Righty-o. Some images blurred so as to not spoil anyone else’s fun LOL.
Here’s the wrong way to do it, obvious in hindsight, with my approach drawn and the embarrassing result of not doing enough geometry and doing too much trig at the bottom (although I’m confident it’s correct…). You can sense from afar my despair at being confronted with the daunting task of simplifying that to the very tidy result they’ve asked for. I see no point in sharing the 4 very messy pages it took to arrive at this, save for mocking its complexity, and by extension, me:
Hey DTC!
Get well soon my dear! I’ve been stupid busy lately and haven’t logged on very often but popped in tonight in a hotel room and saw this post. Will be thinking about you and wishing you a speedy recovery!!
RMC
@DarkSpeedWorks bilateral carpal + cubital tunnel. surgery L arm 1/24/25 but still having major issues, v limited use of either hand, elbow bending not good
surgery follow up today , and R arm surgery possible soon
THANKS ALL can’t computer more right now but LOVE YOU thx @noodle_soup
Not being able to post to the LR is kind of a bummer. Not sure how you are handling not being in the pool. that must really suck. I hope you are coping with this as best you can.
hi…
I’m home from the surgeons office. She called the surgery on my left arm “partially successful” which is I guess what I had been thinking it was because it relieved some pain but replaced it with horrible tingling that we’re trying to control with meds. I will have the same surgery on my right arm on April 4th and we’re hoping it will respond a lot better. She has seen this happen where people who have bilateral carpal tunnel and cubital tunnel have one arm respond well and the other arm respond not so well. She’s seen the horrible tingling lasts for about a year with it decreasing to levels where people can tolerate it or at least they’re not in her office about it anymore.
Renorider, you nailed it with the second approach. First time I tried it I also took your first approach. Not great in an exam hall, and led resulted in a Sliding Doors moment that means I do what I do now instead of having led a completely different life with a different degree, career and family. Weird.
For some of the apparent short cuts:
- The first theta is a reference to the general formula for the area of a sector.
- Angle AOB is obviously 2pi/7. That means that angle AEB has to be half that: AB and E are on a circle with centre O, and there is a geometry theorem that states that the angle that A and B make at O is twice the angle they make on the circle.
- It’s slightly easier to use r rather than U/V, particularly as you know R = 2V.
- working out the area of a triangle using 1/2(a.b.sinC) works a bit better here than 1/2 base x height.
I’m hoping it provided Dr T with a little bit of entertainment!
I missed/forgot this shortcut you used: there is a geometry theorem that states that the angle that A and B make at O is twice the angle they make on the circle so I had to get there the longer way.
And also for the r vs. U/V shortcut, once my head was in the right space, I settled on U/V instead of r (and also 1/2 * b * h instead of 1/2 * a * b * sin(c)) as it avoided the messier expression of r in terms of R and directly yielded the 1/tan(3*pi/7) term without the double angle identity. I think I was so traumatized by the first approach that I wanted that clean tangent term without any sorcery LOL.
But of course, because math is righteous and awesome, there are multiple ways to get there, as you’ve shown!
I’ve offered this problem as a $20 bounty to my employees. No takers yet.
hanging in there
hoping lab work shows something that would crack this, we have illness in mind, positive test for that would be good news as is treatable and would likely solve pain
keep fingers crossed for me, pray, please
(banned)