Two solid cones A and B are placed in a cylindrical tube as shown in the figure. The ratio of their capacities is 2 : 1. Find the heights and capacities of cones. Also, find the volume of the remaining portion of the cylinder.


The diagram is:



Diameter of Cylinder = 6 cm


Radius of cylinder = r = 3 cm


[As radius = diameter/2]


As both cones have equal radius


Radius of cone A = radius of cone B = r = 3 cm


Let the height of cone A be h1 and Cone B be h2


Given,


Ratio of volume of cones is 2 : 1


i.e.



As volume of cone =


where r = base radius and h = height



h1 = 2h2


Now,


Total height of cylinder is 21 cm


h1 + h2 = 21


2h2 + h2 = 21


3h2 = 21


h2 = 7 cm


h1 = 2h2 = 2(7) = 14 cm






We know,


Volume of cylinder = πr2h ,


where r = radius and h = height


Volume of given cylinder = π(3)2(21)



Volume of remaining solid = (Volume of cylinder) – (volume of cone A) – (volume of cone B)


= 594 - 132 - 66 = 396 cm3


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