Question 147781
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):


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/step4.jpg" alt="Photobucket - Video and Image Hosting"><br><br>




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



{{{a^2+b^2=c^2}}} Start with Pythagorean's Theorem



{{{(x-3)^2+48^2=x^2}}} Plug in {{{a=x-3}}} (the length of the vertical leg), {{{b=48}}} (the length of the horizontal leg), and {{{c=x}}}


{{{(x-3)^2+2304=x^2}}} Square 48 to get 



{{{x^2-6x+9+2304=x^2}}} Foil



{{{-6x+9+2304=0}}} Subtract {{{x^2}}} from both sides.



{{{-6x+2313=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=(-2313)/(-6)}}} Divide both sides by {{{-6}}} to isolate {{{x}}}.



{{{x=771/2}}} Reduce.



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Answer:


So the answer is {{{x=771/2}}} 



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.