When a cable is hung between two points part of it sags. The curve that the cabl

Question

When a cable is hung between two points part of it sags. The curve that the cable takes is modeled using catenary curves, which are generated with hyperbolic trig functions. The easiest possible model is
y=acosh(bx)

a and b are constants and cosh(z) is the hyperbolic cosine, "cosh(z) = (e^z + e^(-z)) / 2". The Golden Gate bridge in San Francisco is a suspension bridge whose cables can be modeled with such a function where b=1 / 640. The horizontal distance between the cable support towers is 1280 meters. At the middle of the cable the drop from horizontal (the so-called drape) is 143 m.

1.Determine a

2.Find the length of the cable.

3.On a hot day the metal in the cable expands by about 0.05%. How much more does the cable sag?

Explanation / Answer

cosh(x) function looks like

Note that at point x = 0 the function has value 1 that is its minima

If we multipy this by some parameter 'a' it will shift up or down depending upon sign and value of the parameter a. For instance if the value of a is 2. Then the curve will have its minima at y = 2

For the question given by you we got

y = acosh(x/640)

If we take the height of the ends as zero then the height at the middle will be -143.

Hence at x = 0 the height should be -143

hence a = -143 when height at ends is zero.

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