Question 1168412
Let's break down this problem and find the equation of the parabola.

**1. Understand the Problem**

* The arch of the Bayonne Bridge is in the shape of a downward-opening parabola.
* The origin (0, 0) is at the surface of the water.
* We are given two key dimensions:
    * The width of the arch at the water's surface is 1675 feet.
    * The height of the arch is 325 feet.

**2. Set Up the Vertex**

* Since the parabola opens downward and the origin is at the water's surface, the vertex of the parabola will be at the highest point of the arch.
* The vertex will be located at (1675/2, 325) = (837.5, 325).

**3. General Equation of a Parabola**

The general equation of a parabola with a vertical axis of symmetry is:

y = a(x - h)² + k

Where:

* (h, k) is the vertex of the parabola.
* 'a' determines the direction and width of the parabola.

**4. Plug in the Vertex**

* We know the vertex is (837.5, 325), so h = 837.5 and k = 325.
* The equation becomes:

y = a(x - 837.5)² + 325

**5. Find 'a'**

* We know the parabola passes through the origin (0, 0). Plug in x = 0 and y = 0:

0 = a(0 - 837.5)² + 325
0 = a(837.5)² + 325
-325 = a(837.5)²
a = -325 / (837.5)²
a = -325 / 701406.25
a ≈ -0.00046335

**6. Write the Equation**

* Substitute the value of 'a' back into the equation:

y = -0.00046335(x - 837.5)² + 325

**7. Simplify (Optional)**

* If needed, we can round 'a' to a more manageable value:

y ≈ -0.000463(x - 837.5)² + 325

**Therefore, the equation of the parabola that models the arch of the Bayonne Bridge is approximately y = -0.000463(x - 837.5)² + 325.**