Shadowfax

I have a twin sister. In utero, I had a much better placental connexion and weighed in (or weighed out?) about 50% heavier than her in the midwife's scales. But, normal boy vs girl development being what it is, for all the years that mattered she was about 2 years ahead of me. She was educated in the Arts Block [French and Linguistics BA Edinburgh, as you ask], and worked for several years in the publishing trade. It was, in other words, a long way from Science that she was r'ared. But you can't keep a curious mind from asking scientific questions. A tuthree days ago, she sent me this:

• Here's a mind bender. A woman 5 ft 6 1/2 in tall is walking on open ground at 500feet. Her shadow is 112ft6in long. It's 14 September in northern hemisphere. What time is it? 

That's interesting because it makes you think about which of these data are essential for cracking the puzzle and which are mere padding. Height above sea level, for example, is not relevant and it is only in the metaphorical sense that women throw a longer shadow than men. And we have to deprecate the use of imperial ft and ins because they make the math a bit harder. I straight away converted everything to metres and recalled the Mnemonic of Trig SOH-CAH-TOA. What we need to calculate is the angle on the left of the diagram above formed between two lines ended at The Sister's head and feet.  By extraplolating these behind her to the sun we can calculate Sol's height above the horizon. I've induced kids to use similar trigonometry to estimate the height of trees.

So here we have the sides opposite and adjacent to the angle so we can solve for tan θ = O/A = 1.67m / 34.3m = 0.05 which gives an answer of 2.78 degrees. The sun goes all the way round - 360^o - the earth in 24 hours or 1440 minutes. Throwing the shadow in question, the sun is 2.78/360 * 1440 = 11 minutes before sunset. Which was at 19.24 on the day and at the place where the experiment was carried out.

This is an example of where the calculations are extremely sensitive to the stated conditions. Not in a chaos-theory sense where Hurricane Irma is triggered by a herd of locusts throwing up dust in Mauritania which blows out to sea in a wisp of a vortex. But if you think about it, the angle subtended by my sisters head a moment before sunset is infinitely small and her shadow is infinitely long. Thus a one minute difference about that time will make a huge difference in the shadow length and therefore [what we are driving at] the estimate of what the time is.  As it was, I was ten minutes adrift in my estimate.
Shadowfax is the name of Gandalf's horse and has nothing to do with this at all.