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. 21 cm

Explanation

Let volume of cone A be 2 V and volume of cone B be V .Again, let height of the cone` A= h_(1)`
cm, then height of cone `B= (21 - h_(1)) cm `

Given, diameter of the cone = 6 cm
`therefore ` Radius of the cone `= (6)/(2) = 3 cm`
Now, volume of the cone , `A = 2V = (1)/(3) pir^(2)h = (1)/(3)pi(3)^(2)h_(1)`
`rArr" " V= (1)/(6) pi9h_(1) = (3)/(2) pi(3)^(2)h_(1)`
and volume of the cone , `B = V = (1)/(3)^(2) (21 - h_(1)) = 3pi (21 - h_(1))`
From Eqs. (i ) and (ii) ,
`" "(3)/(2) h_(1) pi = 3 pi (21-h_(1))`
`rArr" " h_(1) = 2 (21-h_(1))`
`rArr" "3h_(1) = 42`
`rArr" " h_(1) = (42)/(3) = 14 cm`
` therefore` " "Height of cone , `B = 21 - h_(1)= 21 - 14 = 7 cm `
Now, " " volume of the cone , `A = 3 xx 14xx (22)/(7) = 132 cm^(3)" "["Using Eq. (i)"]`
and `" ""Volume of the cylinder" = pir^(2)h = (22)/(7) xx9xx7 =66 cm^(3) " "["Using Eq.(ii)"]`
Now, `" ""volume of the cylinder" = pir^(2)h = (22)/(7) (3)^(2) xx 21 = 594 cm^(3)`
`therefore` Required volume of the remaining portion = volume of the cylinder - (Volume of cone A + Volume of cone B)
`= 594 - (132 +66)`
`= 396 cm^(3)`