Two solid cones A and B are placed in a cylindrical tube as shown in the figure below. The ratio of their capacities is 2:1. Find the volume of the remaining portion of the cylinder.
396 cm3
The diameter of the cylinder is 6 cm.
Thus, the radius of the cylinder is 3 cm.
The capacities of A and B are in the ratio 2:1
Radius of cone A
= Radius of cone B = 3 cm
Let the heights of cone A and B be ha and hb respectively.
Assuming the thickness to be negligible, we can say that the volumes of cones A and B are in the ratio 2:1
⇒13π×32×ha13π×32×hb=21
⇒hahb=21
⇒ha=2hb
Also,
ha+hb=21
⇒2hb+hb=21
⇒3hb=21
⇒hb=7 cm
ha=2×hb=2×7=14 cm
Volume of the remaining part of the cylinder
= Volume of the cylinder - Volume of cone A - Volume of cone B
=πr2h−13πr2ha−13πr2hb
=(π×32×21)−(π3×32×14)−(π3×32×7)
=π×9×21−π3×9×(14+7)
=(1−13)×π×9×21
=23×227×9×21
=396 cm3