Question 147777
First let's draw out the problem

<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/tower_1.jpg" alt="Photobucket - Video and Image Hosting"><br><br>


Now take the triangle and make a mirror copy like this:

<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/tower_2.jpg" alt="Photobucket - Video and Image Hosting"><br><br>



Now draw a curve from either endpoint like this:


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/tower_3.jpg" alt="Photobucket - Video and Image Hosting"><br><br>




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=(c^2)/(8s)+s/2}}} where "c" is the length of the chord and "s" is the length of the sagitta.



{{{r=(c^2)/(8s)+s/2}}} Start with the given formula.



{{{r=(96^2)/(8*3)+3/2}}} Plug in {{{c=96}}} and {{{s=3}}}.



{{{r=(9216)/(8*3)+3/2}}} Square 96 to get 9,216



{{{r=(9216)/(24)+3/2}}} Multiply



{{{r=771/2}}} 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.