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

Question

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

GMATinsight · Accepted Answer

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

ENGRTOMBA2018 · Answer

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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}

B is the correct answer.

arhumsid · Answer

Im sorry, i did not get  Radius of Sphere = (1/3)Height .
Could you please explain how can we arrive at above relationship?

Thanks!

GMATinsight · Answer

(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

chesstitans · Answer

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?

KarishmaB · Answer

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)

chesstitans · Answer

sure, even if the ath problem gives a very special scenario, I believe this question emphasizes too much on the 3D geometry.

GMATinsight · Answer

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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Crytiocanalyst · Answer

Radius of the circle = tan 30 * radius of the cone

height of the cone = 12^2 -6^2 =144 -36 =108 =6*rt3

angle inside the triangle = height of the cone / radius of the cone = 60

Since the triangles are equally divided therefore we can use similar triangles and divide the triangle

THerefore IMO 6/rt 3 = 2* rt3 B

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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

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