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What is the radius of the largest sphere that can fit inside a right c

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What is the radius of the largest sphere that can fit inside a right c  [#permalink]

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New post Updated on: 02 Oct 2015, 11:38
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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. \(6\sqrt{3} m\)
Spoiler: OA

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Originally posted by arhumsid on 02 Oct 2015, 10:33.
Last edited by ENGRTOMBA2018 on 02 Oct 2015, 11:38, edited 1 time in total.
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Re: What is the radius of the largest sphere that can fit inside a right c  [#permalink]

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New post 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. \(6\sqrt{3} m\)


Follow posting guidelines (link in my signatures).

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 \(OD = 2\sqrt{3}\)

B is the correct answer.
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10-02-15 1-45-25 PM.jpg
10-02-15 1-45-25 PM.jpg [ 13.71 KiB | Viewed 31077 times ]

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Re: What is the radius of the largest sphere that can fit inside a right c  [#permalink]

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New post 02 Oct 2015, 20:56
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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. \(6\sqrt{3} m\)


Height of Cone = \(\sqrt{(12^2) - (6^2)} = \sqrt{(144) - (36)} = \sqrt{108}= 6\sqrt{3}\)

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  [#permalink]

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New post 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  [#permalink]

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New post 02 Oct 2015, 23:46
2
arhumsid wrote:
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!


(Both) Slant Height = 12
Radius = 6
i.e. base Diameter = 12

i.e. In 2 Dimension the Cone is like an Equilateral Triangle in which we can derive the radius as follows
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Re: What is the radius of the largest sphere that can fit inside a right c  [#permalink]

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New post 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?
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Re: What is the radius of the largest sphere that can fit inside a right c  [#permalink]

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New post 10 Jan 2018, 10:35
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. \(6\sqrt{3} m\)


You can use the properties of equilateral triangle, if you remember them.

Imagine cutting a vertical cross section of the cone with a sphere inscribed in it. The front will look like the figure above (https://gmatclub.com/forum/what-is-the- ... l#p1580927)

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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Re: What is the radius of the largest sphere that can fit inside a right c  [#permalink]

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New post 10 Jan 2018, 11:58
VeritasPrepKarishma wrote:
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. \(6\sqrt{3} m\)


You can use the properties of equilateral triangle, if you remember them.

Imagine cutting a vertical cross section of the cone with a sphere inscribed in it. The front will look like the figure above (https://gmatclub.com/forum/what-is-the- ... l#p1580927)

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.
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Re: What is the radius of the largest sphere that can fit inside a right c  [#permalink]

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New post 11 Jan 2018, 02:22
1
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?


Hi chesstitans

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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Re: What is the radius of the largest sphere that can fit inside a right c &nbs [#permalink] 11 Jan 2018, 02:22
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