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GMAT 1: 630 Q44 V32 GMAT 2: 680 Q47 V35 What is the radius of the largest sphere that can fit inside a right c  [#permalink]

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10 00:00

Difficulty:   75% (hard)

Question Stats: 53% (02:15) correct 47% (02:20) wrong based on 90 sessions

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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. $$6\sqrt{3} m$$

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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.
Renamed the topic
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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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2
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.
Attachments 10-02-15 1-45-25 PM.jpg [ 13.71 KiB | Viewed 43273 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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2
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 = $$(1/3)6\sqrt{3}$$ = $$2\sqrt{3}$$

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GMAT 1: 630 Q44 V32 GMAT 2: 680 Q47 V35 Re: What is the radius of the largest sphere that can fit inside a right c  [#permalink]

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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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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
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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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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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 $$s*\sqrt{3}/6 = 12*\sqrt{3}/6 = 2*\sqrt{3}$$

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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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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 $$s*\sqrt{3}/6 = 12*\sqrt{3}/6 = 2*\sqrt{3}$$

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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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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