A parabolic arch has a span of 10 meters and a maximum height of 4 meters. If the vertex is at the midpoint, what is the equation of the parabola in vertex form?
["Equation of a Parabolic Arch: How to Derive It Given the Span and Height", "A well-designed arch is not only structurally sound but also visually striking. Understanding the mathematical principles behind parabolic arches helps engineers and architects model these forms accurately. In this article, we explore the derivation of the equation of a parabolic arch with a specific span and maximum height, using real-world dimensions to illustrate key concepts.", "### The Parabolic Arch Design", "Suppose we are analyzing a parabolic arch with the following characteristics:", "- The span (horizontal distance across the base) is 10 meters,\n- The maximum height (vertex) is 4 meters,\n- The vertex of the parabola is located at the midpoint of the span.", "This setup ensures symmetry about the vertex, simplifying the equation to parabolic vertex form.", "### Why Vertex Form?", "The vertex form of a parabola’s equation is ideal for this case because it clearly shows the vertex’s location, which sits at the peak of the arch. The standard vertex form is:", "[\ny = a(x - h)^2 + k\n]", "where $(h, k)$ is the vertex. Since the vertex is at the midpoint of the span, and the span is symmetric, the vertex lies at $x = 5$ meters (midpoint of 0 and 10) and $y = 4$ meters (maximum height). Therefore:", "[\n(h, k) = (5, 4)\n]", "### Plugging in the Vertex", "Substitute into the vertex form:", "[\ny = a(x - 5)^2 + 4\n]", "### Finding the Value of (a)", "To determine (a), use a known point on the parabola—specifically, one end of the arch at $(0, 0)$ or $(10, 0)$, since the base spans 10 meters. Let’s use $(0, 0)$:", "[\n0 = a(0 - 5)^2 + 4\n]", "[\n0 = 25a + 4\n]", "Solving for (a):", "[\n25a = -4 \quad \Rightarrow \quad a = -\frac{4}{25}\n]", "### Final Equation in Vertex Form", "Substituting (a = -\frac{4}{25}) into the equation:", "[\ny = -\frac{4}{25}(x - 5)^2 + 4\n]", "This is the equation of the parabolic arch in vertex form, accurately modeling a semi-ellipticalesthetic arch with a 10-meter span and 4-meter height, centered at the vertex.", "### Summary", "- Span: 10 m → midpoint at $x = 5$\n- Vertex height: 4 m\n- Vertex: $(5, 4)$\n- Equation: [\ny = -\frac{4}{25}(x - 5)^2 + 4\n]", "Using this form allows for precise architectural design and engineering calculations, ensuring both beauty and structural integrity in parabolic arches.", "---", "Keywords: parabolic arch equation vertex form, derive parabola equation span height, vertex form parabola, parabolic arch design, architectural parabola, quadratic equation arch modeling"]
