If `pi" cm"^(3)` of metal is stretched to a wire of length 3600 m, then the diameter of the wire will be

Options

A. `(1)/(600)` cm
B. `(1)/(300)` cm
C. `(1)/(200)` cm
D. `(1)/(100)` cm

Answer

To find the diameter of the wire when a volume of \( \pi \, \text{cm}^3 \) of metal is stretched into a wire of length 3600 m, we can follow these steps:

### Step 1: Understand the Volume of the Wire
The volume \( V \) of a cylindrical wire is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height (or length) of the cylinder.

### Step 2: Identify Given Values
From the question, we know:
- Volume \( V = \pi \, \text{cm}^3 \)
- Length \( h = 3600 \, \text{m} \)

### Step 3: Convert Length from Meters to Centimeters
Since the volume is in cubic centimeters, we need to convert the length from meters to centimeters:
\[ h = 3600 \, \text{m} \times 100 = 360000 \, \text{cm} \]

### Step 4: Substitute Values into the Volume Formula
Now, we can substitute the known values into the volume formula:
\[ \pi = \pi r^2 (360000) \]

### Step 5: Simplify the Equation
We can cancel \( \pi \) from both sides:
\[ 1 = r^2 \times 360000 \]
Now, isolate \( r^2 \):
\[ r^2 = \frac{1}{360000} \]

### Step 6: Calculate the Radius
To find \( r \), take the square root of both sides:
\[ r = \sqrt{\frac{1}{360000}} = \frac{1}{600} \, \text{cm} \]

### Step 7: Calculate the Diameter
The diameter \( d \) of the wire is twice the radius:
\[ d = 2r = 2 \times \frac{1}{600} = \frac{2}{600} = \frac{1}{300} \, \text{cm} \]

### Conclusion
Thus, the diameter of the wire is:
\[ \frac{1}{300} \, \text{cm} \]

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