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Question 8
Two solid cones A and B are placed in a cylindrical tube. 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 2V 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,volume of the cone,A2V=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)
V=V
32h1π=3π(21h1)h1=2(21h1)3h1=42h1=423=14 cm

Height of cone,A=h1=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
= (Vol. of the cylinder) – (Vol. of cone A + Vol. of cone B)
=594 cm3(132 cm3+66 cm3)
=396 cm3


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