MHB How can I find the intersection point of two sticks dropped into a well?

[Ackbach]

Here is this week's University POTW:

-----

I have to admit this is something of an experiment. Please put aside your ego, and bear with me. You MUST record (be honest, now!) the amount of time it takes you to solve this problem. Any solution not including the time spent I will consider incorrect, and I will not publish it. Also, to be clear, I will not publish your times. If I get enough submissions, I will publish a summary of the time statistics. Indeed, if you spend time on this problem, but do not submit a solution, I'd love to have that data as well (again, I will not publish the specifics). In other words, this data I am regarding as anonymous.

There is a well, open at the top, of diamater 3 meters. Two sticks, one of length 4 meters, and one of length 5 meters, are dropped into the well, and they land as shown in the following figure:

https://www.physicsforums.com/attachments/3735._xfImport

How high is the intersection point of the two sticks from the bottom of the well?

-----

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[Ackbach]

This problem is on pages 5 and 6 of How to Solve It: Modern Heuristics, by Michalewicz and Fogel.

MarkFL and lfdahl answered this week's POTW correctly. Here is MarkFL's solution:

Spoiler

I will use coordinate geometry to solve the problem. If we orient our coordinate axes such that the lower lect corner of the well is at the origin, then the 5 m stick lies along the line:

$$y_1=-\frac{4}{3}x+4$$

And the 4 m stick lines along the line:

$$y_2=\frac{\sqrt{7}}{3}x$$

Equating the two, we obtain:

$$-\frac{4}{3}x+4=\frac{\sqrt{7}}{3}x$$

And solving for $x$, we find:

$$x=\frac{12}{4+\sqrt{7}}$$

Substituting into either line (I will choose $y_2$) we then find:

$$h=\frac{\sqrt{7}}{3}\cdot\frac{12}{4+\sqrt{7}}=\frac{4\sqrt{7}}{4+\sqrt{7}}$$

I don't really have enough time stats to publish.