# I need help finding a formula!

### I need help finding a formula!

Hello,

I've been stuck with a math problem for a while, so I decided to ask for help. I have a function :

Y = $$\frac{mx}{k + x}$$

25 $$\le$$ m $$\le$$ 45
0 $$\le$$ k $$\le$$ 4000
0 $$\le$$ Y $$\le$$ 25
0 $$\le$$ x $$\le$$ 3000
x and Y are integers

I can calculate Y for any x by doing an experiment. The values of m and k are determined by 6 numbers $$z_{1 }$$,$$z_{2 }$$,$$z_{3 }$$,$$z_{a }$$,$$z_{5 }$$,$$z_{6 }$$. I can set the values of them before the experiment.The sum of these numbers must be equal to 9000 :

$$\sum_{a=1}^{6 } z_{a }$$ = 9000

All z are integers. They are all interchangeable, for example k (1000,1000,1000,1000,1000,4000) = k (1000,1000,1000,1000,4000,1000) = k (1000,1000,1000,4000,1000,1000) and so on.
The value of coefficient k depends on the distribution of numbers z, for example : k (1500,1500,1500,1500,1500,1500) $$\ne$$ k (1,1,1,1,1,8995). I want to know the law how k depends on the distribution of numbers z.
I've found out that k reaches the highest value of roughly 4000 when z are distributed at equal values of 1500. As I change numbers z, k decreases, and k reaches zero if 4 of 6 numbers z are set to zero.
( then the function Y = 25 ) The coefficient m behaves in the same manner, but differs from k in limits.

The first thing I tried to do is to increase one z slightly from the point k (1500,1500,1500,1500,1500,1500) by $$\triangle z_{1 }$$ and find the dependency $$\triangle$$k ($$\triangle z_{1 }$$), but if I increase $$z_{1 }$$ by $$\triangle$$z I must decrease other z by $$\triangle z_{1 }$$ because $$\sum_{a=1}^{6 } z_{n }$$ = 9000.( I can also decrease 2 numbers z by $$\frac{1}{2} \triangle z_{1 }$$ or 1 number z by $$\frac{1}{3} \triangle z_{1 }$$ and the other z by $$\frac{2}{3} \triangle z_{1 }$$ and so on.)

I found that function $$\triangle$$k = $$\triangle$$k( 1500 + $$\triangle$$z,1500 - $$\triangle$$z,1500,1500,1500,1500) approximates to polynom :

$$\triangle$$k = a$$\triangle z^{3}$$ + b$$\triangle z^{2}$$+cz + d

I do not know what to do next. My goal is to find function k = f (z). Any advice?
Guest