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Question 8
Two solid cones A and B are placed in a cylindrical tube as shown in the Fig.12.9. The ratio of their capacities are 2:1. Find the heights and capacities of cones. Also, find the volume of the remaining portion of the cylinder.

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Solution

Let volume of cone A be 2 V and volume of cone B be V. Again, let height of the cone A=h1cm the height of cone B=(21h1)cm.

Given , diameter of the cone = 6 cm
Radius of the cone=62=3 cmNow,volumeofthecone,A=2v=13πr2h=13π(3)2h1V=16π9h1=32h1π ...................(i)
And volume of the cone, B=V=13π(3)2(21h1)=3π(21h1)........(ii)
From Eqs. (i) and (ii)
32h1π=3π(21h1)h1=2(21h1)3h1=42h1=423=14 cmHeight of cone,B=21h1=2114=7 cm
Now, Volume of the cone =A=3×14×227=132 cm3 [Using Eq.(i)]
And Volume of the cone , B=V=13×227×9×7=66 cm3 [UsingEq.(ii)]
Now, volume of the cylinder =πr2h=227(3)2×21=594 cm3
Required volume of the remaining portion = Volume of the cylinder – ( Volume of cone A + Volume of cone B)
= 594 – ( 132 + 66)
=396 cm3

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