How is your mathematics skills?

Question

On a level plot of ground there stand two vertical posts above ground level. From the top of each pole a rope is stretched to an arbitrary point on the other pole. If given the height of the two poles and the two points, how far above ground is the point where the ropes cross?

I would like to thank everyone for their input. There have been two correct answers. However, no one has posted the shorter formula.

Can anyone explain why the distance between the two poles does not change the solution?

Answer 1

If you're assuming the lines are stretched taut, the solution is found using the equations of the lines, solving for y.

Let A = height of Pole A.
Let B = height of Pole B.
Let a = height on Pole B of bottom of rope to top of A.
Let b = height on Pole A of bottom of rope to top of B.

The height where the two ropes cross is:
(AB - ab) / (A + B - a - b).

Guide to proof:
Let d = distance between poles A and B.
One rope goes from (0, A) to (d, a). It has a slope of (a - A) / d and a y-intercept of A. The equation of the line is y = (a - A)x / d + A.
The other rope goes from (d, B) to (0, b). It has a slope of (B - b) / d and a y-intercept of b. The equation of the line is y = (B - b)x / d + b.
Solve the equations simultaneously. The y-coordinate of the solution's ordered pair is the height.

If the ropes have any slack at all, they form catenaries, the cosh function must be used, and I believe you would need to supply a value for the distance between the poles to have any meaningful answer at all.

Answer 2

Let the heights of the poles be A and B, and the points of attachment be a and b respectively to the first and second pole.

Additionally, let the horizontal distance between the poles be d.

Hence, we have points (0, A) and (d,b) for the first pole.

The straight line equation is,

y = (b - A)/(d - 0) x + A ... (1)

Similarly, we have points (0, a) and (d, B) for the second pole.

The straight line equation is,

y = (B - a)/(d - 0) x + a ... (2)

Equating (1) = (2),

(b - A) / d x + A = (B - a) / d x + a

Solving,

(b - A) x + Ad = (B - a) x + ad

x (A + B - a - b) = (A - a)d

x = (A - a)d / (A + B -a -b) ... (3)

Substitute this into equation (1),

y = (b - A) / d [(A - a)d / (A + B - a - b)] + A

y = (b - A)(A - a) / (A + B - a - b) + A

y = (Ab - A^2 - ab + Aa + AB + A^2 - Aa - Ab) / (A + B - a - b)

y = (AB - ab) / (A + B- a - b)

So, the height of the intersecting point, h is given by:

h = (AB - ab)/(A + B - a - b)

It is interesting to note that h is independent of the distance between the poles.

Answer 3

Let us attach a co-ordinate grid to the ground.

Consider the first pole with height h1. The line from the other pole connects it at 0 < p1 < h1. The second pole is m units to the right with height h2 and the line from the other pole connects at 0 < p2 < h2

The points for the first pole in coordinate terms are
(0,0) bottom
(0,p1)
(0,h1) top

For the second
(m,0) bottom
(m,p2)
(m,h2) top

The lines assuming no slack are straight lines containing the middle and opposite top points.

L1 (skipping work)

y = h1 + (p2-h1)/m * x

L2 (also skipping work

y = p1 + (h2-p1)/m * x

We wish to find the point (x',y') where they intersect. y' is the desired height.

Setting equal.

h1 + (p2-h1)/m * x = p1 + (h2-p1)/m * x

and simplifying

x = m*[ p1 - h1 ] / [ p2-h1+p1-h2 ]

substituting for y we get

y = h1 + (p2-h1)/m * [ m*[ p1 - h1 ] / [ p2-h1+p1-h2 ] ]

= h1 + [p1-h1] *[p2-h1] / [ p2-h1+p1-h2 ]

Clearly we need p2 - h1 + p1 - h2 /= 0

If it were 0 then it would mean that the lines are parallel

Also 0 <= [ p1 - h1 ] / [ p2-h1+p1-h2 ] <= 1 otherwise the intersection is not between the two lines and thus they dont intersect.

EDIT Labelled for the whiner.

Answer 4

I'm assuming that you meant to say that we are to neglect the slight curvature of the stretched ropes. Then this can be solved with simultaneous linear equations. Not so hard.

Answer 5

If anyone were to answer that it would sound so complicated because they would have to explain the variables throughout the solution. You would lose track and so would the poor bugger trying to explain it. We need a diagram with variables and so forth. You get an "F" for setting math questions.

Answer 6

I am way to lazy....

But here's a tip.

H1 and H2 equal the pole heights
and I1 and I2 can equal the insertion points.

find the equation.

Answer 7

You need to give height details.

Answer 8

Really Good!

Answer 9

My math skills are prett good...it would help since I am studying math.

Answer 10

Q: How is your mathematics skills?

A: Better than your grammar skills.