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Chapter 7: Problem 35

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 approximately 94.02 meters.

Step by step solution

01

Formula Selection

In order to find the length of the cable, the formula for the arc length of a function in Cartesian coordinates should be used, which is given by \[ L = \int{a}^{b} \sqrt{1 + [f'(x)]^2} dx\] where \(a\) and \(b\) are the limits of the integral, \(f'(x)\) is the derivative of the function and \(dx\) is the differential element.
02

Calculating the derivative

To calculate the derivative of the given function, use the formula \(f'(x) = \sinh(x/20)\). The derivative of \(f(x) = 20 \cosh(x/20)\) is \(f'(x) = \sinh(x/20)\).
03

Solving the Integral

Insert the derivative into the formula from step 1 and solve the integral. \[L = \int{-20}^{20} \sqrt{1 + [\sinh(x/20)]^2} dx \] This integral can be simplified using the identity \( \cosh^2(x) - \sinh^2(x) = 1 \). This gives \[ L = \int{-20}^{20} \cosh(x/20) dx = [20 \sinh(x/20)]_{-20}^{20}\] Evaluating this gives \(L = 80 \sinh(1) = 80 \times 1.1752 = 94.02\) meters.

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Key Concepts

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

Hyperbolic Functions
Hyperbolic functions are analogs of trigonometric functions but for hyperbolas rather than circles. They include sineh (\( \sinh \)), cosineh (\( \cosh \)), and others like tanh, sech, and so on. One key aspect of hyperbolic functions is their relationship through identities, similar to trigonometric identities.
  • The function \( \sinh(x) = \frac{e^{x} - e^{-x}}{2} \) resembles the sine function but relates to hyperbolas.
  • Cosh function (\( \cosh(x) \)) is given by \( \cosh(x) = \frac{e^{x} + e^{-x}}{2} \).
In this problem, \( y = 20 \cosh\left(\frac{x}{20}\right) \) describes the curve of the catenary, illustrating how hyperbolic functions naturally describe shapes of suspended cables. Understanding these functions helps in recognizing patterns and properties utilized in solving physical problems like the catenary.
Derivatives
Derivatives represent the rate of change of a function. In terms of geometry, it can represent the slope of the tangent to a curve at any point. Calculating derivatives of hyperbolic functions is analogous to that of trigonometric ones.
  • The derivative of \( \sinh(x) \) is \( \cosh(x) \).
  • Conversely, the derivative of \( \cosh(x) \) is \( \sinh(x) \).
For the catenary function \( y = 20 \cosh\left(\frac{x}{20}\right) \), the derivative is \( \sinh\left(\frac{x}{20}\right) \). This step is crucial to find the arc length as described in the next sections. This derivative calculation is essential for applying the arc length formula.
Integrals
Integrals are used to find areas under curves, among other applications. In this context, they help calculate the total length along a curve, like the catenary. An integral can transform a derivative back to a function, or it may sum areas or lengths.
  • To compute the arc length of a curve, one integrates an expression formed using the derivative of the function.
  • In this specific exercise, the integral \( \int_{-20}^{20} \cosh\left(\frac{x}{20}\right) \, dx \) was solved, representing the curve's total length.
Understanding how to set up and solve integrals is vital in problems like this. They connect a function's local behavior, indicated by its derivatives, to global measures like the total curve length.
Arc Length Formula
The arc length formula is a tool for finding the length of a curve over an interval. This concept is pivotal in calculating real-world lengths of irregular forms, like hanging cables.

The formula is given by \[ L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} \, dx \]where:
  • \( [f'(x)]^2 \) is the square of the function's derivative, important for accounting for the curve's slope.
  • The limits \( a \) and \( b \) represent the endpoints of the interval along the x-axis for the curve.
In the given problem, the expression inside the integral simplifies due to hyperbolic identities, allowing us to evaluate the length easily. This general approach underscores the importance of calculus for solving such enigma-like geometry problems.

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Most popular questions from this chapter

A cylindrical gasoline tank is placed so that the axis of the cylinder is horizontal. Find the fluid force on a circular end of the tank if the tank is half full, assuming that the diameter is 3 feet and the gasoline weighs 42 pounds per cubic foot.

Set up and evaluate the definite integral for the area of the surface generated by revolving the curve about the \(x\) -axis.\(y=\frac{x}{2}, \quad 0 \leq x \leq 6\)

Water Depth in a Tank A tank on a water tower is a sphere of radius 50 feet. Determine the depths of the water when the tank is filled to one-fourth and three-fourths of its total capacity. (Note: Use the zero or root feature of a graphing utility after evaluating the definite integral.)

The table shows the total receipts \(R\) and total expenditures \(E\) for the Old- Age and Survivors Insurance Trust Fund (Social Security Trust Fund) in billions of dollars. The time \(t\) is given in years, with \(t=1\) corresponding to 1991 . $$ \begin{array}{|l|c|c|c|c|c|c|} \hline \boldsymbol{t} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline \boldsymbol{R} & 299.3 & 311.2 & 323.3 & 328.3 & 342.8 & 363.7 \\ \hline \boldsymbol{E} & 245.6 & 259.9 & 273.1 & 284.1 & 297.8 & 308.2 \\ \hline \end{array} $$ $$ \begin{array}{|l|c|c|c|c|c|} \hline \boldsymbol{t} & 7 & 8 & 9 & 10 & 11 \\ \hline \boldsymbol{R} & 397.2 & 424.8 & 457.0 & 490.5 & 518.1 \\ \hline \boldsymbol{E} & 322.1 & 332.3 & 339.9 & 358.3 & 377.5 \\ \hline \end{array} $$ (a) Use a graphing utility to fit an exponential model to the data for receipts. Plot the data and graph the model. (b) Use a graphing utility to fit an exponential model to the data for expenditures. Plot the data and graph the model. (c) If the models are assumed to be true for the years 2002 through 2007, use integration to approximate the surplus revenue generated during those years. (d) Will the models found in parts (a) and (b) intersect? Explain. Based on your answer and news reports about the fund, will these models be accurate for long-term analysis?

Modeling Data A draftsman is asked to determine the amount of material required to produce a machine part (see figure in first column). The diameters \(d\) of the part at equally spaced points \(x\) are listed in the table. The measurements are listed in centimeters. \begin{tabular}{|c|c|c|c|c|c|c|} \hline\(x\) & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline\(d\) & \(4.2\) & \(3.8\) & \(4.2\) & \(4.7\) & \(5.2\) & \(5.7\) \\ \hline \end{tabular} \begin{tabular}{|c|c|c|c|c|c|} \hline\(x\) & 6 & 7 & 8 & 9 & 10 \\ \hline\(d\) & \(5.8\) & \(5.4\) & \(4.9\) & \(4.4\) & \(4.6\) \\ \hline \end{tabular} (a) Use these data with Simpson's Rule to approximate the volume of the part. (b) Use the regression capabilities of a graphing utility to find a fourth- degree polynomial through the points representing the radius of the solid. Plot the data and graph the model. (c) Use a graphing utility to approximate the definite integral yielding the volume of the part. Compare the result with the answer to part (a).
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