A Question for the Nerds Among You...

So here's the setup...

My dad was cutting hay today on our pivot. Well... half pivot, it's only an 80 (1/4 mile by 1/2 mile) so the sprinkler can't go all the way around, and becomes what is called a "windshield wiper" because it goes from one barrier and then reverses direction to the other like a windshield wiper.

Anyway... he asked me how many acres he was cutting with each pass, something he'd been trying to figure out as he was cutting along in order to determine the half way point. Now I've got to explain one other thing... the proper name of a pivot is actually a "center pivot" which means that the sprinkler pivots around a central point, which is called the pivot point. Now our pivot has 9 towers which means there are 9 arcs equidistant from the center creating 9 concentric half circles radiating from the pivot point. In order to cut the field, you don't go directly back and forth because you'd have to jump the pivot tracks each time. Instead you follow the arcs.

I'm going to simplify the situation because it's not really that important right now to know that I can make approximately 9 windrows per wheel track, what is important is to know that the swather has a 14 foot header which makes each cut a 14 foot swath.

Now to answer his question, I took out my celly and simply divided 43560 (sq feet per acre) by 14 feet (the width of the swath) and that means every 3111 feet you travel you cut one acre. Or in other words, every 59 hundredths of a mile you cut an acre of hay (3111/5280 = .589). However... this doesn't answer his question. It's a helluva lot more complex than that. So I started thinking about it as I was baling this afternoon.

You could do it by a brute force attack. The first, or outside arc, would be calculated simply by using 2*pi*r, where the radius would be 1/4 of a mile or 1320 feet (the distance from the pivot point to the edge of the field). That means that the outside arc makes you travel a distance of 8293.80 feet or 1.57 miles. That means the outside arc has you cut approximately 2.67 acres out of the 70 under the sprinkler. So you take that number and add it to the the next arc which would be 2*pi*1306 feet (1320 minus the 14 foot already cut) and would give you 8217.09 feet or another 2.64 acres for a cumulative total of 5.31 acres on your way to 35, but there's got to be an easier way.

Now if memory serves... one could use a sigma equation to determine the entire distance traveled, where the equation would be E = 2*pi*(1320-(14*n)) where n would range from 0 to 81 (9 swaths per pivot track for 9 towers). Now I haven't run this because I can't remember if I'm doing it correctly.

So here's where you all come in...

Tell me... are my equations wrong? Is there a better mathematical way to calculate what my dad was asking than to do a brute force attack? I haven't run the numbers, but from the look of things, do you suppose that there is a consistent loss of exactly .03 acres every arc and therefore you can extrapolate from there? Kick them thar brains inna gear an give me the answer peeps!

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Since nobody's bothered to even visit lately except the lovely Knight... I'll let you guess what the last mystery lyric was this time too.