What is the length of the uniform wire of diameter 0.4 cm that can be drawn from a solid sphere of radius 9 cm ?

Question

Accepted Answer

To determine the length of the uniform wire that can be drawn from a solid sphere, we need to use the principle of conservation of volume. The volume of the sphere will be equal to the volume of the wire.

1. **Volume of the Sphere:**
   The formula for the volume of a sphere is given by:
   $$
   V_{\text{sphere}} = \frac{4}{3} \pi r^3
   $$
   where $ r $ is the radius of the sphere. Given that the radius of the sphere is 9 cm, we can calculate:
   $$
   V_{\text{sphere}} = \frac{4}{3} \pi (9)^3 = \frac{4}{3} \pi \times 729 = 972 \pi \, \text{cm}^3
   $$

2. **Volume of the Wire:**
   The wire is cylindrical in shape, and the volume of a cylinder is given by:
   $$
   V_{\text{cylinder}} = \pi r^2 h
   $$
   where $ r $ is the radius of the cylinder and $ h $ is the height (or length) of the cylinder. The diameter of the wire is given as 0.4 cm, so the radius is 0.2 cm.

3. **Equating the Volumes:**
   Since the volume of the sphere is equal to the volume of the wire, we have:
   $$
   972 \pi = \pi (0.2)^2 h
   $$
   Simplifying, we get:
   $$
   972 = 0.04 h
   $$
   Solving for $ h $:
   $$
   h = \frac{972}{0.04} = 24300 \, \text{cm}
   $$
   Converting this to meters (since 1 m = 100 cm):
   $$
   h = \frac{24300}{100} = 243 \, \text{m}
   $$

Therefore, the length of the wire is 243 meters, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2 (240 m):** This option is incorrect because the calculation clearly shows that the length of the wire is 243 meters, not 240 meters. The discrepancy arises from incorrect division or conversion.

- **Option 3 (60.75 m):** This option is incorrect as it significantly underestimates the length of the wire. The calculation based on the volume of the sphere and the wire's dimensions does not support this value.

- **Option 4 (60 m):** Similar to Option 3, this option is incorrect because it does not match the calculated length of 243 meters. It is likely based on a misunderstanding or miscalculation of the volume relationship.

In conclusion, the correct answer is Option 1, 243 meters, as supported by the calculations based on the conservation of volume principle.