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.
Answers
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
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