Problem 79 - A Graphical Approach to College Algebra

Question

Solve each problem. Train Curves When curves are designed for trains, sometimes the outer rail is elevated or banked so that a locomotive can safely negotiate the curve at a higher speed. Suppose a circular curve is being designed for a speed of 60 mph. The rational function $$f(x)=\frac{2540}{x}$$ computes the elevation \(y\) in inches of the outer track for a curve with a radius of \(x\) feet, where \(y=f(x) .\) (GRAPH CANT COPY) (a) Evaluate \(f(400)\) and interpret its meaning. (b) Graph \(f\) in the window \([0,600]\) by \([0,50] .\) Discuss how the elevation of the outer rail changes with the radius \(x\) (c) Interpret the horizontal asymptote. (d) What radius is associated with an elevation of 12.7 inches?

Step-by-Step Solution

Answer

(a) 6.35 inches. (b) Elevation decreases as radius increases. (c) Asymptote at \(y=0\). (d) Radius \(\approx 200\) feet.

Step 1: Evaluate f(400)

To evaluate the function for \(f(400)\), we substitute \(x = 400\) into the rational function. Calculate \(f(400) = \frac{2540}{400} = \frac{2540}{400}\).Perform the division: \(f(400) = 6.35\).Interpretation: At a radius of 400 feet, the outer rail should be elevated by 6.35 inches.

Step 2: Graph the function

To graph \(f(x) = \frac{2540}{x}\), plot the function over the domain \([0,600]\) and the range \([0,50]\).Notice that as \(x\) increases, the value of \(f(x)\) decreases, indicating that the elevation decreases as the radius increases.Use graphing software or a calculator for precise plotting within the window specified.

Step 3: Interpret the horizontal asymptote

The function \(f(x) = \frac{2540}{x}\) has a horizontal asymptote at \(y=0\).This means that as the radius \(x\) increases towards infinity, the elevation \(y\) decreases and approaches zero.Interpretation: For very large radius values, the need for elevation diminishes, leading the curve to flatten.

Step 4: Find the radius for an elevation of 12.7 inches

Set \(f(x) = 12.7\) and solve for \(x\):Let's equate \(12.7 = \frac{2540}{x}\).Multiply both sides by \(x\) to get: \(12.7x = 2540\).Divide both sides by 12.7: \(x = \frac{2540}{12.7}\).Perform the division to find \(x \approx 200\).Interpretation: A radius of approximately 200 feet corresponds to an elevation of 12.7 inches.

Key Concepts

Graphing Rational Functions

Graphing rational functions like \(f(x) = \frac{2540}{x}\) involves understanding how the values change as \(x\) varies. A rational function is a ratio of two polynomials, typically presented as \(f(x) = \frac{P(x)}{Q(x)}\). For our exercise, since \(P(x) = 2540\) (a constant) and \(Q(x) = x\), the graph will show how the division of a constant by \(x\) changes with different \(x\) values.
To graph this function between \([0, 600]\) for the range \([0, 50]\), follow these tips:

• First, understand that \(f(x)\) is only defined for \(x > 0\) since \(x\) is in the denominator. Avoid \(x = 0\) as it causes division by zero.
• As \(x\) increases, \(f(x)\) decreases, resulting in a graph that slopes downward from the y-axis.
• Plot key points by choosing strategic \(x\) values (like 100, 200, or 300) to evaluate \(f(x)\) and graph the resulting coordinates.
• Utilize graphing tools or calculators for precision, allowing you to visualize the curve accurately across the specified window.

Remember, the shape of the graph is crucial to interpret how the rail elevation changes relative to the track's radius.

Evaluating Functions

Evaluating a function at a particular value involves substituting the given input into the function and calculating the result. For the rational function \(f(x) = \frac{2540}{x}\), evaluating \(f(400)\) means replacing \(x\) with 400.
This calculation yields:

• \(f(400) = \frac{2540}{400} = 6.35\)

Interpretation of this result is straightforward:

• When the radius of the curve is 400 feet, the necessary elevation for the outer rail is 6.35 inches.

This suggests you need more elevation for smaller radii for safe negotiation of train curves. Always ensure calculations are carefully checked, especially with units of measurement critical in engineering design, like inches and feet.

Horizontal Asymptotes

Horizontal asymptotes in rational functions describe the expected behavior as \(x\) increases towards infinity. The rational function \(f(x) = \frac{2540}{x}\) has a horizontal asymptote at \(y = 0\).
Here's what this concept means:

• As \(x\) (radius) becomes larger, \(f(x)\) approaches zero, indicating the elevation diminishes.
• This aligns with the logic that larger curves require less banking of the outer rail.

The horizontal asymptote represents the flattening trend of elevation for very large radii, emphasizing stability in train navigation over expansive curves. Understanding where horizontal asymptotes exist in a graph aids in predicting long-term behavior beyond the plotted domain.

Radius and Elevation Relationship

The relationship between radius and elevation in designing train curves is inversely proportional, as shown in the function \(f(x) = \frac{2540}{x}\). When tasked to find the radius for a specific elevation \(y\), such as 12.7 inches, rearrange the equation to solve for \(x\).
Follow these steps:

• Set \(f(x) = 12.7\), forming \(12.7 = \frac{2540}{x}\).
• Multiply both sides by \(x\) to eliminate the fraction: \(12.7x = 2540\).
• Divide 2540 by 12.7 to isolate \(x\): \(x \approx 200\) feet.

Interpreting this calculation:

• A track with approximately a 200-foot radius requires 12.7 inches of elevation for safe curve navigation.
• This illustrates the critical balance of radius and elevation in railroad engineering, where smaller curves demand higher elevations to offset the centripetal forces acting on a train.

Understanding this relationship helps in the optimal design of safe and efficient railway systems.