CONCEPT:The question discusses and applies the concepts of Mensuration and Geometry basics


Let us assume ACB=x, therefore BAC=4x,
ABC=x (ABC is Isosceles with AC=AB (radii))

In triangle ABC,x+4x+x=180° , x= 30° ,
Thus, BAC=4*30=120°

Volume is proportional to Cross sectional area(Circle for the entire cylinder and a sector for the shaded part).
This cross sectional area is proportional to angle which is 360 for the complete cylinder and
120 for the sector that represents the cross section area of the triangle.

Hence, Volume proportion of shaded region to entire Cylinder
= Angle of the sector/Angle of the full circle
 =120/360 =1/3 (E)