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.
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 = (21–h1)cm.
Given , diameter of the cone = 6 cm
∴Radius of the cone=62=3 cmNow,volume of the cone,A⇒2V=13πr2h=13π(3)2h1⇒V=16π9h1=32h1π ...(i)
And volume of the cone B
⇒V=13π(3)2(21−h1)=3π(21−h1) ...(ii)
From Eqs. (i) and (ii)
⇒V=V
⇒32h1π=3π(21−h1)⇒h1=2(21−h1)⇒3h1=42⇒h1=423=14 cm
∴Height of cone,A=h1=14 cmHeight of cone,B=(21–h1)=21–14=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