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 [ 36.94 KiB | Viewed 1334 times ]
#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

_________________

Dave de Koos

GMAT aficionado