Catenary arch The portion of the curve y =17/15 - cosh x that lies above the x-axis forms a catenary arch. Find the average height of the arch above the x-axis.

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Identify the function describing the arch: \(y = \frac{17}{15} - \cosh x\). The arch is the portion of this curve where \(y > 0\) (above the x-axis).

Find the points where the arch intersects the x-axis by solving \(\frac{17}{15} - \cosh x = 0\). This will give the interval \([a, b]\) over which the arch lies above the x-axis.

Set up the formula for the average height of the arch over the interval \([a, b]\): \(\text{Average height} = \frac{1}{b - a} \int_a^b \left( \frac{17}{15} - \cosh x \right) \, dx\).

Evaluate the integral \(\int_a^b \left( \frac{17}{15} - \cosh x \right) \, dx\) by integrating each term separately: \(\int_a^b \frac{17}{15} \, dx\) and \(\int_a^b \cosh x \, dx\).

Substitute the limits \(a\) and \(b\) into the evaluated integral, then divide by \((b - a)\) to find the average height of the arch above the x-axis.

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주요 개념

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Catenary Curve

A catenary curve is the shape formed by a flexible chain or cable hanging under its own weight when supported at its ends. It is described by the hyperbolic cosine function, y = a cosh(x/a), and differs from a parabola. Understanding the properties of cosh(x) is essential to analyze the given curve.

Average Value of a Function

The average value of a continuous function f(x) over an interval [a, b] is given by (1/(b - a)) times the integral of f(x) from a to b. This concept helps find the mean height of the arch by integrating the curve's height over the domain where it lies above the x-axis.

Definite Integration and Area Under Curve

Definite integration calculates the exact area under a curve between two points. For the catenary arch, integrating y = 17/15 - cosh(x) over the interval where y ≥ 0 provides the total height accumulation, which is then used to find the average height.