Problem 33 - Essential Calculus

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 cable is represented by the value of the definite integral \(L\), which might require a calculator to evaluate.

Step 1: Calculate Derivative

First, calculate the derivative of the catenary function \(y=20 \cosh \frac{x}{20}\). Using the chain rule, the derivative \(\frac{dy}{dx}\) is \( \cosh \frac{x}{20} \).

Step 2: Compute Function for Integral

Next, substitute this derivative into the arc length formula. This will produce the function inside the integral: \(\sqrt{1 + (\frac{dy}{dx})^2}\). We've got \( \sqrt{1 + (\cosh \frac{x}{20})^2}\).

Step 3: Evaluate Integral

Lastly, evaluate the definite integral of the function from step 2 over the interval [-20, 20]. This will give the length of the cable, which is: \( \int_{-20}^{20} \sqrt{1 + (\cosh \frac{x}{20})^2} dx \).The integral can be a bit complex, numerical methods like a calculator might be needed to evaluate it. Let's denote it as L.