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MTH1030/35 Project 1
1. The cube (40 marks)
In a totally fictitious city there is a strange building in the shape of a cube partially sunk into the ground.
The image on the right shows what you would see if you were looking straight down at the cube from a drone. Imagine that you have been recruited by an architectural visialisation company that specialises in 3D modelling of urban environments. Your task? To reconstruct an accurate 3D model of this cube for an upcoming virtual reality city simulation.
To do this, you need to find the coordinates of the cube’s key points:
The four visible corners of the cube in the top-down view: A, B, C, D.
The points where four of the cube’s edges meets the ground: A′
, B′
, C′
, D′
.
As indicated in the reference diagram, point A is connected to A′
, point B to B′
, and so on.
Also, both points B and D are at the same height above the ground.
However, there’s a catch. The engineer previously assigned to this project quit unexpectedly, leaving behind a set of unorganised, cryptic notes. From what can be deciphered, the cube’s base is aligned with the xy-plane, with the y-axis pointing West and the x-axis pointing North.
Additionally, two crucial corner points, A and C, were already measured before the previous engineer left.
To retrieve the coordinates of A and C you need to enter your 8-digit (Monash) ID into into the following online tool https://www.qedcat.com/cubeAC.html.
Your task
Apart from determining the coordinates of A, B, C, D, A′
, B′
, C′
, D′
, your job also involves cal culating the following:
The lengths of the edges AA′
, BB′
, CC′
, DD′
.
The side length of the cube |AB|.
The area of the base A′B′C
′D′
(not a square!).
The surface area of the part of the cube above ground (xy-plane).
The volume of the part of the cube above ground.
At the end of your report please summarise your results as follows:
A = (∗, ∗, ∗)
B = (∗, ∗, ∗)
C = (∗, ∗, ∗)
D = (∗, ∗, ∗)
A
′ = (∗, ∗, ∗)
B
′ = (∗, ∗, ∗)
C
′ = (∗, ∗, ∗)
D′ = (∗, ∗, ∗)
|AA′
| = ∗m
|BB′
| = ∗m
|CC′
| = ∗m
|DD′
| = ∗m
sidelength = ∗m
area(A′B′C
′D′
) = ∗m2
surface area = ∗m2
volume = ∗m3
Additionally, generate a 3D representation of the structure above the ground, clearly showing all 12 edges.
Important
Accuracy: Some of your coordinates are rounded to four decimal places. However, your final answers should be accurate to at least two decimal places (e.g., 12.5674m ≈ 12.57m). To ensure accuracy, only round final results, NOT intermediate values.
Methodology: Your calculations must exclusively utilise the mathematical tools covered in Week 1 of MTH1030:
The cross product to construct perpendicular vectors.
The cross and box products to compute areas and volumes.
Line equations to determine cube edges and intersections.
These calculations should be carried out in Mathematica. You may find the following in built Mathematica functions useful: Dot, Cross, Norm. Avoid copying and pasting computed numbers, instead use the [[]] syntax to extract elements from lists as much as possible.
Strict Marking Policy: If any error affects subsequent calculations, every resulting mistake will also be penalised. To avoid errors, double-check your work using techniques such as the dot product to confirm right angles. Ensure all results are verified before submission.
You can adapt the following piece of Mathematica code to produce the picture of the sunken cube.
Graphics3D[{{Red, PointSize[Large], Point[{0, 0, 0}],
Point[{0, 0, 1}], Point[{0, 1, 0}], Point[{1, 0, 0}]}, {Red, Thick,
Line[{{0, 0, 0}, {0, 0, 1}}], Line[{{0, 0, 0}, {0, 1, 0}}],
Line[{{0, 0, 0}, {1, 0, 0}}], Line[{{0, 0, 1}, {0, 1, 0}}],
Line[{{0, 0, 1}, {1, 0, 0}}],
Line[{{0, 1, 0}, {1, 0, 0}}]}, {Opacity[0.3], Blue,
Polygon[{{0, 0, 0}, {0, 0, 1}, {0, 1, 0}}],
Polygon[{{0, 0, 0}, {0, 0, 1}, {1, 0, 0}}],
Polygon[{{0, 0, 0}, {0, 1, 0}, {1, 0, 0}}],
Polygon[{{0, 0, 1}, {0, 1, 0}, {1, 0, 0}}]}}, Boxed -> False,
Axes -> True, AxesLabel -> {"x", "y", "z"}]