Problem 37 - Calculus Early Transcendentals: Pearson New International Edition

Question

Call the graph of \(y=b-a \cosh (x / a)\) an inverted catenary and imagine it to be an arch sitting on the \(x\) -axis. Show that if the width of this arch along the \(x\) -axis is \(2 a\) then each of the following is true. (a) \(b=a \cosh 1 \approx 1.54308 a\). (b) The height of the arch is approximately \(0.54308 a\). (c) The height of an arch of width 48 is approximately \(13 .\)

Step-by-Step Solution

Answer

(a) Yes, \(b \approx 1.54308a\). (b) Height \(\approx 0.54308a\). (c) Height for width 48 \(\approx 13\).

Step 1: Understanding the Problem

The problem involves the function \(y = b - a \cosh(x/a)\), which is called an inverted catenary. The function describes an arch sitting on the \(x\)-axis, and we're given conditions about the width of this arch. Specifically, the width is \(2a\), and we are to prove certain results related to the parameters \(a\) and \(b\). Let's break these down step by step.

Step 1: Analyze the Arch Width

Since the arch has a specified width of \(2a\), the endpoints of the arch along the \(x\)-axis are at \(x = -a\) and \(x = a\). At these points, the height above the \(x\)-axis is zero, thus we set \(y = 0\) when \(x = \pm a\). Consequently, \(b - a \cosh(1) = 0\).

Step 2: Solve for \(b\)

From \(b - a \cosh(1) = 0\), we solve for \(b\): \[ b = a \cosh(1) \]The hyperbolic cosine function, \(\cosh(1) = \frac{e + e^{-1}}{2}\), approximates to \(1.54308\). Thus,\[ b \approx 1.54308 a \]This confirms part (a).

Step 3: Determine the Height of the Arch

The maximum height of the arch occurs at \(x = 0\), where the height is \(b - a\). From the previous step, we know \(b = a \cosh(1)\), so the height is:\[ H = b - a = a (\cosh 1 - 1) \approx 1.54308a - a = 0.54308a \]This confirms part (b).

Step 4: Calculate the Height for a Given Width

For an arch width of 48, the relation states \(2a = 48\), leading to \(a = 24\). Using the height formula from Step 3, the maximum height \(H\) is:\[ H \approx 0.54308 \times 24 \approx 13.0339 \]Rounding gives the height as approximately 13, confirming part (c).

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogous to trigonometric functions but are based on hyperbolas instead of circles. These functions include the hyperbolic sine (\(\sinh\)) and hyperbolic cosine (\(\cosh\), which we will focus on). They are defined using the exponential function as follows:

• \(\sinh(x) = \frac{e^x - e^{-x}}{2}\)
• \(\cosh(x) = \frac{e^x + e^{-x}}{2}\)

These functions are useful in various areas of mathematics, including calculus and geometry. In particular, the hyperbolic cosine function is crucial in describing curves like the catenary, which can be seen in structures like bridges and arches in architecture. Understanding these functions allows you to explore how complex curves relate to exponential growth and decay.

Arch Geometry

Arch geometry involves analyzing curves and shapes that form arches, commonly seen in architecture. The inverted catenary function, \(y = b - a \cosh(x/a)\), is a classic example of arch geometry. This particular shape is prevalent in construction because it efficiently distributes weight and withstands tension.
For an arch with a width of \(2a\), the endpoints are positioned at \(x = -a\) and \(x = a\). At these positions, the height from the base (\(x\)-axis) is zero. This ensures structural stability. The maximum height of the arch, situated at the center, is determined by subtracting \(a\) from \(b\), which reflects the vertical distance from the base to the arch's peak.
Understanding this aspect of geometry helps architects and engineers design structures that are not only aesthetically pleasing but also structurally sound.

Calculus Problem Solving

In calculus, problem-solving often involves analyzing and deriving relationships involving functions and their properties. With an inverted catenary, you encounter a blend of algebraic manipulation and function analysis.

• First, identify the important quantities involved, such as width and height.
• Write down the equation of the arch and express it in terms of known parameters like \(a\) and \(b\).
• Solve for unknown variables by substituting conditions provided in the problem, e.g., the endpoints of the arch.

Being methodical and structured, as seen in this problem where we determine \(b\) in terms of \(a\) and use this to find the arch's height at its peak, is key. Calculus not only aids in finding solutions but also provides deeper insight into how mathematical concepts govern real-world structures.

Hyperbolic Cosine

The hyperbolic cosine function, \(\cosh(x)\), plays a pivotal role in describing the shape of an inverted catenary. It is essential because it describes how the curve behaves and impacts the geometry of structures.
Mathematically, \(\cosh(x)\) has properties that make it continuously differentiable and symmetrical, which are beneficial for creating smooth curve transitions in arches. Its formula \(\cosh(x) = \frac{e^x + e^{-x}}{2}\) demonstrates that it remains positive, providing a solid foundation for creating structural arches that maintain their height uniformly across their span.
Moreover, understanding the approximation \(\cosh(1) \approx 1.54308\) is specifically useful in real-world applications where precise measurements are crucial, such as in architecture, where dimensional accuracy leads to stronger, more reliable structures.