A cone of radius 8 cm and height 12 cm is divided into two parts by a plane through the mid-point of its axis parallel to its base. Find the ratio of the volumes of two parts
Solution:
Given, cone of radius 8 cm and height 12 cm
The cone is divided into two parts by a plane through the midpoint of its axis parallel to its base.
We have to find the ratio of the volumes of two parts.
Given, height of the cone, h = 12 cm
Radius of the cone, R = 8 cm
On cutting the cone through the midpoint parallel to the base, we get frustum of a cone and a smaller cone.
Considering similar triangles OMN and OPD,
The corresponding sides are proportional.
MN/PD = OM/OP
From the figure,
MN = radius of cone = 8 cm
PD = radius of smaller cone = r cm
OM = height of the cone = 12 cm
OP = height of the smaller cone = 12/2 = 6 cm (since cone is cut at the midpoint)
So, 8/r = 12/6
8/r = 2
2r = 8
r = 4 cm
Volume of the frustum of a cone = πh/3 [R² + r² + Rr]
= π(6/3)[8² + 4² + 8(4)]
= π(2)[64 + 16 + 32]
= 2π(112)
= 224π cm³
Volume of the smaller cone = (1/3)πr²h
= π(1/3)(4)²(6)
= 2π(16)
= 32π cm³
Volume of smaller cone / volume of frustum of a cone = 32π / 224π
= 32/224
= 1/7
Therefore, the ratio of the volumes of two parts is 1:7
✦ Try This: A cone of radius 6 cm and height 9 cm is divided into two parts by a plane through the mid-point of its axis parallel to its base. Find the ratio of the volumes of two parts.
☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 13
NCERT Exemplar Class 10 Maths Exercise 12.3 Problem 4
A cone of radius 8 cm and height 12 cm is divided into two parts by a plane through the mid-point of its axis parallel to its base. Find the ratio of the volumes of two parts
Summary:
A cone of radius 8 cm and height 12 cm is divided into two parts by a plane through the mid-point of its axis parallel to its base. The ratio of the volumes of two parts is 1:7
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