Explanation:

- First, calculate the volume of the metallic cylinder using the formula:

$$V=\pi \times r^2\times h=\pi \times 3^2\times 120$$

$$V=1080\pi {\text{cm}}^3$$

- After a 10% metal loss, the remaining volume is:

$$0.9\times 1080\pi =972\pi {\text{cm}}^3$$

- Now, this volume is recast into a sphere. Use the sphere volume formula:

$$\frac{4}{3}\pi r^3=972\pi$$

- Solving for $r$:

$$\frac{4}{3}{r}^3=972$$

$$r^3=729$$

$$r=9\text{cm}$$

- Given options: 1) 9, 2) 8, 3) 13.5, 4) 12

- Correct Answer: Option 1 - 9