Formula #1: y=12000e0.001x
The first part is straight forward, but how to solve for x when you know y?
To input values for y and get an output for x we need to use logarithms, or more specifically the natural logarithm for e.
To simplify for explanation purposes; if the formula was y = ex then by taking the ln of each side we see ln(y) = ln(ex), the second part we can solve since we’re actually asking what power must e be raised to in order to equal ex. Obviously it is x. Then, we can solve for x by changing the value of y since now x = ln(y) which is an algebraic function we can plug into Excel.
All we have to do now is deal with the constants associated with the formula (the 18699.88 and the 0.00097763 - 0.001 was just an approx).
y = 12000e0.001x
Now take the natural log of each side and get this:
LN(y/12000) = 0.001x
Then we divide by 0.001 on each side
This is what the formula “ =LN(C8/C2)/D2” does for us where C2 and D2 are our constants and C8 is our changeable value of y.
Formula #2 and Inverse: y=0.003x2 + 10.2x+ 4000
To get the inverse of the formula this is how to proceed.
Looking at y=0.003x2 + 10.2x+ 4000 you can see that by subtracting y from each side you get
From http://mathworld.wolfram.com/QuadraticEquation.html A quadratic equation is a second-order polynomial equation in a single variable
So we see that our formula is now in classic quadratic equation form. THEN we do a quick search of the web to find http://phoenix.phys.clemson.edu/tutorials/excel/algebra.html to remember what we forgot in college Algegra.
Aslo at the site we find http://phoenix.phys.clemson.edu/tutorials/excel/algebra.html an example.
The general quadratic equation is given by .
Its solution is, of course, the quadratic formula, or
Say we are given the quadratic equation and we wish to find both of its solutions. There are always a possible minus and plus when using the quadratic equation, although in our case only one will apply. We can use Excel to help us do this and the screen shot below shows how that is done. (The spaces within the formula were added to make the formula easier to read.)
Of course, to solve another quadratic equation, all one needs to do is change the values of the constants a, b and c in cells A1, B1 and C1. There is no need to alter the formula and the new result is given immediately after the new values are entered!
So in this case a = 0.003, b = 10.2, c = (4000-y), and follow the directions from the information above.