Final answer:
The fraction of water that overflows from the conical vessel can be calculated by finding the difference between the volume of the cone and the volume of the sphere, divided by the volume of the cone. The correct option is not mention.
Step-by-step explanation:
To find the fraction of water that overflows, we first need to find the volume of water in the conical vessel. The volume of a cone can be calculated using the formula V = (1/3) * π * r² * h, where r is the radius and h is the height of the cone. In this case, the radius is 6 cm and the height is 8 cm, so the volume of water is (1/3) * 3.142 * (6 cm)² * 8 cm = 301.44 cm³.
The sphere is lowered into the water until it touches the sides and is just immersed. When the sphere is just immersed, its volume is equal to the volume of water in the cone. Let's assume the radius of the sphere is R cm.
The volume of the sphere can be calculated using the formula V = (4/3) * π * R³. Setting this equal to the volume of water, we have (4/3) * 3.142 * R³ = 301.44 cm³. Solving for R, we find that R ≈ 5.25 cm.
The fraction of water that overflows can be calculated as the difference between the volume of the cone and the volume of the sphere, divided by the volume of the cone. So, the fraction of water that overflows is (301.44 cm³ - (4/3) * 3.142 * (5.25 cm)³) / 301.44 cm³.
The correct option is not mention.