I have three variables $r$, $p$, and $ w$. The $w$ is weight, $r$ is distance, and $p$ is the number of people. Morover, I have $$r = aw^b,~~p = ae^{bw}$$ where $a$ and $b$ are constants.
Given a value of $r$ and $p$, what is the way to calculate $w$ per unit distance per unit people?
Any help would be appreciated.
EDIT:
To make the question clearer:
I have a whole set of data for $r$, $p$, and $w$. I've plotted two scatter-plot graphs, one for $r$ vs $w$ and another for $p$ vs $w$. The above equations are the trendlines for the plots. I want to know if given a value of $r$ and $p$, is it possible to use the trendlines to estimate the value of $w$?
Edit2:
You know $b$ and you have $$r = aw^b,~~p = ae^{bw}.$$
Your question is "Given a value of $r$ and $p$ , what is the way to calculate $w$ per unit distance per unit people?"
Thus given particular values $R$ and $P$ you need to calculate $W$ which fits $R$ and $P$ (i.e. $W, R, P$ must satisfy your equations). Then divide $W$ by $R P$ to get weight per unit distance per unit people.
If you divide your formulae for $r$ and $p$ you get
$$\frac{r}{p} = \frac{aw^b}{ae^{bw}} = \left( \frac{w}{e^w} \right)^b.$$
For a given $R$, $P$ you want to find $W$ for which it is true that
$$\frac{R}{P} = \left( \frac{W}{e^{W}} \right)^b.$$
Subtracting $\frac{R}{P}$ from both sides of this equation you want to find $W$ for which it is true that $$\left( \frac{W}{e^{W}} \right)^b - \frac{R}{P} = 0.$$
In other words, you have to find
$$W = x: f(x) = 0$$ where $$f(x) = \left( \frac{x}{e^x} \right)^b - \frac{R}{P}.$$
You can find $x$ such that $f(x) = 0$ numerically, e.g. via Newton's method.
When you have $W$ you can get weight per unit distance per unit people = $\frac{W}{R P}$.