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# A rectangular solid, x by y by z, is inscribed in a sphere, so that al

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Math Expert
Joined: 02 Sep 2009
Posts: 55803
A rectangular solid, x by y by z, is inscribed in a sphere, so that al  [#permalink]

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24 Dec 2018, 09:33
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Difficulty:

45% (medium)

Question Stats:

63% (02:27) correct 37% (02:49) wrong based on 60 sessions

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Fresh GMAT Club Tests' Question:

A rectangular solid, x by y by z, is inscribed in a sphere, so that all eight of its vertices are on the sphere. If x, y and z are positive integers, and the radius of the sphere is $$\frac{\sqrt{14}}{2}$$, what is the volume of the rectangular solid?

A. 6
B. 11
C. 12
D. 18
E. 22

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Posts: 55803
Re: A rectangular solid, x by y by z, is inscribed in a sphere, so that al  [#permalink]

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24 Dec 2018, 09:33
Bunuel wrote:

Fresh GMAT Club Tests' Question:

A rectangular solid, x by y by z, is inscribed in a sphere, so that all eight of its vertices are on the sphere. If x, y and z are positive integers, and the radius of the sphere is $$\frac{\sqrt{14}}{2}$$, what is the volume of the rectangular solid?

A. 6
B. 11
C. 12
D. 18
E. 22

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Math Expert
Joined: 02 Sep 2009
Posts: 55803
Re: A rectangular solid, x by y by z, is inscribed in a sphere, so that al  [#permalink]

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24 Dec 2018, 09:34
Bunuel wrote:

Fresh GMAT Club Tests' Question:

A rectangular solid, x by y by z, is inscribed in a sphere, so that all eight of its vertices are on the sphere. If x, y and z are positive integers, and the radius of the sphere is $$\frac{\sqrt{14}}{2}$$, what is the volume of the rectangular solid?

A. 6
B. 11
C. 12
D. 18
E. 22

_________________
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Location: India
Concentration: Finance, Marketing
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Re: A rectangular solid, x by y by z, is inscribed in a sphere, so that al  [#permalink]

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24 Dec 2018, 09:54
Given radius of the sphere = $$\sqrt{14}/2$$
Diameter = $$\sqrt{14}$$
Also, diameter of the sphere is the diagonal of the rectangle.
Diagonal of the rectangle = $$\sqrt{x^2+y^2+y^2}$$ = $$\sqrt{14}$$
$$x^2+y^2+y^2$$ = 14.
Since, x, y, and z are all positive integers.
Only possible value is 1,2 and 3.
As $$1^2+2^2+3^2 = 14$$
Hence the volume = 1*2*3 = 6.

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Re: A rectangular solid, x by y by z, is inscribed in a sphere, so that al  [#permalink]

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25 Dec 2018, 00:59
the longest diagonal of the rectagular solid would be equal to the diameter of the sphere

radius is sqrt14/2 so diameter would be sqrt 14

sqrt of ( l^2+b^2+h^2) = sqrt 14

since given that l,b,h are all integers so value of l,b,h = 1,2,3 whose square 1^2+2^2+3^2 = 14
so area would be 1*2*3= 6 IMO A

Bunuel wrote:

Fresh GMAT Club Tests' Question:

A rectangular solid, x by y by z, is inscribed in a sphere, so that all eight of its vertices are on the sphere. If x, y and z are positive integers, and the radius of the sphere is $$\frac{\sqrt{14}}{2}$$, what is the volume of the rectangular solid?

A. 6
B. 11
C. 12
D. 18
E. 22

_________________
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Re: A rectangular solid, x by y by z, is inscribed in a sphere, so that al   [#permalink] 25 Dec 2018, 00:59
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