Problem 55 - College Algebra with Corequisite Support

Question

For the following exercises, use a calculator to find the answer. A suspension bridge can be modeled by the quadratic function \(h(x)=.0001 x^{2}\) with \(-2000 \leq x \leq 2000\) where \(|x|\) is the number of feet from the center and \(h(x)\) is height in feet. Use the TRACE feature of your calculator to estimate how far from the center does the bridge have a height of 100 feet.

Step-by-Step Solution

Answer

The bridge is 100 feet high at 1000 feet and -1000 feet from the center.

Step 1: Understand the Problem

We are given the quadratic function \(h(x) = 0.0001x^2\) representing the height of a suspension bridge. We need to find the value of \(x\) for which the height \(h(x)\) is 100 feet by using a calculator and the TRACING feature.

Step 2: Set the Function for Height 100 Feet

To find when the height of the bridge is 100 feet, we set the equation \(h(x) = 100\). The equation becomes \(0.0001x^2 = 100\). Our goal is to solve for \(x\) within the given domain \(-2000 \leq x \leq 2000\).

Step 3: Isolate x using a Calculator

Using your calculator, solve the equation \(0.0001x^2 = 100\) by isolating \(x\). Divide both sides by 0.0001 to get \(x^2 = \frac{100}{0.0001}\) which simplifies to \(x^2 = 1000000\).

Step 4: Solve for x

Now solve for \(x\) by taking the square root of both sides. Thus, \(x = \pm \sqrt{1000000}\). The solution simplifies to \(x = \pm 1000\).

Step 5: Verify Using Calculator's TRACE Feature

Using the TRACE feature, graph the function \(h(x) = 0.0001x^2\), and check the points where the graph meets the line \(h(x) = 100\), confirming the solutions \(x = 1000\) and \(x = -1000\). This ensures the answers are within the boundaries of \(-2000 \leq x \leq 2000\).

Key Concepts

Suspension Bridge

A suspension bridge is a type of bridge in which the deck is hung below suspension cables on vertical suspenders. This design is not only visually appealing but also structurally efficient, allowing it to span longer distances without requiring multiple piers. In a simplified mathematical model, the curve of the cables can be represented by a quadratic function, which resembles a parabola. In our exercise, the quadratic function given is \[h(x) = 0.0001x^2\] where \(x\) represents the horizontal distance from the center of the bridge, and \(h(x)\) represents the height at that point. The parameters of the function help dictate the overall shape and size of the bridge. By studying this mathematical model, one can better appreciate the elegance and engineering that go into designing these majestic structures.

Graphing Calculator

A graphing calculator is a powerful tool in understanding and visualizing mathematical functions, especially quadratic functions. This calculator type allows you to input equations and immediately see their graphical representations. When approaching problems involving complex functions like that of a suspension bridge, a graphing calculator can significantly simplify the process.
The calculator provides several features that aid in solving equations:

• Plotting functions
• Zooming in and out to see detailed behavior near specific points
• Providing a numerical table that corresponds to values of \(x\) and \(h(x)\)

With these capabilities, it's easier to understand how the function behaves across a specified range, making the graphing calculator an invaluable resource in both study and application.

Trace Feature

The trace feature on a graphing calculator is extremely helpful when analyzing graphs of functions. This tool allows you to move along a plotted graph and see the exact coordinates at any point. When dealing with a model like the suspension bridge's quadratic function, using the trace feature can aid in determining precise values where the function satisfies particular conditions, such as reaching a specific height.
For example, if you need to find where the bridge attains a height of 100 feet, you can graph the height function \(h(x) = 0.0001x^2\) and use the trace to pinpoint the \(x\)-values where the graph touches the line \(h(x) = 100\). This enables you to visually confirm the solutions obtained by solving the equation analytically. It turns a sometimes abstract concept into something more tangible and straightforward.

Solving Equations

Solving equations is a fundamental aspect of mathematics that involves finding the values that make an equation true. In our suspension bridge example, we are tasked with finding the values of \(x\) for which the height \(h(x)\) is 100. This involves setting the equation \[0.0001x^2 = 100\] and manipulating it to isolate \(x\).
The steps involve:

• Dividing both sides by the coefficient of \(x^2\), which is 0.0001, leading to \(x^2 = 1000000\)
• Taking the square root of both sides to solve for \(x\), resulting in \(x = \pm 1000\)

Thus, the solutions \(x = 1000\) and \(x = -1000\) indicate the points where the bridge reaches a height of 100 feet from the center. Understanding each step in this process helps in tackling similar problems with confidence.