As others have suggested: You are permitting spurious accuracy to intrude on your notion of how to anchor. It ain't all that complicated :-). "More (scope) is better" is always warring with "that's enuff!" and which of them must win will always be determined by the particular circumstances obtaining at the time you anchor. And on how circumstances are expected to change while you are there.
Obviously you can always change your scope to accommodate changing circumstances, so the "7-5-3 rule" is always just an approximation that gives you a place to start.
The fact that you are asking the question suggests that you are fairly new to the game
, and it's right that you should seek to learn from others. So FWIW here is my take: Say you have a five foot draft
. You should obviously never anchor where you boat (as it swings to the anchor) will have less than, say, 3 feet of water under the keel
at the lowest state of tide during your stay. So don't anchor where the water within the boats "circle of swing" will be less than 8 feet deep during your stay. If it's gonna get choppy, then add a coupla extra feet for bouncing up and down in the waves.
If your total rode is, say, 300 feet, then at a scope of 7 you can't anchor in water more than 40 ft deep at the HIGHEST state of tide you expect during your stay since 40 x 7 = 280. So there you have the limits (as defined by the contours on your chart) of the places you can anchor. No less than 8 feet at low water, no more than 40 feet at high water. That's really easy to remember. No math required :-)!
Once you are within those limits, set your hook on the longest scope you can. I don't bother with measuring in feet. Fathoms (the reach between your hands when you stretch our your arms) is close enuff. If I want, say, 180 feet of scope, I just haul 30 fathoms of rode out on deck
, lower the hook till it's "up'n'down", and pay out hand over hand as we back down. Paying out hand over hand means that I can feel what the hook's doing on the bottom. When I feel it "bite", I pay out more rapidly so the rode goes slack and permits the hook to lie still till all the rode is out. Then I belay the rode and back down hard to ensure the hook has indeed set.
I really don't care what length of rode is out at that point, because it's so easy to adjust it as circumstances change. How d'ye know that circumstances are changing? Well, that's what you keep an anchor watch for :-)
Now, there is more stuff to learn about, but that will get you off to a reasonable start :-)