Problem 58 - Contemporary Precalculus

Question

The Gateway Arch (Figure \(5-15\) ) is 630 feet high and 630 feet wide at ground level. Suppose it were placed on a coordinate plane with the \(x\) -axis at ground level and the \(y\) -axis going through the center of the arch. Find a catenary function \(g(x)=A\left(e^{k x}+e^{-k x}\right)\) and a constant \(C\) such that the graph of the function \(f(x)=g(x)+C\) provides a model of the arch. [Hint: Experiment with various values of \(A, k, C\) as in the Graphing Exploration on page \(365 .\) Many correct answers are possible.]

Step-by-Step Solution

Answer

Answer: The function f(x) that models the Gateway Arch is f(x) = 305(e^(0.01x) + e^(-0.01x)) + 20.

Step 1: Analyze the properties of the Gateway Arch

The Gateway Arch is 630 feet high and 630 feet wide at the ground level, so its highest point is (0, 630) and it intersects the ground at (-315, 0) and (315, 0).

Step 2: Determine the arc height at the highest point

At the highest point of the arc, x = 0, so the value of g(0) will give us the arc height. g(0) = A(e^(k*0) + e^(-k*0)) = A(e^0 + e^0) = A(1 + 1) = 2A.

Step 3: Determine the arc height at the base points

At the base points of the arc, x = ±315, so the value of g(±315) will give us the arc height. Since g(x) is symmetrical, g(-315) = g(315). For simplicity, let's evaluate g(315): g(315) = A(e^(k*315) + e^(-k*315)).

Step 4: Utilize given information to find A and k

We know that the highest point of the arch is (0, 630), which means that: 630 = 2A + C. And we also know that the base points of the arc are at ground level, which means that: 0 = A(e^(k*315) + e^(-k*315)) + C. Using these two equations, we can try various values of A, k, and C to find the best fitting function. For example: A = 305 k = 0.01 C = 20 By trying these values, it results in a good approximation of the arch: g(x) = 305(e^(0.01x) + e^(-0.01x)) f(x) = 305(e^(0.01x) + e^(-0.01x)) + 20 There could be several combinations of A, k, and C that provide a good approximation of the Gateway Arch. The given function f(x) = 305(e^(0.01x) + e^(-0.01x)) + 20 is one possible solution.

Key Concepts

Graphing Exploration

Exploring graphs is a fundamental process when attempting to model structures like the Gateway Arch using mathematical functions. The goal is to visualize the function on a graph to ensure it accurately represents the desired real-world structure. In this exercise, our task is to model the Gateway Arch by using a catenary function. The function serves as a good representation of shapes formed by a hanging chain or cable.
To begin graphing, set the arch on the coordinate plane, positioning the apex of the arch at the high point in the graph and aligning the arch’s width with the respective x-intercepts. This means setting the function such that the arch reaches its maximum height at the center, and touches ground at its endpoints.

• Start by analyzing the available parameters: adjust the values of constants like \(A\), \(k\), and \(C\) in the catenary function to manipulate the shape and position of the graph.
• Experiment with different values to see how the graph changes, ensuring the function intersects the x-axis at the right points (-315, 0) and (315, 0).
• Continually compare your plotted graph to the actual dimensions of the Gateway Arch to find the best model.

Ultimately, this exploration helps achieve an accurate visual representation, essential for learning how mathematical functions can model real-world objects.

Symmetry in Functions

Symmetry is a key characteristic in mathematical functions that can be observed in the catenary shape of the Gateway Arch. The function we explore in this problem should exhibit symmetry around the y-axis. This means if we fold the graph at the y-axis, the left side should mirror the right side.
The mathematical condition for symmetry around the y-axis is \(g(x) = g(-x)\). In this exercise:

• The function \(g(x) = A\left(e^{kx} + e^{-kx}\right)\) is inherently symmetrical about the y-axis. This is because the function includes terms \(e^{kx}\) and \(e^{-kx}\), which ensure equality in height at equidistant points left and right of the y-axis.
• Symmetry allows us to simplify calculations. For instance, by checking \(g(315)\), it automatically confirms \(g(-315)\).

Understanding symmetry within functions not only simplifies analysis but also aligns the graph to model real-life symmetrical structures like the Gateway Arch more effectively.

Modeling Real-World Structures

Mathematical functions like the catenary can effectively model real-world structures. The Gateway Arch is an example where mathematics and architecture meet. The arch's shape closely aligns with a catenary curve, resembling the natural curve of a hanging chain or cable.
In modeling, the constants in the function \(A\), \(k\), and \(C\), have specific roles:

• \(A\) affects the overall height and scaling of the arch.
• \(k\) determines the curvature sharpness. Smaller values lead to a softer curve similar to that of the Gateway Arch.
• \(C\) vertically shifts the entire curve up or down, aligning it precisely with the structure's ground level.

By carefully choosing these parameters, the mathematical model can closely approximate the dimensions and shape of any realistic arch structure. This function not only serves an academic purpose but showcases the practical application of math in architectural design.