both sides:
\[ r = \sqrt{0.09} = 0.3 \, \text{cm} \]
Now, the diameter (D) of the wire is:
\[ D = 2r = 2 \times 0.3 \, \text{cm} = 0.6 \, \text{cm} \]

### Final Answer:
The diameter of the wire is **0.6 cm**.