The largest multi-story building built by the Anasazi Indians called the Pueblo Bonito. The shape of its foundation is nearly parabolic, with a base (at the top of the photo) of about 500 feet and a height of about 300 feet. 1. Sketch the multistory building on a coordinate plane with the base of the building on the x-axis and the y-axis exactly in the middle of the building. The y-intercept is (0,-300). 2. Find the x-intercepts. 3. Substitute the three points from Exercises 1 and 2 into the model y = ax^2+ bx +c to obtain a system of linear equations. 4. Solve the linear system to find a quadratic model for the data.
To solve this problem, let's go through each step one by one.
1. Sketch the multistory building on a coordinate plane:
The shape of the foundation is nearly parabolic, with a base of about 500 feet and a height of about 300 feet. The y-axis will be exactly in the middle of the building, and the y-intercept is given as (0, -300).
To sketch the building on a coordinate plane, plot the points (0, -300), (250, 0), and (-250, 0). These represent the y-intercept and the x-intercepts (steps 2).
2. Find the x-intercepts:
To find the x-intercepts, we need to find the points where the graph intersects the x-axis. Since the y-coordinate of these points is 0, the x-intercepts will have the form (x, 0).
The given information tells us that the base of the building is about 500 feet, and the shape is nearly parabolic. Since the shape is symmetrical and the y-axis is in the middle of the building, the x-intercepts must be symmetric about the y-axis.
Therefore, the x-intercepts will be at (250, 0) and (-250, 0).
3. Substitute the points into the model y = ax^2 + bx + c:
We have three points: (0, -300), (250, 0), and (-250, 0). Substituting these points into the equation y = ax^2 + bx + c will give us a system of linear equations.
For point (0, -300):
-300 = a(0^2) + b(0) + c
This simplifies to: c = -300
For point (250, 0):
0 = a(250^2) + b(250) + (-300)
For point (-250, 0):
0 = a((-250)^2) + b(-250) + (-300)
4. Solve the linear system to find a quadratic model:
Using the two equations obtained from points (250, 0) and (-250, 0), we can solve for the unknowns 'a' and 'b' since we already found 'c' as -300:
For point (250, 0):
0 = a(250^2) + b(250) + (-300)
62500a + 250b - 300 = 0
For point (-250, 0):
0 = a((-250)^2) + b(-250) + (-300)
62500a - 250b - 300 = 0
Simplifying, we get the following system of linear equations:
62500a + 250b = 300
62500a - 250b = 300
Solving this linear system will give us the values of 'a' and 'b', which we can substitute back into the equation y = ax^2 + bx + c to obtain the quadratic model for the data.
Please note that solving this linear system involves performing some algebraic operations, which might be best done manually or using a computer algebra system.