It is supposed to be a cube that is sticking out of the ground. Let’s call it the Cube. The

right photo shows what you see when you zoom in on the Cube from above using Google

Earth (

Google Earth). The left photo was taken from the ground. In

this photo the corner of the Cube visible at the top is the corner labeled A in the satellite

photo.

Google Earth has a feature that allows you to get a 3D view of prominent buildings in

various cities (check out the Eifel tower in Paris, for example). Imagine that you are

the person in charge of creating a 3d image of the Cube for Google Earth. For this

you need to find 3d coordinates of the corners of the top square A,B,C,D, as well as

the coordinates of the points A0,B0,C0,D0 at which four edges of the Cube intersect the

ground. Here A is supposed to be connected to A0 by such an edge (see the left photo)

and similarly for B and B0, C and C0, and D and D0.

You are not the first person to attempt finding these coordinates. In fact, the last person

who tried just got fired yesterday because he was not able to do the job. You’ve just

taken over and you are supposed to base your work on your predecessors notes.

His notes are a mess, but it is clear that he started by letting the ground coincide with

the x, y-plane and that the y-axis is supposed to point West and the x-axis North. It is

not clear where exactly the origin of his coordinate system is, but wherever it is, in this

coordinate system A = (13.0492, 30.9498, 9.01115) and C = (1.97687, 4.68868, 17.3632).

You may assume that B and D are at equal height above the ground. Finally, it is also

clear that all these numbers are measurements in meters. Off you go!

Wait, apart from finding those eight coordinates I would also like you to figure out how

long one of the sides of the Cube is; what the lengths of the segments AA0,BB0,CC0,DD0

are; what the area of the quadrilateral A0,B0,C0,D0 is; and what the volumes of the parts

of the Cube above and below the ground are.

10-May-2010 4

School of Mathematical Sciences Monash University

At the end of this question please summarize your results as follows:

A = (13.0492, 30.9498, 9.01115)

B = (?, ?, ?)

C = (1.97687, 4.68868, 17.3632)

D = (?, ?, ?)

A0 = (?, ?, ?)

B0 = (?, ?, ?)

C0 = (?, ?, ?)

D0 = (?, ?, ?)

|AA0| =?m

|BB0| =?m

|CC0| =?m

|DD0| =?m

sidelength =?m

area(A0B0C0D0) =?m2

volume above =?m3

volume below =?m3

(50 marks): For any single one of these vectors/values that you get wrong

2 marks will be deducted. Tough rules: one mistake that implies other

mistakes will result in every single one of these mistakes being penalized

in this way—Google Earth only accepts results that work! Please make

absolutely sure that you double-check your results before you submit your

report!

Hint: Since we are dealing with a cube that is full of right angles, to quickly check

whether your coordinates for the different points make sense, calculate some the distances

between these points and check that some of the edges defined by them meet at

right angles (using the dot product).