A cone of radius 4 cm is divided into two parts by drawing a plane through the midpoint of its axis and parallel to its base. Compare the volumes of the two parts
Solution:
Given, a cone of radius 4 cm
Cone is divided into two parts by drawing a plane through the midpoint of its axis and parallel to its base.
We have to compare the volumes of the two parts.
Let h be the height of the cone.
On cutting the cone through the midpoint of its axis and parallel to the base, we obtain frustum of a cone and a smaller cone
Considering the similar triangles OAB and DCB,
The corresponding sides are proportional.
So, OA/DC = OB/DB
From the figure,
OA = radius = 4 cm
DC = radius of smaller cone = r
OB = height of the cone = h
DB = height of smaller cone = h/2
Now, 4/r = h/(h/2)
4/r = 2
2r = 4
r = 4/2
r =2 cm
Volume of the frustum of a cone = πh/3 [R² + r² + Rr]
= (π/3)(h/2)[(4)² + (2)² + 4(2)]
= (π/3)(h/2)(16 + 4 + 8)
= (π/3)(h/2)(28)
Volume of smaller cone = (1/3)πr²h
= (1/3)π(2)²(h/2)
= (4π/3)(h/2)
Volume of smaller cone/volume of the frustum of a cone = (4π/3)(h/2) / (π/3)(h/2)(28)
= 4/28
= 1/7
Therefore, the ratio of the volume of smaller cone to the volume of the frustum of a cone is 1:7
✦ Try This: The height of a cone is 10 cm. The cone is divided into two parts using a plane parallel to its base at the middle of its height. Find the ratio of the volume of the two parts.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 13
NCERT Exemplar Class 10 Maths Exercise 12.3 Sample Problem 4
A cone of radius 4 cm is divided into two parts by drawing a plane through the midpoint of its axis and parallel to its base. Compare the volumes of the two parts
Summary:
A cone of radius 4 cm is divided into two parts by drawing a plane through the midpoint of its axis and parallel to its base. The volume of the two parts is in the ratio 1: 7
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