SOLUTION: Two vertical poles have heights 6ft and 12ft. A rope is stretched from the top of each pole to the bottom of the other. How far above the ground do the ropes cross? Please spell ou

Question 178260This question is from textbook Geometry
: Two vertical poles have heights 6ft and 12ft. A rope is stretched from the top of each pole to the bottom of the other. How far above the ground do the ropes cross? Please spell out steps! This question is from textbook Geometry

Found 2 solutions by EMStelley, jim_thompson5910:
Answer by EMStelley(208) (Show Source): You can put this solution on YOUR website!
Hey, not trying to take your question off the list, but I just wanted to you to know that I've been thinking about it for quite a while and I feel like you need at least one more piece of information, such as the distance (on the ground) between the two poles. I could be wrong, but thought I'd give you my two cents.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
If we draw a picture, we get

Now let the horizontal distance from the poles be the value "a" (this variable is needed, but as you will soon see, it does not change the final answer). Also, label the top and bottom of both poles with coordinates (draw some coordinate axis if necessary)

So to find the height of the point where the ropes cross, we first need to find the equations of the lines (the red and blue lines)

Equation of the first line from (0,12) to (a,0) (red line)

Slope:

So the slope of the red line is

Start with the slope-intercept general equation

Plug in , and (since the red line goes through the point (0,12))

Multiply

Simplify

So the first equation (the equation of the red line) is

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Equation of the second line from (0,0) to (a,6) (blue line)

Slope:

So the slope of the blue line is

Start with the slope-intercept general equation

Plug in , and (since the blue line goes through the point (0,0))

Multiply

Simplify

So the second equation (the equation of the blue line) is

-----------------------------------------------------------

Now equate the two equations and

Multiply EVERY term by the LCD "a" to clear the fractions.

Add 12x to both sides.

Combine like terms.

Divide both sides by 18 to isolate "a".

Reduce. Note: this tells us that no matter what the value of "a" is, the crossing point will ALWAYS be of the horizontal length

Go back to the second equation

Plug in

Multiply

Reduce

So the point of intersection of the red and blue lines is for any value of "a"

Since the y-coordinate is 4, this means that the height of the crossing point is 4 ft.

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Answer:

So the ropes cross 4 ft above the ground regardless of the horizontal distance between the two poles.
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