It is a variation problem.
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?
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=1.16A0.6
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.
In this problem, "y" represents the length of the river, while "x" represents the area of the basin. You correctly identified that the equation relating the length of the river to the area of the basin is y = k * x^0.6.
To determine the value of "k," you can use the information given that the Rio Grande is 3034 kilometers long and drains a basin of about 500,000 square kilometers. Plugging in these values into the equation, you have:
3034 = k * (500,000)^0.6
To solve for "k," you need to isolate it. Start by dividing both sides of the equation by (500,000)^0.6:
k = 3034 / (500,000)^0.6
Once you calculate this, you will find the value of "k," which represents the constant of variation in the equation.
Regarding your confusion about the dependent variable, in this case, the length of the river (y) is the dependent variable because it depends on the area of the basin (x). The equation is expressing how the length of the river varies as a function of the area of the basin.
Now, to find the approximate area of land drained by the Nile, which is 6700 kilometers long, you can use the same equation and solve for "x." Plugging in the length of the Nile (6700) into the equation, you get:
6700 = k * x^0.6
Since you know the value of "k" from the previous calculation, you can rearrange the equation to solve for "x." Raise both sides of the equation to the power of (1/0.6):
(x^0.6)^(1/0.6) = (6700)^0.6
Simplifying, you have:
x = (6700)^0.6 / k
Substitute the value of "k" that you previously calculated, and you can solve for "x" to find the approximate area of land drained by the Nile.