The problem describes two parabolic arches that support the second span of the Bluewater Bridge in Sarnia, Ontario. We are asked to write a function that models the shape of one of the arches.

Let's extract the key information:

• The arches are 281 meters apart (this represents the span or the distance between the two bases of the parabolas).
• The height of the arch at its highest point (vertex) is 71 meters above the river.

To model the arch, we can assume that the parabolic function is of the form:

$y = a(x - h)^2 + k$

Where:

• $(h, k)$ is the vertex of the parabola.
• $a$ is a constant that determines the width and direction of the parabola (since this is an arch, it opens downwards, so $a$ will be negative).
• $x$ is the horizontal distance, and $y$ is the height.

Step 1: Define the vertex

Since the arches are symmetrical and the span is 281 meters, the vertex of the arch will be located at the midpoint between the two bases. Thus, the horizontal coordinate of the vertex is at: $h = 0 \quad \text{(we set the center of the arch at } x = 0\text{)}$ The maximum height is given as 71 meters, so: $k = 71$

Thus, the equation becomes: $y = a(x - 0)^2 + 71 \quad \text{or simply} \quad y = ax^2 + 71$

Step 2: Determine $a$

The two bases of the arch are 281 meters apart, so the arch reaches the ground (where $y = 0$) at $x = \pm 140.5$ meters (half of 281 meters).

Substitute $y = 0$ and $x = 140.5$ into the equation: $0 = a(140.5)^2 + 71$ Solve for $a$: $a(140.5)^2 = -71$ $a = \frac{-71}{140.5^2}$

Let's calculate $a$.The value of $a$ is approximately $-0.0036$.

Thus, the equation of the parabolic arch is:

$y = -0.0036x^2 + 71$

This function models the shape of the arch, where $x$ is the horizontal distance from the center of the arch (in meters), and $y$ represents the height above the river (in meters).

Would you like further details or explanations?

Here are 5 related questions for further exploration:

1. How would the equation change if the height of the arch increased to 80 meters?
2. What if the distance between the arches was only 250 meters—how would that affect the equation?
3. Could we use a different function type, such as a cubic, to model the arch?
4. What does the negative value of $a$ signify in this context?
5. How would you determine the maximum height and span if only the equation was provided?

Tip: The sign of the coefficient $a$ in a quadratic function determines the direction in which the parabola opens—negative for downward-facing parabolas, as is the case for an arch.