Going off on a tangent.....

[BobG]

Eddie has an interesting wind guage that he can see from his office window. The wind guage is a helium balloon anchored to a wire. The ends of the wire are anchored to pegs in the ground (the wire is longer than the distance between the pegs). If there's no wind, the balloon sits midway between the pegs, pulling upward on the wire. If one were looking at the the wind guage, the wire and the ground between the pegs would form a triangle. When the wind blows, the triangle can tilt one way or the other. Depending upon the speed and direction of the wind, the balloon can slide along the wire, creating a triangle of a different shape. In fact, if the wire blows hard enough in the proper direction, the balloon can lie flat on the ground, so the wires just lay on top of each other, one side obviously much shorter than the other.

Eddie watches this every day, and eventually starts recording details to see if he can figure out what each angle means in terms of wind speed. The very first day, he notices something very interesting.

If he adds the tangents of all 3 angles in the triangle formed by the balloon wires and the ground between the pegs, they add up to 6.5.

What product is obtained if the tangent of all 3 angles are multiplied together?

Bonus questions:

If the pegs are 100 inches apart, what is the minimum length of wire needed (plus or minus 1 inch) for the sum of the tangents to equal 6.5?

The maximum length (plus or minus 1 inch)?

[Gokul43201]

product of tgts = 6.5 too !

[Gokul43201]

Bonus : min length, about 302" ? EDIT : 301"

don't believe there's a max !

[BobG]

Wow, that one lasted a long time.

I figured at least a few might know the first one without even having to think about it and not be that hard for those who didn't know to figure out, which is why I added the bonus.

Except the 302" is the max, not the min.

[Gokul43201]

Except the 302" is the max, not the min.

I think not...

Take a really long rope, so that the length of rope >> 100".

When the rope falls flat on the ground (super strong wind) the sum of tgts = 0 (this is indep't of the rope length).

When the rope makes an isoceles triangle (no wind) the sum of tgts --> infinity, because of 2 nearly right angles at the base.

Since I can get SUM = 0 in one position and SUM = inf. at another, there must be an intermediate position, where SUM = 6.5.

So there is no maximum length...unless there's an error in my argument.

[BobG]

Because the balloon can slide along the wire, the two sides can be different lengths.

If the sides of the triangle are almost on the ground, one of the angles is almost 180, the other two are barely over 0. In other words angles in a triangle can range from >0 to <180. You can never have more than one angle 90 degrees or greater in a triangle. If 90 degrees, tangent is undefined, so that eliminates him. If greater than 90, tangent is negative, which eliminates all of those triangles.

For all isoceles triangles, the wire can range from >100" to infinity. Only one of those lengths give you tangents with a sum of 6.5, which is what you found. It doesn't account for the triangles with legs of different lengths in which all of the angles are less than 90 degrees.

It's probably also worth noting that the angle between the two legs has to be less than 90 degrees, which reduces the possibilities even further.

[Gokul43201]

I'm not trying to argue here...but you have neither shown where my proof is wrong, nor have you shown why 302" is the max length.

What if I tell you that I've just drawn a triangle where the rope legth is 500" and the SUM = 6.5 ?

[BobG]

If you said the wire were 500"....

I would say the shortest leg had to be >240" because if one leg were 240", you would have a 100-240-260 triangle (a 5-12-13 triangle multiplied by 20). You would have one angle equal to 90 degrees and the tangent would be undefined. If one leg were shorter than 240", one angle would be greater than 90 degrees, meaning the product of the tangents would be negative.

If a 240.000001 - 259.999999 - 100 triangle, the angles opposite those sides would be 67.38, <90, 22.62 (approximately). The sum of the tangents is astronomical since one angle is so close to 90.

The smallest angles you can get with the 500" wire would be an isoceles triangle of sides 250-250-100. That gives you about 78.46, 78.46, 23.07 for the opposite angles. The tangent of 78.46 is about 4.898, the tangent of 23.07 is about .426. The sum is about 10.22 - too high.

The lowest possible sum of tangents for any triangle that includes a 78.46 degree angle is 7.35 and that would be for an isoceles triangle in which the 78.46 degree angle were opposite the base.

[Gokul43201]

You're right. The error in my argument is that there is a non-analytic point (when one base angle is 90) between the 2 end cases I described, about which point the SUM flips from -inf. to + inf.

Sorry for troubling the hell out of you...my bad.

[BobG]

No problem. I was just pleased to find a challenging one.

I got this from a slide rule group, except they didn't ask for the minimum - not that it's not doable, it's just that the minimum isn't that fun on a slide rule. Finding the max is almost a trial and error method - kind of visually zeroing in on the right answer.

You lose that on the minimum. I wound up narrowing the answer down to a manageable range and plugging numbers into the cosine law in a trial and error method. Fortunately, I'm really good at guessing and got the answer on my second try. If I hadn't been quite so good at narrowing the range, it would have taken me at least three tries.

[Hurkyl]

Going off on a tangent is always a bad sine...

[Gokul43201]

It took me 4 tries (of the base angle) 65, 70, 71, and 70.7 deg. Of course, the tangent of this angle is just the solution of a cubic.

[BobG]

Answer to minimum: 162.5 inches.

If the sum of the tangents is anything more than 6.5, the problem gets a lot tougher. The shortest length won't be for an isoceles triangle. (Actually, at 6.5, the middle is starting to bulge, which is the reason for the plus or minus 1 inch - when I first solved this, I figured the minimum length as just under 163 inches.)