Question

A cone is having its base radius 12 cm and height 20 cm. If the top of this cone is cut in to form of a small cone of base radius 3 cm is removed, then the remaining part of the solid cone becomes a frustum. Calculate the volume of the frustum.​

Answer 1

Given data : A cone is having its base radius 12 cm and height 20 cm. If the top of this cone is cut in to form of a small cone of base radius 3 cm is removed, then the remaining part of the solid cone becomes a frustum.

To find : Calculate the volume of the frustum.

Solution : Now according to given, and figure

For cone :

⟹ Radius ( r ) = QB = 12 cm

⟹ Height ( h ) = OQ = 20 cm

For small cone :

⟹ Radius ( r' ) = PD = 3 cm

⟹ Height ( h' ) = OP = ?

Let, CD parallel to AB, and OP perpendicular to CD and OQ perpendicular to AB,

Here, we know from ∆OPD and ∆ OQB

⟹ ∠ POD = ∠ QOB and ∠ OPD = ∠ OQB

By AA similarity theorem ∆ OQB ~ ∆ OPD

Now, we know, by similar triangle property,

Here, we know by ∆ OQB ~ ∆ OPD

⟹ QB/OQ = PD/OP

⟹ 12/20 = 3/OP

⟹ 12 * OP = 3 * 20

⟹ 12 * OP = 60

⟹ OP = 60/12

⟹ OP = 5 cm

Hence, height of small cone is 5 cm.

Now, a small cone is removed from the cone hence, remaining part of the solid cone becomes a frustum.

Hence, height of frustum is PQ,

Let , height of frustum be H,

• H = OQ - OP = 20 - 5 = 15 cm

Now,

⟹ Volume of frustum = (1/3) * π * H * [r² + r'² + (r * r')]

⟹ Volume of frustum = (1/3) * 22/7 * 15 * [12² + 3² + (12 * 3)]

⟹ Volume of frustum = 1/3 * 330/7 * [144 + 9 + 36]

⟹ Volume of frustum = 110/7 * [153 + 36]

⟹ Volume of frustum = 110/7 * 189

⟹ Volume of frustum = 110 * 27

⟹ Volume of frustum = 2970 cm³

Answer : Volume of the frustum is 2970 cm³.

Learn more :

In a right triangle ABC, Right angled at C in which AB=13cm,BC=5cm, determine the value of cos²B+sin²A

https://brainly.in/question/38873876?