Explanation:

- We start by calculating the volume of the metallic sphere using the formula \( \frac{4}{3} \pi r^3 \), where \( r \) is the radius.

- Given that the radius of the sphere is 12 cm, the volume becomes \( \frac{4}{3} \pi (12)^3 = \frac{7236}{3} \pi \) cm³.

- This volume will be equal to the volume of the water displaced in the cylindrical vessel.

- For the cylindrical vessel, use the formula for the volume \( \pi r^2 h \), where \( r \) is the radius of the base of the cylinder and \( h \) is the height (or increase in water level).

- The radius of the cylinder is \( \frac{32}{2} = 16 \) cm.

- Solving \( \pi (16)^2 h = \frac{7236}{3} \pi \), we find \( h = \frac{7236}{768} \).

- Simplifying gives \( h = 9.4375 \) cm.

- Thus, Option 2: 9 cm is present but the closest estimate.