Converting tabulated data to equation of multiple variables

Question

I've come across a mathematical problem while working on my school project. I've got tables for airplane takeoff distance which is dependent on airport altitude, temperature and mass of an airplane. The goal is to make one single equation which describes the tables instead of searching and interpolating result in tables. I've plotted XY graph for given (4) temperatures and applied best fit lines (X being altitude, Y distance), which results in exponential trendline equations showing that with variable temperature, the coefficient changes and exponential part remains "same". So I made XY for that coefficient and have put that equations together. My biggest concern is if that process (not sure how exactly name it) is anywhere close to correct solution. I'd be grateful for every suggestion (e.g. different software to use, decimal precision) Thank you :)) excel regression

Answer 1

From the good work you did, you basically showed that $$d=(a+b\, t)\,e^{c\,p}$$ with $a\approx 359$, $b\approx 3.20$, $c\approx 0.00086$.

Using these estimates, you could use a nonlinear regression using all the data points and get $$\begin{array}{clclclclc} \text{} & \text{Estimate} & \text{Standard Error} & \text{Confidence Interval} \\ a & 357.995 & 1.71386 & \{354.531,361.459\} \\ b & 3.19337 & 0.02991 & \{3.13291,3.25382\} \\ c & 0.0000873449 & \approx 0 & \{0.000086066,0.0000886239\} \\ \end{array}$$

I hope and wish that I properly copied the numbers. But, this is the idea : good preliminary work, analysis of the results for each isotherm, conclude and go to the full nonlinear regression.

Edit

In comments, you mention that you have these charts for different weights of the airplane. So, for each weight $w$, process the data in the same way as above to get for each of them $$d_w=(a_w+b_w\, t)\,e^{c_w\,p}$$ I suppose that you can directly fit the $d_w$ chart using as initial estimates the results of the first calculation.

Analyze the dependency of each parameter $(a_w,b_w,c_w)$ to $w$.

Assuming that they could be linear, use all data to fit $$d=\big((a_0+a_1w)+(b_0+b_1w)t\big)\,e^{(c_0+c_1w)\,p}$$