Assignment - Modelling our 5x5x5 Rubik's cube
Here are some photos of a cube shaped exhibition area that is part of the Melbourne Museum on Exhibition Street.
The photo on the right shows the satellite view of the cube using the amazing free program Google Earth. The photo on the left was taken from the ground.
Google Earth allows you to view many prominent buildings in 3d. Imagine that you are the maths wiz at Google whose job it is to create a 3d image of our cube for Google Earth. This means that you have to find coordinates of the corners of the square A, B, C, D visible from the top, as well as the coordinates of the points A', B', C', D' at which four edges of the Cube meet the ground. As indicated in the first picture, A is supposed to be connected to A' by such an edge, and similarly for B and B', C and C', and D and D'.
You are new to this job and you are supposed to base your work on your predecessor's work (who got fired yesterday). His notes are a terrible mess, but it is clear that he started modelling the cube by letting the (fiat) ground coincide with the xy-plane and having the y-axis point West and the x-axis point North. His notes don't tell you where the origin of his coordinate system is supposed to be, but wherever it may be, in this coordinate system A = (42.5062, 34.7033, 8.79659) and C = (32.7431, 8.65133,16.9498). Furthermore, we know that B and D are at equal height above the ground, and that all these numbers are measurements in meters.
Your job also involves figuring out how long the sides of the cube are; how long the four segments AA', BB', CC', DD' are; how large the area of the quadrilateral A', B', C', D' is; and, finally, what the volume of the part of the cube that sits above the ground is.
At the end of your report please summarize your results as follows:
A = (42.6062, 34.7033, 8.79659)
B = (*, *, *)
C = (32.7431, 8.65133, 16.9498)
D = (*,*,*)
A' = (*, *, *)
B' = (*, *, *)
C' = (*, *, *)
D' = (*, *, *)
|AA'| = *m
|BB'| = *m
|CC'| = *m
|DD'| = *m
sidelength = *m
area(A'B'C' D') = *m2
volume = *m3
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!
There are different approaches to solving this set of problems, including some that don't involve the new mathematical tools that we introduced at the beginning of this unit. However, to make this assignment into a meaningful exercise you should use our new tools in your solutions:
- The dot product for calculating angles and checking quickly whether an angle is a right angle.
- The cross product for calculating vectors that are perpendicular to two given vectors (or a plane) in space and for calculating areas of parallelograms, triangles, and other plane figures in space.
- Line equations for describing lines in space (here, edges of cubes).