Question

MODELING WITH MATHEMATICS The Gateshead Millennium Bridge spans the River Tyne. The arch of the bridge can be modeled by a parabola. The arch reaches a maximum height of 50 meters at a point roughly 63 meters across the river. Graph the curve of the arch. What are the domain and range? What do they represent in this situation?

Short answer

The equation modeling the bridge is \(y = -0.032(x-31.5)^2 + 50\). The domain is [0, 63] which represents the width of the bridge across the river, and the range is [0,50] representing the height of the arch from the water level to the highest point.

Step-by-step solution

Define the function

First, define a quadratic function to model the bridge. A parabolic curve is typically modeled by the equation \(y = ax^2 + bx + c\). In this case, we know the vertex of the parabola (the highest point of the bridge, which is the middle of the bridge), which is at (31.5, 50). This is because the bridge is 63 meters wide and the maximum point is in the middle. The standard form equation of a parabola with the vertex at (h, k) is \(y = a(x-h)^2 + k\). So the equation of the bridge is \(y = a(x-31.5)^2 + 50\).

Find the value of 'a'

We know that the bridge starts and ends at the water level, i.e., y = 0. At this point, the bridge is 63 meters wide. So we set x = 0 and x = 63 in our equation and solve for 'a'. We get a = -0.032. So, the equation modeling the bridge is \(y = -0.032(x-31.5)^2 + 50\). This equation can be used to graph the curve.

Graph the function and find the domain and range

Next, graph the function using the derived equation. The domain of the bridge (x-values) will be [0, 63], as the width of the bridge is 63 meters. The range (y-values) will be [0, 50], since the height of the bridge varies from 0 at the water level to 50 meters at its highest point.

Interpret the domain and range

The domain, which consists of x-values, represents the width of the bridge across the river. In this case, it means that the bridge spans 63 meters across the River Tyne. The range, consisting of the y-values, represents the height of the arch of the bridge above the water level. It means the height of the bridge varies from 0 meters at the river's edge to 50 meters at the highest point of the arch.

Key concepts

Quadratic Functions

Quadratic functions are at the heart of parabolic shapes, such as the majestic arch of a bridge. These functions have the general form of a second-degree polynomial, \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, with \(a \eq 0\). The graph of a quadratic function creates a U-shaped curve called a 'parabola.' This shape is symmetric around a line known as the axis of symmetry, which corresponds to the vertex—or in the context of our bridge, the highest point of the arch.

In real-world modeling, like that of the Gateshead Millennium Bridge, the quadratic function enables us to represent the curved structure mathematically, so we can predict and calculate various aspects of the bridge's design, like its highest point and the width it spans. As in the exercise, finding the right quadratic equation is a journey of determining the curve's specific characteristics, like its vertex and the points where the bridge meets the water level, symbolized as \((h, k)\) and the zeros of the function.

Domain and Range of a Function

The concepts of 'domain' and 'range' are fundamental in understanding the behavior of functions, including quadratic ones. The domain of a function contains all the possible input values (typically \(x\)-values), while the range is all the resultant output values (typically \(y\)-values) after applying the function to the domain.

In the context of our bridge model, the domain defined the actual width the bridge spans across the river. It is the set of all horizontal distances we can measure from one edge of the bridge to the other—starting at zero, where the bridge meets the water, and ending at the bridge's total width. The range, on the other hand, described the vertical variation in the height of the bridge, from the lowest point at water level (0 meters) to the highest point of the arch (50 meters).

The domain and range provide us with a clear picture of the boundaries within which the bridge exists. They ensure that any predictions or calculations we carry out using the quadratic function stay within these physical boundaries—after all, the bridge cannot span a width beyond its actual construction, nor can the arch be taller than its highest point.

Graphing Parabolas

Graphing parabolas is a visual representation of quadratic functions. It is not only a powerful tool to understand the function's characteristics but also critical when applying concepts to real-life structures like bridges. A parabola can open upwards or downwards, determined by the sign of the coefficient \(a\) in the quadratic function—Illustrated by the negative coefficient in the bridge's equation, revealing an arch that opens downwards.

To graph a parabola, identifying the vertex is paramount, as it acts as the turning point of the curve. In the case of the Gateshead Millennium Bridge, the vertex was the highest point of the arch. With the vertex, the zeros—where the parabola intersects the \(x\)-axis—and the axis of symmetry identified, a complete picture of the parabola starts to emerge, anchored by these critical points.

Graphing the bridge's parabola involves plotting these known points onto a coordinate system and drawing the curve that represents the quadratic equation. This graph provides not just aesthetic pleasure but also offers insights into the parabola's behavior—like how wide and how high the arch actually is—and serves as a testament to the applicability of quadratic functions in the physical world.