SOLUTION: 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 i

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Question 147768: 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?

(I made a diagram, but I don't really understand what the bell rope and belfry floor is. My answer was approx. 3.4 feet, but that doesn't sound correct to me. I found it by using triangles with x, 2x, and x square root 3.)

Found 3 solutions by stanbon, jim_thompson5910, scott8148:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website!
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?
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This one is hard to explain so be patient.
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Draw a rectangle and label the vertices a, b, c, d, starting at the
lower left corner and ending at the lower right corner.
Draw the diagonal ac
ae and ce are the length of the rope so triangle aec is isosceles
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Draw a right triangle on top of the rectangle using bc as the base.
Label the top-most point of the right triangle "e".
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bc = ad ; ad is the floor of the belfry and is 4 ft wide
ab and be are pieces of the rope which is hanging from point e.
ab = cd = (1/4 ft) or 3 inches
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Let be = ce = x (the length of the rope)
be = x-(1/4) ft
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Use Pythagoras to find x :
(x - (1/4)^2 + 4^2 = x^2
x^2 - (1/2)x + 1/16 + 16 = x^2
(1/2)x = 16 + 1/16
x = 32 + 1/8
x = 32 1/8 ft (length of the rope)
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Cheers,
Stan H.

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.

note: 32 feet 1 and a half inches is equal to 32 1/8 feet

Answer by scott8148(6628)   (Show Source):
You can put this solution on YOUR website!
"A bell rope, passing through the ceiling above, just barely reaches the belfry floor"
__ so the length of the rope is the height of the ceiling

the distance from the rope to the wall, the ceiling height minus 3 inches, and the length of the rope form a right triangle
__ with the rope as the hypotenuse

let x="length of rope" (and height of ceiling) in feet

by Pythagoras __ 4^2+(x-.25)^2=x^2 __ 16+x^2-.5x+.0625=x^2 __ subtracting x^2-.5x __ 16.0625=.5x __ dividing by .5 __ 32.125=x