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Numerade Educator

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Problem 44 Hard Difficulty

(a) The figure shows a telephone wire hanging between two poles at $ x = -b $ and $ x = b $. It takes the shape of a catenary with equation $ y = c + a \cosh (\frac{x}{a}) $.Find the length of the wire.
(b) Suppose two telephone poles are 50 ft apart and the length of the wire between the poles is 51 ft. If the lowest point of the wire must be 20 ft above the ground, how high up on each pole should the wire be attached?

Answer

a) Length of the wire $=2 a \sinh (b / a)$
b) The wire should be attached 24.36 $\mathrm{ft}$ above the ground

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Video Transcript

he is clear. So when you read here, so we're given why is equal to Steve us. Any closure of X divided by a and the length of the wire we're integrating from negative B to be square root of one plus hyperbolic function of sign that's divine of fighting square defense. This is equal to from negative B to B square root of caution square that's divided but a defense This is equal to from my gonna be to be waas the next departing by a defense. This is equal to hey, sign hyperbolic function. Next Divided by Ang, we're finding from negative beat B because evil tu tu es the hyperbolic function of Sign E over a. That's part A. And for part B, we know the wire is start that X is equal to zero. That's when it's the lowest. So why a zero when we plug it and become C plus A. And since the wire is 20 feet high with the lowest, the lowest point, C plus eight is equal to 20 were given that the wire ISS 51 feet. So we get to a fine high public function being divided by a is equal to 51 since we calculated this in part A in the distance between the two poles is 50 so to be is equal to 50. That makes be equal to 25. You put this into our equation to a the high public function of sign 25 divided, but a is equal to 51. You need a find for a we could use a calculator and we get around 72.4. So to find how high the pole should be attached, begin why off B is equal to see less a caution, the divine a frightening. We just plug things in and it becomes 20 minus a less a caution. You divide it by any we may be equal 25 a equal 72 point fly. We plug these both values insight and we get 24 0.3 six feet approximately