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: \(y=-\frac{7}{1000}(x+300)(x-300)\) According to the model, how far apart are the legs of the arch?

Short answer

The legs of the arch are 600 feet apart according to the model.

Step-by-step solution

Understanding the Problem

The problem provides a mathematical function that models the shape of the Gateway Arch in St. Louis, and we are asked to determine how far apart the legs of the arch are. In mathematical terms, we are finding the difference between the x-values where the quadratic function intersects the x-axis (i.e., the 'roots' or 'zeros' of the function). These points occur where \(y = 0\).

Setting up the Equation

To find where the legs of the arch are (the x-intercepts), we need to set \(y\) equal to zero in the function provided and solve for \(x\). So, the resulting equation that needs to be solved is \(0=-\frac{7}{1000}(x+300)(x-300)\)

Solving the Equation

The equation set in step 2 is a quadratic equation and it is already factored. So we can set each factor equal to zero and solve for \(x\). Setting \(x + 300 = 0\) gives \(x = -300\) and setting \(x - 300 = 0\) gives \(x = 300\).

Calculating the Distance

The distance between the legs of the arch is the absolute difference in the x-values. Thus, Distance = \( |x_1 - x_2| = |-300 - 300| = 600\) feet.