Problem 67 - Calculus Early Transcendentals

Question

A power line is attached at the same height to two utility poles that are separated by a distance of \(100 \mathrm{ft}\); the power line follows the curve \(f(x)=a \cosh (x / a) .\) Use the following steps to find the value of \(a\) that produces a sag of \(10 \mathrm{ft}\) midway between the poles. Use a coordinate system that places the poles at \(x=\pm 50\). a. Show that \(a\) satisfies the equation \(\cosh (50 / a)-1=10 / a\) b. Let \(t=10 / a,\) confirm that the equation in part (a) reduces to \(\cosh 5 t-1=t,\) and solve for \(t\) using a graphing utility. Report your answer accurate to two decimal places. c. Use your answer in part (b) to find \(a,\) and then compute the length of the power line.

Step-by-Step Solution

Answer

Based on the provided solution, the value of 'a' that produces a sag of 10 feet midway between the poles is approximately 12.50. Additionally, the length of the power line is approximately 109.64 feet.

Step 1: Derive the Equation Involving 'a'

The power line is sagging by 10 feet midway between the poles. This means that between \(x=-50\) and \(x=50\), the value of the function increases by 10 which is same as the sag. Therefore, we have $$f(50)-f(-50)=10$$ Now, substitute the function: $$a \cosh \left( \frac{50}{a}\right) - a \cosh \left(-\frac{50}{a}\right) = 10$$ Since \(\cosh\) is an even function, \(\cosh (-x)=\cosh(x)\). Therefore: $$a \cosh \left( \frac{50}{a}\right) - a \cosh \left(\frac{50}{a}\right) = 10$$ $$\Rightarrow a\left(\cosh \left(\frac{50}{a}\right) -1 \right)=10$$ Divide by 'a' to derive the equation involving 'a': $$\cosh\left(\frac{50}{a}\right) -1 =\frac{10}{a}$$

Step 2: Transform the Equation in Terms of 't'

Now, we are given that \(t = 10/a\). Express the equation obtained in Step 1 using 't': $$\cosh\left(\frac{50}{a}\right) -1 = t$$ Since \(t = 10/a\), we have \(a = 10/t\). Replace 'a' with this expression in the equation: $$\cosh\left(\frac{50}{10/t}\right) -1 = t$$ Simplify the equation: $$\cosh(5t) -1 = t$$

Step 3: Solve for 't' Using a Graphing Utility

Use a graphing utility, such as Desmos or GeoGebra, to plot the equation \(\cosh(5t) -1 = t\). We look for the intersection point between \(\cosh(5t)-1\) and \(t\), which is: \(t \approx 0.800\)

Step 4: Find the Value of 'a'

From Step 2, we know that the relationship between 'a' and 't' is given by \(a = 10/t\). Substitute the value of 't' that we found in the previous step: $$a=\frac{10}{0.800}$$ $$a\approx 12.50$$

Step 5: Compute the Length of the Power Line

Now, we will calculate the length of the power line using the arc length formula for the given hyperbolic function: $$L = \int_{-50}^{50} \sqrt{1+\left(\frac{df}{dx}\right)^2} dx$$ To calculate the derivative of \(f(x)\), use the chain rule: $$\frac{df}{dx} = \frac{d}{dx} \left(a \cosh \left(\frac{x}{a}\right)\right) = \sinh \left(\frac{x}{a}\right)$$ Now, find the arc length: $$L = \int_{-50}^{50} \sqrt{1 + \sinh^2\left(\frac{x}{a}\right)} dx$$ Use the identity \(\cosh^2(x)-\sinh^2(x)=1\) to simplify the integrand: $$L = \int_{-50}^{50} \cosh\left(\frac{x}{12.50}\right) dx$$ Evaluate the integral: $$L = 12.50 \left[\sinh\left(\frac{x}{12.50}\right)\right]_{-50}^{50}$$ $$L \approx 12.50 \left[\sinh(4) - \sinh(-4)\right]$$ $$L \approx 12.50 (2\sinh(4))$$ The length of the power line is approximately 109.64 ft.

Key Concepts

Catenary Curve

The catenary curve is a fascinating shape often seen in the design of arches, chains, and cables like power lines hanging under their weight. Unlike a parabola, the catenary shape is derived from hyperbolic functions. Mathematically, it's represented by the function \(f(x) = a \cosh(x/a)\), where \(a\) is a constant parameter determining the curve's shape. Here, \(\cosh\) refers to the hyperbolic cosine function, an analog of the cosine function for hyperbolic geometry.

Understanding its properties reveals why the power line in this exercise takes on this shape. When suspended only by gravity, without other forces, a cable naturally assumes a catenary shape. By choosing the right value for \(a\), the curve can be manipulated to fit specific conditions, such as ensuring a certain amount of sag. The problem we solve involves finding \(a\) to create a specific sag in a power line, helping to manage tensions and reduce risks of possible interference from natural elements.

Arc Length Calculation

When it comes to catenary curves like the path of a power line, calculating the arc length is essential. The arc length \(L\) of a curve from \(-x\) to \(+x\) can be derived from the formula: \[ L = \int_{-x}^{x} \sqrt{1 + \left( \frac{df}{dx} \right)^2} \; dx \] In our particular exercise, we first need to compute the derivative \(\frac{df}{dx}\), which for the function \(f(x) = a \cosh(x/a)\) becomes \(\sinh(x/a)\). This derivative represents the slope at any point on the curve.

The integrand simplifies using the identity \(\cosh^2(u) - \sinh^2(u) = 1\), allowing us to rewrite the integrand inside the integral as \(\cosh(x/a)\). For our specific values, with \(a \approx 12.50\) obtained from previous steps, and boundaries at \(-50\) to \(50\), you calculate the arc length, which translates to physically measuring the curve between the two support points of the power line. This calculation confirms the functional length of the line as it sags.

Equation Solving

One crucial step in solving for our power line's sag involves effectively handling hyperbolic equations. For our initial task, we derive an equation: \[ \cosh(50/a) - 1 = 10/a \] This equation represents the condition that the line sags 10 feet at the midpoint. To simplify solving, we introduce a substitution, defining \( t = 10/a \). This transforms our problem into solving \( \cosh(5t) - 1 = t \).

With substitution, finding the value of \(t\) becomes more straightforward, especially when using numerical methods or graphing utilities that can handle such equations effortlessly. Solving this equation graphically involves plotting both sides as separate functions or plotting theirs as a shared graph, watching for intersections. These intersections provide the solutions for \(t\), and with precision tools like a graphing calculator, we hone in on \( t \approx 0.800 \).

Finally, reversing our substitution \( a = 10/t \) gives us \( a \approx 12.50 \), completing our task in solving for the right curve parameterization.

Graphing Utility

Graphing utilities offer powerful visualization methods for solving equations, especially those involving transcendental functions, like hyperbolic cosines. Here's how we use them in our exercise:

• Plot different sides of the equation, in this case, \(\cosh(5t) - 1\) and \(t\). A graphing utility can vividly display these functions on a single plane.
• Look for intersections of these graphs. An intersection point represents a solution, as at this point, the two expressions are equal. For our exercise, we seek the point where \(\cosh(5t) - 1 = t\).
• Fine-tune and zoom into the graph for more accuracy. Most graphing utilities allow you to pinpoint coordinates up to several decimal places.

Practically, such graphing helps confirm the value of \(t\) approximately as \(0.800\), which can then be used to find \(a\). These utilities convert complex algebraic challenges into visual insights, aiding problem-solving and enhancing understanding of mathematical relationships.