Originally posted by MathurineCable hanging under their own weight take the shape of a catenary, which is expressed mathematically as the hyperbolic cosine function:
A cable, 16 metres in length, hangs between two pillars that are both 15 metres high. The ends of the cable are attached to the tops of the pillars. At its lowest point, the cable hangs 7 metres above the ground.
[b]How far are the two pillars apart?[/b]
f(x) = a*cosh(x/a) = (a/2)*(e^(x/a) + e^(-x/a))
We centre the low point of the catenary over the origin (x = 0), at a height of 7. The pillars lie at x = -(d/2) and x = (d/2). Subbing in (0,7), we get:
7 = (a/2)*(e^0 + e^0)
7 = a
Now, subbing in (d/2, 15), we get:
15 = (7/2)*(e^(d/14) + e^(-d/14))
(30/7) = e^(d/14) + e^(-d/14)
e^(d/14)*(30/7) = e^(2d/14) + e^(0)
e^(2d/14) - (30/7)*e^(d/14) + 1 = 0
e^(d/14) = (30/7) +/- (SQRT((30/7)^2 - 4))/2 = k
d/14 = ln(k)
d = 14*ln(k) = +/- 19.541 (approx.)
"d" cannot be negative (although in this case it doesn't matter since the negative value gives an answer identical to the positive value), so the distance between the pillars must be +19.541 (approx.). However, since the cable is only 16 metres long, it is not possible to string it up between the two pillars. (I think the constants need some adjustment 😉)
Originally posted by PBE6Maybe it's a big rubber band cable🙂
Cable hanging under their own weight take the shape of a catenary, which is expressed mathematically as the hyperbolic cosine function:
f(x) = a*cosh(x/a) = (a/2)*(e^(x/a) + e^(-x/a))
We centre the low point of the catenary over the origin (x = 0), at a height of 7. The pillars lie at x = -(d/2) and x = (d/2). Subbing in (0,7), we get:
7 = (a/2)*(e^0 ...[text shortened]... ble to string it up between the two pillars. (I think the constants need some adjustment 😉)
Originally posted by geepamoogleGeez! Two goof-ups in a row. I was really off yesterday. 😳
I've seen one like this before, and you don't need anything other than your basic add/subtract.
The 16 metre cable hangs down 8 metres, and hence there is only one easy-to-find solution (no special math functions needed).
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Originally posted by PBE6your hilarious😲🙄
Cable hanging under their own weight take the shape of a catenary, which is expressed mathematically as the hyperbolic cosine function:
f(x) = a*cosh(x/a) = (a/2)*(e^(x/a) + e^(-x/a))
We centre the low point of the catenary over the origin (x = 0), at a height of 7. The pillars lie at x = -(d/2) and x = (d/2). Subbing in (0,7), we get:
7 = (a/2)*(e^0 ...[text shortened]... ble to string it up between the two pillars. (I think the constants need some adjustment 😉)
2 edits
Originally posted by PBE6I think PBE6 got the correct answer since the question asks how far apart the pillars are.
However, since the cable is only 16 metres long, it is not possible to string it up between the two pillars. (I think the constants need some adjustment 😉)
The answer, as stated by PBE6, is that it's impossible to string up a 16 metre cable "BETWEEN" 2 pillars and still have it be 7 metres above the ground at its lowest point.
2 pillars touching each other do not allow anything to be "BETWEEN" them.