The following pictures show a curious Rubikscubelike structu

The following pictures show a curious Rubik\'s-cube-like structure that greets visitors The following pictures show a curious Rubik\'s-cube-like structure that greets visitors to a Museum.

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 (flat) 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 B = (59.3506, 36.4925, 13.8151), B0 = (57.8976, 32.7136, 0), D = (88.3907, 25.3267, 13.8151). 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 = (?, ?, ?)

B = (59.3506, 36.4925, 13.8151)

C = (?, ?, ?)

D = (88.3907, 25.3267, 13.8151)

A\'= (?, ?, ?)

B\'= (57.8976, 32.7136, 0)

C\'= (?, ?, ?)

D\'= (?, ?, ?))

|AA\'| = ?m

|BB\'| = ?m

|CC\'| = ?m

|DD\'| = ?m

sidelength = ?m

area(A\'B\'C\'D\') = ?m2

volume = ?m3

Solution

z coordinate of A\',B\',C\',D\' = 0
z coordinate of A,B,C,D = 13.8151

Side length BB\' = sqrt ((59.3506- 57.8976)^2 + (36.4925-32.7136)^2 + (13.8151-0)^2 )
               = 14.39612
Since it is cube , all side lengths are equal :
AA\' = BB\' = CC\' = DD\' = side length = *m = 14.39612 Answer

Area = (14.39612)^2 = 207.24827

Volume = (14.39612)^3 = 2983.57098

Coordinates:
A = (59.3506 , 36.4925- 14.39612, 13.8151) = 59.3506 , 22.09638, 13.8151)

B = (59.3506, 36.4925, 13.8151)

C = (88.3907, 25.3267-14.39612, 13.8151) = (88.3907, 10.93058, 13.8151)

D = (88.3907, 25.3267, 13.8151)

A\' =(59.3506- (59.3506- 57.8976), 22.09638-1.453, 0) = (57.8976, 20.64338,0)

Note that: 59.3506- 57.8976 = 1.453

B\' = (57.8976, 32.7136, 0)

C\' =(88.3907- (59.3506- 57.8976), 10.93058-1.453 ,0) = (86.9377, 9.47758 ,0)

D\' =(88.3907- (59.3506- 57.8976), 25.3267-1.453 ,0) = (86.9377, 23.8737, 0)

Hope this helps!