davedekoos wrote:
rajarshee wrote:
A cube with each side = 1 is half filled with water. Now it is held in such a manner that only one of vertices is touching the floor and the diagonal of the cube is perpendicular to the floor.
What is the surface area of the water?
(1) √2
(2) 2
(3) √3
(4) 2√2
(5) 1
This problems takes some mental geometrical gymnastics of Olympic proportions. And this is almost certainly not a real GMAT question... But it's a really interesting fun problem, so here goes.
When you orient a cube so that one vertex is touching the ground and the diagonal is perpendicular to the ground, you'll basically get two triangular pyramids (one pointing up on top and the other pointing down on the bottom), with a triangular hexagonal hybrid shape in the middle. (What do you mean you have no idea what I'm saying??
)
I've drawn some different perspectives of a cube here to help
Attachment:
Wireframe cube 2.png
#1 and #2 show the cube oriented the way we want with a slight perspective to show where the water line would be.
#3 shows the cube directly from the side, with the water line exactly halfway.
#4 and #5 show a more common view of a cube in isometric view. This time the water line is shown at an angle (it's the same water line from #2 and #3), and at this angle we can see more clearly the shape that the surface of the water will have.
#6 and #7 show the cube from the top, looking down towards the ground, so we can see the full shape of the surface of the water.
As we can see it is a hexagon.
The hexagon is formed by joining the midpoint of the edges that do not connect to the vertices at the top or at the bottom. Since each side of the cube is 1, the length of each side of the hexagon is \(1/\sqrt{2}\). From there the area of the hexagon is \(\frac{3\sqrt{3}(side^2)}{2}\)
So the area of the hexagon, which is the surface area of the water is \(\frac{3\sqrt{3}}{4}\)
This is not one of the answer choices, but I feel pretty good about it. And seeing as this isn't a real GMAT question anyway, who knows where it and the answer choices came from?
Anyone want to have a go and see if I made an error?
At least some kudos for the drawing though...
Cheers
Do you know what the volume of the water would be in the shape you're working with? I bet the problem is that your solution has a different water volume than 0.5 units^3.
Anyways, for the answer, I assumed the water level would be less than half of the cube when the cube is tipped. If this is wrong, I'm sorry, it's 3 AM and I'm very tired. Either way, if the water level is less than half, if the water level is just low enough, you can use pyramidal formulas. The volume of a pyramid is always 1/3(a*a*h). If you take 1/2 as your water volume, you get 1/2 = 1/3(a*a*h), which becomes (3/2) = a*a*h. From here, I engage luck mode and hope that because we've seen nothing but clean numbers so far, our height and our final answer will be clean as well. That is, no roots. From this, we see that only B and E are applicable. E sets off red flags due to the 1, so I opted for B. B ended up being the answer.
Please note that my methodology and assumptions are pretty bad if you're sitting the gmat. I just wanted to see what happens if I use a line of assumptions to reach an answer.
If anyone has a real method to solve this question in a reasonable amount of time, please let me know. I'm curious!
close
30 Oct 2015, 16:42
71%
(02:12)
wrong 
