# John's question at Yahoo! Answers: the rate of change of pollution with respect to distance

#### MarkFL

Staff member
Here is the question:

The air pollution y(in parts per million) x miles away is given by y+2xy+x^2y=600.?

The air pollution y(in parts per million) x miles away is given by y+2xy+x^2y=600. Find the rate of pollution 10 miles away.
A. 1.0020
B. -1.997
C. -0.9016
D. 3.0924
E. None of the above

I have no idea how to do this but it has something to do with derivative because that's what were doing
I have posted a link there to this thread so the OP may view my work.

#### MarkFL

Staff member
Hello John,

We are given:

$$\displaystyle y+2xy+x^2y=600$$

If we solve for $y$, we find:

$$\displaystyle y\left(x^2+2x+1 \right)=600$$

$$\displaystyle y(x+1)^2=600$$

$$\displaystyle y=600(x+1)^{-2}$$

Differentiating with respect to $x$, we obtain:

$$\displaystyle \frac{dy}{dx}=-1200(x+1)^{-3}$$

Hence:

$$\displaystyle \left.\frac{dy}{dx} \right|_{x=10}=-\frac{1200}{1331}\approx-0.9016$$

Thus, C.) is the correct answer.