Two solid cones A and B are placed in a cylindrical tube as shown in the Fig.12.9. 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
Solution:
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
✦ Try This: A solid iron cuboidal block of dimensions 4.4 m x 2.6m x lm is recast into a hollow cylindrical pipe of internal radius 30 cm and thickness 5 cm. Find the length of the pipe.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 13
NCERT Exemplar Class 10 Maths Exercise 12.3 Problem 8
Two solid cones A and B are placed in a cylindrical tube as shown in the Fig.12.9. 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
Summary:
Two solid cones A and B are placed in a cylindrical tube as shown in the Fig.12.9. The ratio of their capacities are 2:1. The heights of cones are 14 cm and 7cm. The capacities of cones are 132 cm³ and 66 cm³. Also, the volume of the remaining portion of the cylinder is 396 cm³
☛ Related Questions:
- An ice cream cone full of ice cream having radius 5 cm and height 10 cm as shown in the Fig.12.10. C . . . .
- Marbles of diameter 1.4 cm are dropped into a cylindrical beaker of diameter 7 cm containing some wa . . . .
- How many spherical lead shots each of diameter 4.2 cm can be obtained from a solid rectangular lead . . . .