SOLUTION: 1. A bell rope, passing through the ceiling above, just barely reaches the belfry floor. When one pulls the rope to the wall, keeping the rope taut, it reaches a point that is thre

Algebra ->  Trigonometry-basics -> SOLUTION: 1. A bell rope, passing through the ceiling above, just barely reaches the belfry floor. When one pulls the rope to the wall, keeping the rope taut, it reaches a point that is thre      Log On


   

Question 147777: 1. A bell rope, passing through the ceiling above, just barely reaches the belfry floor. When one pulls the rope to the wall, keeping the rope taut, it reaches a point that is three inches above the floor. It is four feet from the wall to the rope when the rope is hanging freely. How high is the ceiling?

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First let's draw out the problem
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Now take the triangle and make a mirror copy like this:
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Now draw a curve from either endpoint like this:

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Notice how a circular sector is formed. If we take the rope and swing it 360 degrees, it will form a circle where the rope is the radius.

Also, if we add the 2 lengths 4 feet and 4 feet to get 8 feet or 96 inches, this forms a chord on the circle. Furthermore, the length 3 inches forms the sagitta of the circle (ie the length that is perpendicular to a chord).


Now it turns out that if we know the length of a chord of a circle and its corresponding sagitta, then we can find the radius of the circle. The radius can be found through the formula

r=%28c%5E2%29%2F%288s%29%2Bs%2F2 where "c" is the length of the chord and "s" is the length of the sagitta.


r=%28c%5E2%29%2F%288s%29%2Bs%2F2 Start with the given formula.


r=%2896%5E2%29%2F%288%2A3%29%2B3%2F2 Plug in c=96 and s=3.


r=%289216%29%2F%288%2A3%29%2B3%2F2 Square 96 to get 9,216


r=%289216%29%2F%2824%29%2B3%2F2 Multiply


r=771%2F2 Combine the fractions and simplify


So the radius is r=385.5 which means that the length of the rope is 385.5 inches or 32 feet 1 and half inches. Since the rope "barely reaches the belfry floor", this means that the length of the rope and the height of the ceiling are really close. So we can say that the height of the ceiling is approximately 385.5 inches or 32 feet 1 and half inches.