Lesson A cylindrical glass filled with water and then tilted to a horizon

A cylindrical glass filled with water and then tilted to a horizon

Problem 1

   

   

In Figure, rectangle ABCD represents the tilted glass (or, more precisely, 
its axial section).
Blue line ED represents the horizontal surface of liquid in the glass.


Triangle AED is a right-angled triangle with angles AED and ADE of 45°,
so triangle AED is the isosceles right-angled triangle.


Thus,  AE = AD = 6 cm.


Red line EF in Figure is the trace of a plane parallel to the glass edge.

This plane defines a cylinder AEFD.

The surface (the plane) ED cuts cylinder AEFD in two parts, AED and EFD.

Due to symmetry, the parts AED and EFD have equal volumes, 
each of which is half the volume of the cylinder AEFD.


Cylinder AEFD has the radius of 3 cm and the height AE = AD = 6 cm, 
so the volume of cylinder AEFD is

     =  = .
   
   

    


    Figure. The tilted glass with water ABCD




Therefore, the volume of liquid poured from the glass is 

     =  =  = 84.82293 cm^3.


The volume of the water remaining in the glass is

     - 84.82293 =  - 84.82293 = 311.01741 - 84.82293 = 226.1945,


or about  226.2 cm^3  (or 226.2 mL).


ANSWER.  The volume of water remaining in the glass is about 226.2 cm^3 (or 2226.2 mL).