It's been a long time since I last studied maths but have a problem that's a little but too much for my out-of-practice mind! Maybe someone can help me with the problem? Forgive me for not explaining it very well, it's pushing my understanding!!
Problem scenario (if interested I'm programmatically trying to control some hardware in Java. The hardware has a motor that rotates at seconds-per-rotation 'Y'. It has *two* factors controlling the speed, x1 and x2. I need 3 formulas that approximate the speed:
1) Y in terms of x1 and x2
2) x1 in terms of y (WITHOUT a given x2)
3) x2 in terms of y (WITHOUT a given x1)
The complications are:
Problem 1: for a given Y there are many x1/x2 combinations that would work. So for the last two formulas there are many possibilities, and I suggest a relationship below to come to a solvable equation for 2 and 3. The reason being, I'll need to find an x1 and x2, given a Y. And there are infinite possibilities)
Problem 2: For a the last two equations, a given Y has up to two possible values of x.
So here are the bounds:
0 <= x1 <= 1
0 <= x2 <= 1
0.94 <= Y <= 16
Which leads me to an adequate equation 3, which will be used to find equation 2.
x2 = (y - 0.94) / (16 - 0.94)
This doesn't hold true for equation one, but is an acceptable solution for finding a suitable x2 for a given y. To find x1 for that Y is then fairly trivial. The problem is that it doesn't help find an answer for equation 1 as it doesn't always hold true.
I've done some playing around and have come to some formulas which hold quite true (remember tho, they're approximations) for certain values of x1 and x2. The problem I need to solve is finding equations 1 and 2 above.
When x1 = 0, y in terms of x2:
y = x2^2 + 0.2x2 + 0.94
When x1 = 1, y in terms of x2:
y = 15.07x2^2 - x2 + 0.94
When x2 = 0, y in terms of x1:
y = 17.5x1^2 - 3.5x1 +1.96
When x2 = 1, y in terms of x1:
y = 1.2x1^2 -0.22x1 + 0.94
So to repeat the problem, I need to know:
1) y in terms of x1 and x2
2) x1 in terms of y
And bare in mind that the figures I've arrived at are all approximations and may not meld well. Would anyone be kind enough (or feel challenged/bored enough!) able to solve these for me? Solving quadratics has never been a strong point of mine, and this is just a bit too complicated!!
Thanks in advance