/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 Length of a Catenary Electrical ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 31

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.

Short Answer

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

Step by step solution

01

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 \(\frac{dy}{dx} = \sinh(\frac{x}{20})\).
02

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 \(L = \int_{-20}^{20} 20\cosh(\frac{x}{20}) dx\).
03

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Lifting a Chain In Exercises \(25-28\) , consider a 20 -foot chain that weighs 3 pounds per foot hanging from a winch 20 feet above ground level. Find the work done by the winch in winding up the specified amount of chain. Run the winch until the bottom of the chain is at the 10 -foot level.

Verifying a Formula (a) Given a circular sector with radius \(L\) and central angle \(\theta\) (see figure), show that the area of the sector is given by $$S=\frac{1}{2} L^{2} \theta .$$ (b) By joining the straight-line edges of the sector in part (a), a right circular cone is formed (see figure) and the lateral surface area of the cone is the same as the area of the sector. Show that the area is \(S=\pi r L,\) where \(r\) is the radius of the base of the cone. (Hint: The arc length of the sector equals the circumference of the base of the cone.) (c) Use the result of part (b) to verify that the formula for thelateral surface area of the frustum of a cone with slant height \(L\) and radii \(r_{1}\) and \(r_{2}\) (see figure) is \(S=\pi\left(r_{1}+r_{2}\right) L .\) (Note: This formula was used to develop the integral for finding the surface area of a surface of revolution.)

Volume of a Fuel Tank \(A\) tank on the wing of a jet aircraft is formed by revolving the region bounded by the graph of \(y=\frac{1}{8} x^{2} \sqrt{2-x}\) and the \(x\) -axis \((0 \leq x \leq 2)\) about the \(x\) -axis, where \(x\) and \(y\) are measured in meters. Use a graphing utility to graph the function and find the volume of the tank.

Suspension Bridge A cable for a suspension bridge has the shape of a parabola with equation \(y=k x^{2} .\) Let \(h\) represent the height of the cable from its lowest point to its highest point and let 2\(w\) represent the total span of the bridge (see figure). Show that the length \(C\) of the cable is given by $$ C=2 \int_{0}^{w} \sqrt{1+\left(4 h^{2} / w^{4}\right) x^{2}} d x $$

Finding Distance Using Two Methods In Exercises 1 and \(2,\) find the distance between the points using (a) the Distance Formula and (b) integration. $$ (1,2), \quad(7,10) $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.