Two balls of different material, but the same radius, float on the surface of a pool filled with fresh water of density 1,000 kg/m3. If ball A has 75% of its volume above the surface of the water, and ball B has 25% of its volume above the water, what is the ratio of the mass of ball A to that of B?

To solve this problem, we need to understand the principle of buoyancy, which states that the buoyant force (the force that makes the balls float) is equal to the weight of the fluid displaced by the object.

1. Let's denote the volume of each ball as V. Since both balls have the same radius, their volumes are equal.

2. The volume of water displaced by each ball is equal to the volume of the ball that is submerged. For ball A, 25% of its volume is submerged, so the volume of water displaced is 0.25V. For ball B, 75% of its volume is submerged, so the volume of water displaced is 0.75V.

3. The weight of the water displaced by each ball is equal to the volume of water displaced multiplied by the density of water (1000 kg/m^3) and the acceleration due to gravity (approximately 9.8 m/s^2). So, the weight of the water displaced by ball A is 0.25V * 1000 kg/m^3 * 9.8 m/s^2, and the weight of the water displaced by ball B is 0.75V * 1000 kg/m^3 * 9.8 m/s^2.

4. According to the principle of buoyancy, the weight of the water displaced by each ball is equal to the weight of the ball. So, the weight of ball A is 0.25V * 1000 kg/m^3 * 9.8 m/s^2, and the weight of ball B is 0.75V * 1000 kg/m^3 * 9.8 m/s^2.

5. The mass of each ball is equal to its weight divided by the acceleration due to gravity. So, the mass of ball A is (0.25V * 1000 kg/m^3 * 9.8 m/s^2) / 9.8 m/s^2 = 0.25V * 1000 kg/m^3, and the mass of ball B is (0.75V * 1000 kg/m^3 * 9.8 m/s^2) / 9.8 m/s^2 = 0.75V * 1000 kg/m^3.

6. The ratio of the mass of ball A to that of ball B is (0.25V * 1000 kg/m^3) / (0.75V * 1000 kg/m^3) = 0.25 / 0.75 = 1/3.

So, the ratio of the mass of ball A to that of ball B is 1:3.

The ratio of the mass of ball A to that of ball B can be determined by understanding the principle of buoyancy. According to Archimedes' principle, the buoyant force on an object is equal to the weight of the fluid displaced by the object.

For an object floating in equilibrium, the buoyant force equals the weight of the object. Therefore, the weight of the water displaced by each ball equals the weight of the ball itself.

The volume of water displaced by each ball is proportional to the volume of the ball that is submerged. Therefore, ball A, with 25% of its volume submerged, displaces 25% of its volume in water, and ball B, with 75% of its volume submerged, displaces 75% of its volume in water.

The weight of the water displaced by each ball is the volume of the water displaced multiplied by the density of the water (1,000 kg/m^3) and the acceleration due to gravity (approximately 9.8 m/s^2).

Therefore, the weight (and hence the mass) of ball A is 25% of the weight (and mass) of ball B.

So, the ratio of the mass of ball A to that of ball B is 25:75 or 1:3.