A sphere of radius 9 cm is melt to form a cylindrical wire of radius 3 cm. Find the length of the wire.

Options

A. 96 cm
B. 108 cm
C. 120 cm
D. 135 cm

Answer

To solve the problem of finding the length of a cylindrical wire formed by melting a sphere, we can follow these steps:

### Step 1: Calculate the Volume of the 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. Here, the radius of the sphere is 9 cm.

Substituting the radius into the formula:
\[ V = \frac{4}{3} \pi (9)^3 \]

### Step 2: Calculate \( 9^3 \)
First, we need to calculate \( 9^3 \):
\[ 9^3 = 729 \]

### Step 3: Substitute Back to Find the Volume of the Sphere
Now substituting \( 729 \) back into the volume formula:
\[ V = \frac{4}{3} \pi (729) = \frac{2916}{3} \pi = 972 \pi \text{ cm}^3 \]

### Step 4: Set Up the Volume of the Cylinder
The volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height (or length in this case) of the cylinder. The radius of the cylinder is given as 3 cm.

### Step 5: Substitute the Radius of the Cylinder
Substituting the radius into the volume formula:
\[ V = \pi (3)^2 h = \pi (9) h = 9 \pi h \]

### Step 6: Equate the Volumes
Since the volume of the sphere is equal to the volume of the cylinder (as the sphere is melted to form the cylinder), we can set the two volumes equal to each other:
\[ 972 \pi = 9 \pi h \]

### Step 7: Cancel \( \pi \) from Both Sides
We can cancel \( \pi \) from both sides:
\[ 972 = 9h \]

### Step 8: Solve for \( h \)
Now, divide both sides by 9 to find \( h \):
\[ h = \frac{972}{9} = 108 \text{ cm} \]

### Conclusion
Thus, the length of the wire formed is:
\[ \text{Length of the wire} = 108 \text{ cm} \]

### Final Answer
The correct option is **108 cm**.