A parabolic arch has a span of 120 feet and a maximum height...

Name: Problem 12, Chapter 4: Algebra and Trigonometry
Uploaded: 2021-06-30T17:33:12-08:00
Duration: 6 min 32 s
Channel: Andrew Elmer
Description: A parabolic arch has a span of 120 feet and a maximum height of 25 feet. Choose suitable rectangular coordinate axes and find the equation of the parabola. Then calculate the height of the arch at points 10 feet, 20 feet, and 40 feet from the center.

A parabolic arch has a span of 120 feet and a maximum height of 25 feet. Choose suitable rectangular coordinate axes and find the equation of the parabola. Then calculate the height of the arch at points 10 feet, 20 feet, and 40 feet from the center.

Key Concepts

Evaluating the Parabolic Equation

After the complete equation of the parabola is derived, it can be used to find the y-values (heights) for specific x-values (distances from the center). This evaluation process involves substituting the x-values into the equation. The resulting y-values give the heights of the arch at those points, which is fundamental for practical applications like design and structural analysis.

Determining the Parameter Using a Known Point

Once the vertex form is established, a known point on the parabola can be substituted into the equation to solve for the coefficient a. This process involves plugging in the x and y values of the point and isolating a. The method is essential for tailoring the general equation to represent the specific dimensions and curvature of an arch or other parabola-defined structure.

Vertex Form of a Parabolic Equation

A parabola can be expressed in the vertex form, y = a(x - h)² + k, where (h, k) is the vertex. This form is particularly useful when the highest or lowest point (vertex) is known. It lays out the structure of the parabola, indicating not only its direction and width through the parameter a but also its precise position in the coordinate system.

Coordinate Axes Selection

When modeling a parabolic arch, choosing an appropriate coordinate system is essential. By aligning the coordinate axes with the symmetry of the parabola, the problem becomes simpler. Typically, the vertex of the parabola is set at the origin or along a convenient axis, which simplifies the algebra required to derive the equation.

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