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

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New post 24 Dec 2018, 09:33
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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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Re: A rectangular solid, x by y by z, is inscribed in a sphere, so that al  [#permalink]

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

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

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New post 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.

A is the answer.
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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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New post 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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