- #1

edge333

- 16

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## Homework Statement

I have developed an equations for calculating some pollutant concentration as a function of x and y. I'm trying to simplify the problem so that I can write the equation for w

_{p}as a function of x. All variables except x and w

_{p}are known.

## Homework Equations

[itex]c(x,y=w_{p})=0.01 \cdot c_{peak} = c_{peak} \cdot \left[ exp \left( -\frac{ u \left( w_{p} \right)^{2}}{ 4 E_{y}x}\right) + exp \left( -\frac{ u \left( w_{p} - y_{1} \right)^{2}}{ 4 E_{y}x} \right) + exp \left( -\frac{ u \left( w_{p} - y_{2} \right)^{2}}{ 4 E_{y}x} \right) + exp \left( -\frac{ u \left( w_{p} - y_{3} \right)^{2}}{ 4 E_{y}x} \right) + exp \left( -\frac{ u \left( w_{p} - y_{4} \right)^{2}}{ 4 E_{y}x}\right) \right][/itex]

## The Attempt at a Solution

Basically, all I've gotten to simplify is that c

_{peak}cancels such that:

[itex] 0.01 = \left[ exp \left( -\frac{ u \left( w_{p} \right)^{2}}{ 4 E_{y}x}\right) + exp \left( -\frac{ u \left( w_{p} - y_{1} \right)^{2}}{ 4 E_{y}x} \right) + exp \left( -\frac{ u \left( w_{p} - y_{2} \right)^{2}}{ 4 E_{y}x} \right) + exp \left( -\frac{ u \left( w_{p} - y_{3} \right)^{2}}{ 4 E_{y}x} \right) + exp \left( -\frac{ u \left( w_{p} - y_{4} \right)^{2}}{ 4 E_{y}x}\right) \right][/itex]

Is there a way of simplifying the exponential terms so that I can take the natural log of the right side of the equation and solve for w

_{p}as a function of x?