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 147781: 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 draw a straight line from the top of the tower to the floor. Denote the length as "x":

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Now pull back the rope, this will angle the rope

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Draw a triangle between the endpoint of the drawn rope to the rope's resting position. Since the horizontal length between the resting position and the new position is 4 feet (48 inches) be sure to draw that in:

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Now draw a rectangle right underneath the triangle like this. Take note that the height of this rectangle is 3 inches. Also, notice how the vertical leg of the triangle is x-3 inches (subtract the height 3 from the rope length x to get x-3 inches):

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So we now have a triangle. We can use Pythagorean's Theorem to find the length of the rope (which is the hypotenuse)


a%5E2%2Bb%5E2=c%5E2 Start with Pythagorean's Theorem


%28x-3%29%5E2%2B48%5E2=x%5E2 Plug in a=x-3 (the length of the vertical leg), b=48 (the length of the horizontal leg), and c=x

%28x-3%29%5E2%2B2304=x%5E2 Square 48 to get


x%5E2-6x%2B9%2B2304=x%5E2 Foil


-6x%2B9%2B2304=0 Subtract x%5E2 from both sides.


-6x%2B2313=0 Combine like terms on the left side.


-6x=0-2313 Subtract 2313 from both sides.


-6x=-2313 Combine like terms on the right side.


x=%28-2313%29%2F%28-6%29 Divide both sides by -6 to isolate x.


x=771%2F2 Reduce.


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Answer:

So the answer is x=771%2F2


Which approximates to x=385.5


So the length of the rope (and the consequently the height of the ceiling) is 385.5 inches which is 32 1/8 feet or 32 feet 1 and a half inches.