Introduction:
In this task, the building with a roof structure shown above is initially assumed to have a rectangular base of 150m long and 72m wide. The maximum height of the structure should not exceed 75% of its width for stability or be less than half the width for aesthetic purposes. Moreover, the minimum height of a room in a public building is 2.5 m.
Under this condition, several questions about the modifications of the structure of the building and the office block inside it are investigated, thus for convenience, this building is modeled in a cross coordinates for investigation.
The questions will be discussed again when the rectangular base is change to be 150m wide and 72m long without changing the other sets.
Part I: Modeling the structure in cross coordinates
When the base of the structure is 150m long and 72m wide, the height of the structure should no exceed 54m for stability or 36m for aesthetic according to the requirements.
Calculation:
72m x 0.75=54m
72m x 0.5=36m
Because we model the structure in the cross coordinate, the function of the roof structure is modeled to be a quadratic function with the equationaccording to the shape of the structure. And because the open side of the function is pointed downward, therefore a is smaller than 0 and the highest point of the equation is (0, c).
Next, we can create a model for the curved roof structure, which has the minimum height of 36m, thus the highest point of the curve is (0, 36).
On the basis of these data, we can draw a sketch of the curve of roof structure as shown below.
By using the graph and the data, we can find the equation of the parabola which goes through point A, B and C.
Calculation:
Substitute x=0, y=36 into the equation, we can get
Substitute x=36, y=0 and x=-36 y=0 into the equation, we can get
After solving the 2x2 system, we can get
Finally, we can get that the equation of the parabola is...