Problem 31 - Calculus of a Single Variable

Question

Length of a Catenary Electrical wires suspended between two towers form a catenary (see figure) modeled by the equation $$ y=20 \cosh \frac{x}{20}, \quad-20 \leq x \leq 20 $$ where \(x\) and \(y\) are measured in meters. The towers are 40 meters apart. Find the length of the suspended cable.

Step-by-Step Solution

Answer

The length of the suspended cable is \(40\sinh(20)\) metres.

Step 1: Find the derivative of the function

The first step in finding the length of the curve is to take the derivative of the function. In this case, the derivative of \(y = 20 \cosh(\frac{x}{20})\) with respect to \(x\) can be found using the chain rule. It is \(\frac{dy}{dx} = \sinh(\frac{x}{20})\).

Step 2: Insert the derivative into the formula

The formula for the length of the curve is \(L = \int_a^b \sqrt{1 + (f'(x))^2} dx\). Substituting the derivative into the formula gives \(L = \int_{-20}^{20} \sqrt{1 + \sinh^2(\frac{x}{20})} dx\). The key here is to remember that \(\cosh^2(x) - \sinh^2(x) = 1\), so we can rewrite the integrand as \(\sqrt{\cosh^2(\frac{x}{20})}\), which simplifies to \(\cosh(\frac{x}{20})\). Hence, the integral becomes \(L = \int_{-20}^{20} 20\cosh(\frac{x}{20}) dx\).

Step 3: Evaluate the integral

The last step is to evaluate the integral from -20 to 20. We are integrating 20 times the hyperbolic cosine, which is 20 times the hyperbolic sine. The integral of \(\cosh(x)\) is \(\sinh(x)\), so the evaluated integral is \(20[\sinh(20) - \sinh(-20)]\). It simplifies to \(40\sinh(20)\) metres.

Key Concepts

Hyperbolic Functions

Hyperbolic functions play a crucial role in this exercise. They are analogs to the trigonometric functions but for a hyperbola rather than a circle. The hyperbolic cosine function, denoted as \(\cosh(x)\), is defined as \(\cosh(x) = \frac{e^x + e^{-x}}{2}\). This function arises naturally in the study of catenary curves, as seen with the suspended cable between two towers.

• Understanding \(\cosh(x)\): It has a similar shape to the parabola but is distinctively more gradual at the extremes.
• Derivatives and \(\sinh(x)\): The derivative of the hyperbolic cosine, \(\cosh(x)\), is the hyperbolic sine, \(\sinh(x)\), defined as \(\sinh(x) = \frac{e^x - e^{-x}}{2}\).

Charts and graphs illustrate the bell-like shape of the curve, making it ideal for modeling the natural shape of a suspended cable, giving us insights into engineering and architecture applications.

Integral Calculus

Integral calculus is the method used to find the length of the suspended cable. It involves calculating the accumulation of quantities, which in this exercise is the length of a curve represented by a function.Essentially, one must integrate the function derived from the hyperbolic cosine, using the formula for the arc length of a curve.

• Arc Length Formula: To find the length \(L\) of a curve defined by a function \(f(x)\), we use \(L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx\).
• Substituting and Simplifying: By substituting the known derivative into this formula, you streamline the integration process.

Integral calculus, therefore, becomes the tool to determine the real-world applications of the catenary model.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. In the context of this exercise, it allows us to find the derivative of the composite hyperbolic cosine function used in modeling the catenary. The chain rule can be summarized as follows:

• Basic Principle: If a function \(y = g(f(x))\), then the derivative \(\frac{dy}{dx} = g'(f(x)) \cdot f'(x)\).
• Application in Catenary: Here, for \(y = 20\cosh\left(\frac{x}{20}\right)\), identify \(f(x) = \frac{x}{20}\) and differentiate accordingly.

Using the chain rule, we manage to compute \(\frac{dy}{dx} = \sinh\left(\frac{x}{20}\right)\), essential for the subsequent integral calculation.

Curve Length

The concept of curve length is pivotal in applications involving shapes and structures, like the catenary shape of the cable. To find the curve length accurately, calculus provides the arc length formula, crucial for measuring the piecewise sum of infinitesimally small segments of a curve.The comprehensive process involves:

• Formula Utilization: Applying \(L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx\) accommodates the undulating form of the hyperbolic function.
• Simplification: Through identity \(\cosh^2(x) - \sinh^2(x) = 1\), simplify the integrand to \(\cosh\left(\frac{x}{20}\right)\).
• Final Calculation: Integrating this function from the given limits results in a succinct evaluation, in this case, \(40\sinh(20)\) meters.

This methodology not only provides practical results for real-world structures but also enriches the students' grasp of integral calculus.