Final answer:
To find the radius of each ball dropped into the vessel, we need to calculate the volume of the water and set up an equation using the volume of the balls. By applying the appropriate formulas and solving the equation, we find that the radius of each ball is (3/4)^(1/3) cm.
Step-by-step explanation:
To find the radius of each ball, we need to first calculate the initial volume of the water in the vessel. The volume of a cone can be calculated using the formula V = (1/3)πr^2h, where r is the radius and h is the height. Substituting the given values, we have V = (1/3)π(5^2)(8) = 400π cm^3.
Since one-fourth of the water flows out of the vessel when the balls are added, the remaining volume of water is three-fourths of the initial volume. So, the volume of water remaining is (3/4)(400π) = 300π cm^3.
The volume of each spherical ball can be calculated using the formula V = (4/3)πr^3, where r is the radius. We can set up an equation: 100(4/3)πr^3 = 300π. Solving for r, we get r = (3/4)^(1/3) cm.