• edge333
In summary, the conversation is about simplifying an equation for calculating pollutant concentration as a function of two variables, in order to solve for one of the variables. The attempt at solving involves cancelling out a common term and considering the use of natural logs, but it is determined that logs may not be helpful in this case.
edge333

## 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 wp as a function of x. All variables except x and wp are known.

## Homework Equations

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

## The Attempt at a Solution

Basically, all I've gotten to simplify is that cpeak cancels such that:

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

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 wp as a function of x?

edge333 said:

## 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 wp as a function of x. All variables except x and wp are known.

## Homework Equations

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

## The Attempt at a Solution

Basically, all I've gotten to simplify is that cpeak cancels such that:

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

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 wp as a function of x?
WOW!

Logs aren't of much help in working with sums.

The only slight help I can see is that all the terms on the right have a common factor of $\displaystyle\ \exp\left(\frac{-u(w_p)^2}{4E_y\,x}\right)\ .$

## 1. What does "simplifying additive exponential terms" mean?

Simplifying additive exponential terms refers to the process of combining and reducing mathematical expressions that involve addition and exponents. This is done to make the expression easier to understand and work with.

## 2. Why is it important to simplify additive exponential terms?

Simplifying additive exponential terms is important because it allows us to solve complex equations and problems more efficiently. It also helps us to better understand the relationships between different terms and variables in a mathematical expression.

## 3. How do I simplify additive exponential terms?

• Combine like terms by adding or subtracting coefficients that have the same base and exponent.
• Use the laws of exponents to simplify terms with the same base but different exponents.
• If there are parentheses, use the distributive property to remove them.
• If there are negative exponents, use the rule x-n = 1/xn to rewrite them as positive exponents.
• Finally, combine any remaining like terms to get the simplified expression.

## 4. Can I use a calculator to simplify additive exponential terms?

Yes, you can use a calculator to simplify additive exponential terms. However, it is important to understand the steps involved in simplifying the terms by hand in order to ensure that the calculator is giving you the correct answer.

## 5. When do I use simplifying additive exponential terms in real life?

Simplifying additive exponential terms is used in various fields such as physics, engineering, and finance. It can help in solving problems related to population growth, compound interest, and exponential decay. Additionally, simplifying expressions is a fundamental skill used in higher-level mathematics courses and can be applied to real-life problem-solving situations.

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