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
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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) (Show Source): You can put this solution on YOUR website!
First let's draw out the problem
Now take the triangle and make a mirror copy like this:
Now draw a curve from either endpoint like this:
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
where "c" is the length of the chord and "s" is the length of the sagitta.
Start with the given formula.
Plug in and .
Square 96 to get 9,216
Multiply
Combine the fractions and simplify
So the radius is 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.
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