Weekly Trivia Question #4.5

[mrbiggs]

This time it actually is a question, the key part being the singular form of "question". But it's a math question that I don't know the answer to, and I was interested if someone here had the ability to get it.

First off-NO CALCULUS for this question!

NO

CALCULUS

I usually wouldn't say that, but this problem is very easy with calculus, and then it just isn't fun, is it? Of course "no calculus" isn't very specific, so I'll put here what you can do:

Basic Algebra

Greek Geometry (Trigonometry (sine, cosine), and conics (parabolae, for ex.) are fair game)

Imaginary Numbers, and vector multiplication

Synthetic division, and methods like it

Finite and Infinite series, so long as their evaluation does not involve limits

Logarithms and Exponents

Please don't use anything whose proof requires calculus.

(These restrictions are not really arbitrary, they are what is normally taught in basic high school math courses.)

Because of these restrictions, if you get the answer, you're going to have to show your work. . PLEASE don't post the answer if you got it by cheating; it's much easier to get an answer when you know what you're looking for. Also, no guess and check. For example, the equation xsin(x) + 2x^2=0 is irreducable by the methods above (so far as I know), so in such a situation, you're not allowed to just guess until you get x.



OK, so now that we got that out of the way, the question is:

What is the length of a line segment connecting the center of a circle to a chord, when the curve splits the area of the circle into a 3:1 ratio, and the radius of the circle is 1?

OR, another way of saying it is:

If you have a cylindrical gas tank (Radius one foot) (mounted horizontally), and you put in the dip stick, how far down will it go when the tank is 1/4 full?


Two more things, I'd like to credit a caller into Car Talk with this question, and also, I'll say again, please don't post the answer until you have the proof. It's okay if you say that you can get to whatever using whatever method, but don't ruin it. Please.

[Black Jeebus]

What happened to actual trivia portions of the weekly trivia questions. It's much too late for math.

[mrbiggs]

Yeah, I'm sorry, I'll get back to those next week. This was just a busy week, so I did this instead.

[nctpbsoya]

If I understand the problem correctly:
the length of the line segment connecting the center of a circle to the cord which cuts the circle in a 3:1 ratio when the radius of the circle is 1 is 1/2.
Proof:
Since the ratio of the segment of the circle which is cut is 3:1, the angle subtended by the resulting arc is 120°. If the line segment is drawn from the center to the chord, this divides that 120° angle in half, making the resulting triangle a 30-60-90 triangle. For this triangle, the side opposite the 30° angle is 1/2 of the hypotenuse (1), therefore, the line segment is 1/2.

[mrbiggs]

It cuts the area, not the diameter.

[mollusk]

I have an answer in the form

1+sin(pi/4 -0.5*cos(-cos(-cos(-cos( ...(1)))))).

No calculus was involved in the solution.

It is pretty easy to get a parameteric equation for an angle that you have to solve for in the form

theta/2 + cos(theta)*sin(theta)/2 = pi/8.

If you can find theta, then the answer is 1+sin(theta) for the dipstick length or sin(theta) for the distance from the center to the chord.

Using a trig identity this can be rewitten as

2*theta + sin(2*theta)=pi/2.

Setting theta = pi/4 +delta, the resulting algebraic expression is

2*delta + cos(2*delta)=0 => 2*delta = -cos(2*delta).

The last expression can be solved by fixed point iteration. For example start with 2*delta = 1 and compute -cos(1). Use that as the new value and repeat, i.e. calculate -cos(-cos(1)). The resultant converges to the value of 2*delta that satisfies the equation. Now back substitute to find theta and hence the required length. The convergence is indepedent of the inital seed value in the iteration, so no guessing of the solution is involved. It takes and infinite number of -cosine evaluations to get the exact answer, however. My computer uses finite digit arithmetic, so it will machine converge in a finite number of iterations.

There might be a more elegant solution, but I don't see it straightaway.

BTW, the equation x*sin(x)-2*x^2=0 mentioned earlier has the obvious solution x=0.

[Westley]

Holy s#it I don't understand a single word that any of you said. Cool!

[mrbiggs]

Wow. Nicely done. Definate rep point.

But if someone can get an actual value, rather than an infinite series, then that'll get you a rep point as well.

[Westley]

This is why I fence as a hobby. fencing=no annoying complicated math. I'm allergic.

[mollusk]

There is math in fencing. You gotta be able to count to 5 and 15 (and 10 for us old geezers).

[Westley]

Aye. There's logic in fencing, especially Epee, which is why I love it.