Q 41.
A rod of radius 7 cm and 150 cm in length is melted and converted into a wire of 6 meter length of uniform thickness. The diameter of the wire is
01)
3.5 cm
02)
7 cm
03)
1.75 cm
04)
None of these

To solve this problem, we need to use the principle of volume conservation. When the rod is melted and converted into a wire, the volume of the rod remains equal to the volume of the wire. Let's break this down.

### Step 1: Volume of the Rod
The rod is cylindrical with:
- Radius = 7 cm
- Length = 150 cm

The volume \( V_{rod} \) of a cylinder is given by the formula:
\[
V_{rod} = \pi r^2 h
\]
where:
- \( r \) is the radius of the rod,
- \( h \) is the height (or length) of the rod.

Substituting the values:
\[
V_{rod} = \pi (7)^2 \times 150 = \pi \times 49 \times 150 = 7350 \pi \, \text{cm}^3
\]

### Step 2: Volume of the Wire
The wire is also cylindrical with:
- Length = 6 meters = 600 cm (since 1 meter = 100 cm),
- Let the radius of the wire be \( r_w \).

The volume \( V_{wire} \) of the wire is:
\[
V_{wire} = \pi r_w^2 \times 600
\]

### Step 3: Equating the Volumes
Since the volumes are equal, we set the two expressions equal to each other:
\[
7350 \pi = \pi r_w^2 \times 600
\]
Canceling \( \pi \) from both sides:
\[
7350 = 600 r_w^2
\]
Now, solve for \( r_w^2 \):
\[
r_w^2 = \frac{7350}{600} = 12.25
\]
Taking the square root of both sides:
\[
r_w = \sqrt{12.25} = 3.5 \, \text{cm}
\]

### Step 4: Diameter of the Wire
The diameter of the wire is twice the radius:
\[
\text{Diameter} = 2 \times 3.5 = 7 \, \text{cm}
\]

### Conclusion
The diameter of the wire is 7 cm. Hence, the correct answer is:
**02) 7 cm**

Would you like more details or have any questions?

Here are five related questions to further explore:
1. What is the volume of a sphere with the same radius as the rod?
2. How would the solution change if the length of the rod was halved?
3. What if the wire was melted and converted back into a new rod of different dimensions?
4. How is the surface area affected during the melting process?
5. What would happen if the wire was stretched instead of melted?

**Tip:** In volume conversion problems, always check if the units are consistent throughout the problem.

A rod of radius 7 cm and 150 cm in length is melted and converted into a wire of 6 meter length of uniform thickness. The diameter of the wire is 01) 3.5 cm 02) 7 cm 03) 1.75 cm 04) None of these

Solve a problem involving a rod of 7 cm radius and 150 cm length melted into a wire of 6 meters. This detailed solution explores the principle of volume conservation and uses cylindrical geometry formulas. Suitable for Grades 9-12 students.

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