Question

Two solid cones A and B are placed in a cylindrical tube such that their points touch each other inside the tube. The height of the cylinder is 21 cm, and the base diameter is 6cm. The ratio of their capacities is 2:1. Find the heights and capacities of the cones. Also find the volume of the remaining portion of the cylinder.

Answer 1

Let height of the cone 1 be 'h' cm and the height of the cone 2 be (21 cm - h) .
Volume of cone 1/Volume of cone 1 = 2/1
⇒ {1/3πr²h}/{1/3πr²(21 cm - h)} = 2/1
⇒ 42 cm - 2h = h
⇒ 3h = 42 cm
⇒ h = 42/3
⇒ h = 14 cm
Height of the 1 st cone is 14 cm and the height of the 2nd cone is 21 - 14 = 7 cm.
Now,
Volume of cone 1 = 1/3*22/7*3*3*14
= 132 cm³ or Capacity = 0.132 litres
Volume of cone 2 = 1/3*22/7*3*3*7
= 66 cm³ or capacity = 0.066 litres
Volume of cylinder = πr²h
= 22/7*3*3*21
= 594 cm³
Remaining volume = 594 - 198
= 396 cm³