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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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