To vertical poles have heights of 7 feet and 12 feet....

Question

To vertical poles have heights of 7 feet and 12 feet. A rope is stretched from the top of each pole to the bottom of the other. Exactly how far above the ground do the ropes cross?

Answer 1

let 5 = distance between bases

let base of 7 foot pole be the origin and find equations of the two lines

one line has y intercept = 7, and slope = -7/5

equation is y = -7x/5 + 7

other line has y intercept = 0 and slope = 12/5

equation is y = 12x/5

the two lines cross when 12x/5 = -7x/5 +7

19x/5 = 7

x = 35/19 and y = 12(35/19)/5 = 12(7/19 = 84/19

they cross at (84/19, 35/19) = (4 8/19, 1 16/19)

they cross at almost 2 feet above ground.

try another distance between the bases, say 10 feet, then

one line has y intercept 7 and slope = -7/10

equation y= -7x/10 +7

2nd line has y intercept = 0, slope = 12/10 = 6/5

equation y = 6x/5

they cross where 6x/5 = -7x/10 + 7

19x/10 = 7

x = 70/19 y = 6(70/19)/5 = 6(14)/19 = 84/19 = 4 8/19 feet above ground

farther the bases area apart, the higher the intersection of ropes, but they can't be higher than 7 feet