Question

Use the following information. The Gateway Arch in St. Louis, Missouri, has the shape of a catenary (a U-shaped curve similar to a parabola). It can be approximated by the following model, where \(x\) and \(y\) are measured in feet. \(.\) Source: National Park Service Gateway Arch model: \(y=-\frac{7}{1000}(x+300)(x-300)\) How high is the arch?

Short answer

The height of the Gateway Arch is 630 feet.

Step-by-step solution

Recognize The Quadratic Function

Examine the model \(y=-\frac{7}{1000}(x+300)(x-300)\) and realize that it is in fact a quadratic function in the form \(y = ax^2 + bx + c\).

Find the x-coordinate of the vertex

The maximum point of a quadratic function is at its vertex. The x-coordinate of the vertex can be given by \( -b/(2a) \). However, recognizing that the equation is already factored, it becomes obvious that the x-coordinate for the vertex is 0 (where the factors (x + 300) and (x - 300) equal to zero).

Find the y-coordinate of the vertex

Substitute the x-coordinate into the original function to find the y value (or height). Calculating this gives \(y = -\frac{7}{1000}(0 + 300)(0 - 300)\) which gives \(y = 630\) feet.