The dist. between the towers of the main span of the Bridge is about 1280m; sag of cable halfway between the towers on a cold day is about 143m.
a) How long is the cable assuming it has an approx. parabolic shape? Represent the cable as a parabola of the form y = kx^2; determine k to @ least 1 dec. place.
b) On a hot summer day the cable is about 0.05%longer. By how much does the sag increase? Assume no movement of towers.
2006-11-27 18:34:19 · 4 answers · asked by mathstinks 1 in Science & Mathematics ➔ Mathematics
For a), we are asked to assume this is a parabola, even though on a real bridge the cable does not assume such a shape.
To solve for k, put in
143 = k * 640^2
k = 143 / 640^2
Do the computation yourself to at least one decimal place.
For b), we need to be able to compute the length of a section of a parabola. For that, we need the derivative of the curve,
f'(x) = 2x
The formula for the length is the integral of
sqrt(1 + 4x^2)
over the desired range. To try to integrate this, we can use trigonometric substitution, defining an angle theta such that
2x = tan(theta)
x = tan(theta) / 2
Now we take dx/d(theta) = sec^2(theta) / 2
sqrt(1 + 4x^2) = sec^2(theta) / 2
2006-11-27 19:04:53 · answer #1 · answered by ? 6 · 1⤊ 0⤋
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2016-12-10 17:34:18 · answer #2 · answered by Anonymous · 0⤊ 0⤋
y = 143/640^2x^2
2006-11-27 18:45:57 · answer #3 · answered by jacinablackbox 4 · 0⤊ 0⤋
it is not a parabolic function but it is an hyperbolic cosine function
2006-11-27 18:43:24 · answer #4 · answered by gjmb1960 7 · 0⤊ 0⤋