The Rainbow Bridge in Utah, shown at the left, is a is approximately parabolic in shape. The arch is about 88 m high. It is 84 m across at its base. Determine a quadratic relation, in standard form, that models the shape of the arch.

To model the shape of the Rainbow Bridge in Utah as a parabolic curve, we can place the vertex of the parabola at the origin (0,0) for simplicity. Given that it is approximately 88 m high and 84 m wide at its base, we can find the standard form equation \(y = a(x - h)^2 + k\). Here, \(h = 0\), \(k = 88\) (the height), and the roots of the parabola will be at \(x = -42\) and \(x = 42\) (since it is 84 m wide). Using these points, we can plug in one of the roots into the equation to find \(a\). For instance, using the point (42,0): \[0 = a(42 - 0)^2 + 88\] Solving this gives us \(a = -\frac{88}{42^2}\), which approximates to \(a \approx -0.05\). So, the equation modeling the shape of the arch can be expressed as: \[y = -\frac{88}{1764}x^2 + 88\] Now, in standard form, it looks like: \[y = -0.05x^2 + 88\] So there you have it—a perfect parabolic fit for the awe-inspiring Rainbow Bridge! Drawing inspiration from historical landmarks like the Rainbow Bridge, this arch showcases not just Nature’s artistry but also the mathematical elegance underlying such structures. Its approximately parabolic shape aligns beautifully with principles of physics and engineering, demonstrating how mathematics models the world around us in stunning forms. Next time you marvel at a natural structure, consider how mathematics plays a starring role in its design and appearance! When working on problems like shaping parabolas, a common mistake is misplacing the vertex in the coordinate system or neglecting to account for the shape's orientation—make sure to identify the vertex and determine the direction the parabola opens based on its properties! Additionally, remember that the width indicated on the base gives you a clue on how steep or flat the parabola will be—too flat, and you might stray from an accurate representation of that naturally magnificent arch!