Exactly how far above the ground do the ropes cross?

Question

Two vertical poles have heights of 7 ft and 12 ft. A rope is stretched from the top of each pole to the bottom of the other.

Answer 1

Say the separation of the two poles is d.  Let the base of the 7-ft pole be the origin of a coordinate system and the y-axis run vertically along it.

 

 

Then one of the ropes go from (0,7) to (d,0).  The other go from (0,0) to (d,12).

 

 

Find the linear equations for both ropes and set them equal to find the height where they cross. 

 

 

 

Ans: (7 x 12)/(7 + 12) = 84/19 ft