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Hemisphere within a cube ! - experts help please

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Hemisphere within a cube ! - experts help please

by kaps786 » Tue Aug 30, 2011 1:00 am
The base of a hemisphere is inscribed in one face of a cube and the entire hemisphere in contained within the volume of the cube. What is the ratio of the radius of the hemisphere to the length of a side of the cube?



A. sqrt (2) : Pi

B. 1 : 2

C. 2 : Pi

D. 1: sqrt (2)

E. 2:1


OA follows....




OA is B


In general what is the key concept to keep in mind in attempting these complicated figure within figure kind of questions...that the GMAT sometimes throws up...some key points will be useful

Thanks
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by Geva@EconomistGMAT » Tue Aug 30, 2011 1:13 am
kaps786 wrote:The base of a hemisphere is inscribed in one face of a cube and the entire hemisphere in contained within the volume of the cube. What is the ratio of the radius of the hemisphere to the length of a side of the cube?



A. sqrt (2) : Pi

B. 1 : 2

C. 2 : Pi

D. 1: sqrt (2)

E. 2:1


OA follows....




OA is B


In general what is the key concept to keep in mind in attempting these complicated figure within figure kind of questions...that the GMAT sometimes throws up...some key points will be useful

Thanks
Many of these 3-dimensional problems actually test a 2D concept, and are easily converted into a simpler 2D form. The 3rd dimension is irrelevant: the hemisphere will extend "upwards" by a length of r, while the cube will extend "upwards' by the length of the side of the cube, which is clearly greater than r (because the base of the cube inscribes the circular base). So the volume of the hemisphere is clearly encased in the volume of the cube, and we can ignore the 3rd dimension, and just focus on what is essentially a circular base inscribed in a square base of the cube.

If a circle is inscribed in a square, it's diameter is equal to the side of the square, so 2r = side. If the radius is 1, then the diameter of the circle is 2, and that diameter will be equal to the sides of the square. Answer is B, taking care to avoid the trap answer E which uses the same ratio in reverse order.
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