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The radii of the frustum of a right circular cone are in the ratio 2 : 1. What is the ratio of the volume of the frustum of the cone to that of the whole cone?
- a)1 : 8
- b)1 : 4
- c)3 : 4
- d)7 : 8
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
The radii of the frustum of a right circular cone are in the ratio 2 :...
Shortcut:
Volume of small cone/Volume of large cone = (r/R)3
⇒ 1 - Volume of small cone/Volume of large cone = 1 - (1/2)3
⇒ (Volume of large cone - Volume of small cone)/Volume of large cone = 1 - 1/8
∴ Volume of frustum/Volume of large cone = 7/8
Volume of small cone/Volume of large cone = (r/R)3
⇒ 1 - Volume of small cone/Volume of large cone = 1 - (1/2)3
⇒ (Volume of large cone - Volume of small cone)/Volume of large cone = 1 - 1/8
∴ Volume of frustum/Volume of large cone = 7/8
Detailed Solution:
Let the larger radii and smaller radii be R and r cm respectively
For the cone, height = rtan α
⇒ Height of the frustum, h = Rtanα - rtanα = (R - r)tanα
Height of the full cone, H = Rtanα
⇒ Ratio = Volume of the frustum ∶ Volume of the cone = [1/3 × π h × (R2 + r2 + Rr)] ∶ [1/3 × π R2 H]
⇒ [(R - r) × tanα × (R2 + r2 + Rr)] ∶ [R2 × Rtanα]
⇒ [(R - r) × (R2 + r2 + Rr)] ∶ R3
⇒ R3 - r3 ∶ R3 = 23 - 13 ∶ 23 [∵ a3 - b3 = (a - b) × (a2 + b2 + ab)] and [R ∶ r = 2 ∶ 1]
⇒ 7 ∶ 8
∴ The required ratio = 7 : 8
Let the larger radii and smaller radii be R and r cm respectively
For the cone, height = rtan α
⇒ Height of the frustum, h = Rtanα - rtanα = (R - r)tanα
Height of the full cone, H = Rtanα
⇒ Ratio = Volume of the frustum ∶ Volume of the cone = [1/3 × π h × (R2 + r2 + Rr)] ∶ [1/3 × π R2 H]
⇒ [(R - r) × tanα × (R2 + r2 + Rr)] ∶ [R2 × Rtanα]
⇒ [(R - r) × (R2 + r2 + Rr)] ∶ R3
⇒ R3 - r3 ∶ R3 = 23 - 13 ∶ 23 [∵ a3 - b3 = (a - b) × (a2 + b2 + ab)] and [R ∶ r = 2 ∶ 1]
⇒ 7 ∶ 8
∴ The required ratio = 7 : 8
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The radii of the frustum of a right circular cone are in the ratio 2 : 1. What is the ratio of the volume of the frustum of the cone to that of the whole cone?a)1 : 8b)1 : 4c)3 : 4d)7 : 8Correct answer is option 'D'. Can you explain this answer?
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The radii of the frustum of a right circular cone are in the ratio 2 : 1. What is the ratio of the volume of the frustum of the cone to that of the whole cone?a)1 : 8b)1 : 4c)3 : 4d)7 : 8Correct answer is option 'D'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about The radii of the frustum of a right circular cone are in the ratio 2 : 1. What is the ratio of the volume of the frustum of the cone to that of the whole cone?a)1 : 8b)1 : 4c)3 : 4d)7 : 8Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The radii of the frustum of a right circular cone are in the ratio 2 : 1. What is the ratio of the volume of the frustum of the cone to that of the whole cone?a)1 : 8b)1 : 4c)3 : 4d)7 : 8Correct answer is option 'D'. Can you explain this answer?.
The radii of the frustum of a right circular cone are in the ratio 2 : 1. What is the ratio of the volume of the frustum of the cone to that of the whole cone?a)1 : 8b)1 : 4c)3 : 4d)7 : 8Correct answer is option 'D'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about The radii of the frustum of a right circular cone are in the ratio 2 : 1. What is the ratio of the volume of the frustum of the cone to that of the whole cone?a)1 : 8b)1 : 4c)3 : 4d)7 : 8Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The radii of the frustum of a right circular cone are in the ratio 2 : 1. What is the ratio of the volume of the frustum of the cone to that of the whole cone?a)1 : 8b)1 : 4c)3 : 4d)7 : 8Correct answer is option 'D'. Can you explain this answer?.
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The radii of the frustum of a right circular cone are in the ratio 2 : 1. What is the ratio of the volume of the frustum of the cone to that of the whole cone?a)1 : 8b)1 : 4c)3 : 4d)7 : 8Correct answer is option 'D'. Can you explain this answer?, a detailed solution for The radii of the frustum of a right circular cone are in the ratio 2 : 1. What is the ratio of the volume of the frustum of the cone to that of the whole cone?a)1 : 8b)1 : 4c)3 : 4d)7 : 8Correct answer is option 'D'. Can you explain this answer? has been provided alongside types of The radii of the frustum of a right circular cone are in the ratio 2 : 1. What is the ratio of the volume of the frustum of the cone to that of the whole cone?a)1 : 8b)1 : 4c)3 : 4d)7 : 8Correct answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
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