## Introduction

The Environmental Protection Agency is a body given the responsibility of coming up with legislation and procedures whose sole purpose is to protect the environment and human health. The policies target reversal of already dilapidated localities and preserving existing natural environment.

All policies in place ensure that there’s Environmental sustainability i.e. the current generation uses the available environmental resources without compromising the requirements of the future generations (Vezzoli & Manzini, 2008).

The agency in conjunction with partners from governmental and nongovernmental organization periodically conducts environmental assessment, research on environmental preservation, and education of the general public on the importance of environmental conservation.

Mathematical concepts including, but not limited to algebra are used to design models, calculate and estimate changes in factors that that may degrade or regenerate vital components of the environment.

## Objective

To demonstrate the use of algebraic functions- linear, constant and midpoint function- in environmental applications.

## Algebraic Function

A function is an equation that describes a relationship between two variables – an independent and independent variable (Research and Education Association, 2004). The variables themselves represent environmental variables

### Linear functions

Linear equations are used to obtain environmental solutions by plotting and analyzing lines representing environmental data. Thus knowing the model is sufficient to find the missing values and predict a trend or pattern.

They have straight line graphs and take the form y = f(x) = a + bx; where ‘x’ in the independent variable, ’y’ dependent variable and ‘a’ and ‘b’ are constants. The notation f(x) is used to clarify which function depends on the other.

#### Example

In a nuclear radiation process, if the frequency of the outgoing radiation is known the energy of the radiation can be calculated and hence safe distance for locating radiation experimentation facilities can be determined. This follows the linear relation

E= E_{0 }+ hf

where h is constant f frequency of radiation E_{0} is energy due to natural radiation.

The equation below is used in determining if urban drainage designs are up to standard to prevent spillage and pollution that may result from a given amount of rainfall. The runoff coefficient accounts for infiltration and other forms of rainfall losses that may occur.

Q = kCiA

Where: Q – peak flow m^{3}/s., k – conversion factor equal to 1.008, C – dimensionless runoff coefficient, i – rainfall intensity mm/hr, A – catchment area in acres or ha.

### Constant function

Constant functions don’t change i.e. they are fixed. They take the form of y= k; where k is a constant

The environmental agency requires funding to run its daily operations..assume that it costs $7000 per month, this can be represented as

C= $7000 where C is the cost

The natural proportion of the atmospheric gases without human interference is approximately constant. To test the level of pollution in a given locality, the variation of the proportion can be ascertained.

### Midpoint function

The midpoint function is used to obtain averages of two but of the same type of environmental variable. They have a general form of y=(X1 + X2)/2 where X1 and X2 are the two extreme variables under consideration.

For example the average rise in the global temperature as result of increased greenhouse gas emission in a year would given by

(T_{H} +T_{L}) /2 =T_{A}; where T_{H }is the highest temperature, T_{L} is the lowest temperature and T_{A} is the average temperature.

## References

Research and Education Association (2004) *Math Made Nice-n-easy: III. Factoring, ratios, linear equations, proportions, variations, functions*. New Jersey: Research & Education Association.

Vezzoli, C. & Manzini, E. (2008) *Design for environmental sustainability. *London: Springer.