It is a variation problem.

Question

I know what the answer is and how to get to the answer I just don't know how I got it.
From measurements on many rivers, geographers find that the length of a river that drains a particular "basin" of land is approximately proportional to the 0.6 power of the area of the basin. The Rio Grande is 3034 kilometers long, and drains a basin of about 500,000 square kilometers.
To get the particular equation expressing river length in terms of basin area, I know that the equation is going to be y=k*x^0.6 -> 3034=500000^0.6*k which results with y=1.16*x^0.6.
But why does y have to be 3034? I thought y had to be the dependent variable, and I thought the basin depended on how long the river is, not vice versa.
Also, if the river in the world is the 6700 kilometer Nile, approximatley what area of land does the Nile drain? (still the same question)
I think it is a logs question but how would you solve it?

Answer 1

First the dependent and independent variables are mathematical terms to express a relation. In real life, depending on which information is available, we can evaluate either one, directly or indirectly.
So if y is know, we can calculate x, and vice versa.

Now that you have established k, assuming the same value of k applies to the Nile, then the area A of the drainage basin would be given by:
6700=1.16A^0.6
As you said, take log on both sides:
0.6logA = log(6700/1.16)
I get A=1,859,351.
Check the numbers and units.