nusmavrik wrote:
What is the difficulty level - hard, medium or simple?
The dimensions of a rectangular solid are 4 inches, 5 inches, and 8 inches. If a cube, a side of which is equal to one of the dimensions of the rectangular solid, is placed entirely within the sphere just large enough to hold the cube, what the ratio of the volume of the cube to the volume within the sphere that is not occupied by the cube?
(A) 10:17
(B) 2:5
(C) 5:16
(D) 25:7
(E) 32:25
The dimensions of a rectangular solid are 4 inches, 5 inches, and 8 inches. If a cube, a side of which is equal to one of the dimensions of the rectangular solid, is placed entirely within the sphere just large enough to hold the cube, what the ratio of the volume of the cube to the volume within the sphere that is not occupied by the cube?
(A) 10:17
(B) 2:5
(C) 5:16
(D) 25:7
(E) 32:25
I find this question very ambiguous. Why was the diameter chosen as 4?
firstly the question says: their is a cuboid - 4*5*8
If a cube, a side of which is equal to one of the dimensions of the rectangular solid => this means that a cube can have a side = 4 or 5 or 8
Now this cube is placed entirely withing a sphere, just large enough to hold it => their can be 3 spheres with diameters as =>
\sqrt{4}, \sqrt{5}, \sqrt{8},
How is 4 decided to be the length of the side of the cube?
Now the ratio asked is = (a^3/(4/3*22/7*(\sqrt{3}/2*a)^3-a^3))
Here a^3 will cancel out, hence ratio is independent of length of the cube any ways.