An iron sphere of radius 27 cm is melted to form a wire of length 729 cm. The radius of wire is

Options

A. 6 cm
B. 9 cm
C. 18 cm
D. 36m

Answer

To find the radius of the wire formed by melting an iron 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. Given that the radius of the sphere is 27 cm, we can substitute this value into the formula:
\[ V = \frac{4}{3} \pi (27)^3 \]

### Step 2: Calculate \( 27^3 \)
First, we need to calculate \( 27^3 \):
\[ 27^3 = 27 \times 27 \times 27 = 19683 \]
Now substitute this back into the volume formula:
\[ V = \frac{4}{3} \pi (19683) \]

### Step 3: Simplify the volume expression
Now we can simplify the volume:
\[ V = \frac{4 \times 19683}{3} \pi = 26244 \pi \, \text{cm}^3 \]

### Step 4: Set up the volume of the wire
The volume of the wire can be calculated using the formula for the volume of a cylinder:
\[ V = \pi r^2 h \]
where \( r \) is the radius of the wire and \( h \) is the height (or length) of the wire. Given that the length of the wire is 729 cm, we can write:
\[ V = \pi r^2 (729) \]

### Step 5: Equate the volumes
Since the volume of the melted sphere is equal to the volume of the wire, we can set the two volume expressions equal to each other:
\[ 26244 \pi = \pi r^2 (729) \]

### Step 6: Cancel out \( \pi \)
We can cancel \( \pi \) from both sides of the equation:
\[ 26244 = r^2 (729) \]

### Step 7: Solve for \( r^2 \)
Now, divide both sides by 729 to isolate \( r^2 \):
\[ r^2 = \frac{26244}{729} \]

### Step 8: Calculate \( \frac{26244}{729} \)
Now we perform the division:
\[ r^2 = 36 \]

### Step 9: Solve for \( r \)
Finally, take the square root of both sides to find \( r \):
\[ r = \sqrt{36} = 6 \, \text{cm} \]

### Final Answer
The radius of the wire is \( 6 \, \text{cm} \).

---