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Re: What is the radius of the largest sphere that can fit inside a right c
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02 Oct 2015, 11:02
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arhumsid wrote:
What is the radius of the largest sphere that can fit inside a right circular cone of base radius \(6 m\) and slant height \(12 m\)?
A. \(2 m\) B. \(2\sqrt{3} m\) C. \(4\sqrt{3} m\) D. \(6 m\) E. \(6\sqrt{3} m\)
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This problem is easy to solve when you convert the seemingly "3D" question into a representative 2D problem of maxium radius of the circle inside a triangle. The radius will be maximum when the triangle is equilateral triangle and the circle will be the incircle of the triangle.
As shown in the attached figure, OE=OD=radius of the incircle and triangle ABC is an equialteral triangle with OD \(\perp\) BC. Thus the triangle ODC is a 30-60-90 triangle and as BD=DC=6, \(\frac{OD}{CD} = \frac{1}{\sqrt{3}}\), giving you \(OD = 2\sqrt{3}\)
Radius of Sphere = (1/3)Height = \((1/3)6\sqrt{3}\) = \(2\sqrt{3}\)
Answer: option B
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Re: What is the radius of the largest sphere that can fit inside a right c
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02 Oct 2015, 23:14
GMATinsight wrote:
arhumsid wrote:
Radius of Sphere = (1/3)Height = \((1/3)6\sqrt{3}\) = \(2\sqrt{3}\)
Im sorry, i did not get Radius of Sphere = (1/3)Height. Could you please explain how can we arrive at above relationship?
Thanks!
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Re: What is the radius of the largest sphere that can fit inside a right c
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10 Jan 2018, 03:27
GMATinsight, Bunuel it seems that this math problem is beyond the scope of GMAT geometry because the problem requires outside knowledge. Do you agree with me?
The radius of the cone is 6 so the base of the triangle is 12. The slant height of the cone is 12 so both other sides are also 12. Hence it is an equilateral triangle. In an equilateral triangle, the inradius is \(s*\sqrt{3}/6 = 12*\sqrt{3}/6 = 2*\sqrt{3}\)
Answer (B)
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The radius of the cone is 6 so the base of the triangle is 12. The slant height of the cone is 12 so both other sides are also 12. Hence it is an equilateral triangle. In an equilateral triangle, the inradius is \(s*\sqrt{3}/6 = 12*\sqrt{3}/6 = 2*\sqrt{3}\)
Answer (B)
sure, even if the ath problem gives a very special scenario, I believe this question emphasizes too much on the 3D geometry.
Re: What is the radius of the largest sphere that can fit inside a right c
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11 Jan 2018, 02:22
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chesstitans wrote:
GMATinsight, Bunuel it seems that this math problem is beyond the scope of GMAT geometry because the problem requires outside knowledge. Do you agree with me?
This question may be solved by the properties that GMAT expects from tests takers hence I wouldn't discount the possibility of such a question to appear in gMAT however I will assign a very low probability for such question to appear in exam as it's too complex and also uses a solid rarely used by GMAT I.e. Cone
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16 Jun 2019, 23:35
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53% (02:15) correct 
