SOLUTION: A glass of water 10 cm high, with a circumference of 6π cm, is 1/4 full. A metal sphere of radius 3 cm is dropped into the glass. (It just fits). What is the increase in the heigh

The radii of the glass and the sphere both are the same, 3 cm.

The level of water in the glass originally was 2.5 cm (0.25 of 10 cm).


Hence, the volume of water in the glass initially was   =  =  =  cubic cm.


The volume of the sphere is    =  =  =  cubic cm.


Let assume for a minute that the water level after submerging sphere is at the top of the sphere,

and evaluate the volume of this cylinder. This volume is   V =  =  = .


Now, since   +  =  +  =   is  GREATER THAN  V = ,

the volume of water in the glass after submerging sphere is OVER the top of the sphere.


In other words, the sphere is FULLY SUBMERGED into water.


Now it is easy to evaluate the water level rise:  it equals to the volume of the sphere divided by the area of the bottom of the glass


      delta_H =  =  = 4 centimeters.    ANSWER.


Notice that  2.5 cm + 4 cm = 6.5 cm  is greater than the sphere diameter of 6 cm.

It CONFIRMS that the sphere is fully submerged.