Question

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.

Answer 1

Given, two solid cones A and B are placed in a cylindrical tube.

Ratio of the volume of cones A and B are 2:1

We have to find the height and capacities of cones and the volume of the remaining portion of the cylinder.

Volume of cone = (1/3)πr²h

From the figure,

Radius of cone A = 6/2 = 3 cm

Radius of cone B = 6/2 = 3 cm

Let the height of cone A be h₁

So, height of cone B = 21 - h₁

Volume of cone A = (1/3)π(3)²h₁

= 3πh₁ cm³

Volume of cone B = (1/3)π(3)²(21 - h₁)

= 3π(21 - h₁)

= 63π - 3πh₁ cm³

Given, volume of cone A : volume of cone B = 2:1

So, volume of cone A = 2 × volume of cone B  .........(1)

3πh₁ = 2(63π - 3πh₁)

 3πh₁ = 126π - 6πh₁

3πh₁ + 6πh₁ = 126π

9πh₁ = 126π

9h₁ = 126

h₁ = 126/9

h₁ = 42/3 cm

h₁ = 14 cm

Height of cone B = 21 - 14 = 7 cm

Volume of cone A = 3(22/7)(42/3)

= (22)(6)

= 132 cm³

From (1),

Volume of cone B = Volume of cone A/2

= 132/2

= 66 cm³

Volume of cylinder = πr²h

Given, r = 6/2 = 3 cm

h = 21 cm

Volume of cylinder = (22/7)(3)²(21)

= (22)(9)(3)

= 22(27)

= 594 cm³

Volume of the remaining portion = volume of cylinder - volume of cone A - volume of cone B

= 594 - 132 - 66

= 594 - 198

= 396 cm³

Therefore, the volume of the remaining portion is 396 cm³.

Answer 2

Let volume of cone A be 2 V and volume of cone B be V. Again, let height of the cone A = h cm, then height of cone B = (21 – h₁ ) cm