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# Application of Functions in the Real World

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## Gajane Gopal

on 20 June 2013

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#### Transcript of Application of Functions in the Real World

Did you say functions?
Here are some functions that will be discussed:

1) Rational Functions

2) Logarithmic Functions

3) Trigonometric Functions

What is a Rational Function?
A rational function is a function that can be represented through an equation in this form:

f(x) = p(x)
g(x)

***both p(x) and g(x) are polynomial functions, where g(x) cannot equal to 0.
Some Real - Life Applications of
Rational Functions
There are many situations in the real world where an application of a rational function is used as a representation. There are three well - known situations that follow under this category of rational functions.
Using Rational Functions in the Medical Field
A rational function can be used to determine the concentration of a certain drug after a fixed amount of hours. This is the equation used to model this read world topic.
This topic is significant in our society because it is required to calculate the concentration of a drug injected in a patient in order to perform medical procedures. This is important in the medical field to be aware of how far the drug has affected the patient before starting treatment because the outcome can be influenced by the concentration of the drug. For example, surgeries and operations can only begin after a concentration of a drug reaches a desired level. Therefore, this function is helpful for doctors to determine after how many hours they should start surgery based on the concentration level they prefer.
1) Medical Dosage

2) Cost of Living

3) Economic Production of Goods

Let's focus on how rational functions can be used in the medical field
C(t) = 4t
t^2 + 2

Why is this important you ask?
Let's solve a real world problem!
What will the concentration, in ppm, of the drug be after 4 hours?

Use the given formula: where C = concentration in ppm and t = number of hours

C(t) = 4t
t^2 + 2
= 4 (4)
(4)^2 + 2
= 16
18
= 8
9

From calculating, we know that after 4 hours of injecting the drug, the concentration of the patient's blood will be 8 ppm.
9
Visual Representation

Let's visually see what the graph of this rational equation will look like.
Properties of This Graph :
Domain: {xER}

Range: {yER | y > 0 }

No x-intercept or y - intercept, passes through (0, 0)

Increasing Intervals: [0,2]

Decreasing Interval: [2, ~ ]

Now let's do the same for another function like logarithmic functions
What is a logarithmic function?
A logarithmic function can be described as the inverse of an exponential function. This is the logarithmic expression that represents the inverse of an exponential expression:

y = log x , b > 0 , b cannot equal 1
b

** When the graph of an exponential function is reflected over the line y = x, the inverse of the graph is formed which is in this case referred to as the graph of the logarithmic function.
Some Real- Life Examples of Logarithmic Functions

Logarithmic Functions can be used to do various, significant calculations in the real world. Here are some real-world scenarios which require the application of a logarithmic function.

1) Calculating pH levels
2) Richter Scale
3) Decibles for sound intensity
4) Finanical Applications
Let's look more into how logarithmic functions are used for pH levels
But how does this help in the real world?
Calculating pH with a logarithmic function
A logarithmic function can be used to calculate the pH of a substance based on the concentration of hydrogen ions. This is the logarithmic equation that is used for determining pH.

pH = - log [H+],
where H+ is equal to the concentration of hydrogen ions
Calculating the pH of a substance is significant for various purposes such as chemical, industrial, and medical. Individuals with a profession in chemistry can use the hydrogen ions of a substance to calculate pH so that they can distinguish an acidic solution from a basic solution. It is also necessary for the individuals in the industry to be aware of the pH of a solution to prevent hazards. Physicians also have to calculate the pH of a patient's blood for medical procedures. If the pH is not at desired level, either too high or too low, doctors have to consider treatments because it is not normal or healthy to have a low or high pH level.
Let's try calculating pH using a real world situation
What will the pH of a substance be if the concentration of the hydrogen ions is 0.001?
Let's use the logarithmic function given:

pH = - log [H+]
= - log [0.001]
= - (-3)
= 3
From solving the logarithmic equation, we can conclude that the pH of a substance with hydrogen ions that have a concentration of 0.001 is 3.
Visual Representation
This is the visual representation of the logarithmic equation used to calculate pH
Properties of this graph:
Domain : {xER | x > 0 }

Range : { yER | 0 <= y <=14 }

X - intercept : (1,0) , no y-intercept

Asymptote : x = 0

Increasing Interval: none

Decreasing Interval: always

What is a trigonometric function?
A trignometric function are the functions used to determine an angle. The most common trigonometric functions are sine, cosine and tangent. A trignometric function follows the form of this equation:

f(x) = a sin[k(x-d)] +c or f(x) = a cos[k(x-d)] +c

How does calculating daylight hours help us?
Let's do a calculation for daylight hours

Using the given formula to calculate daylight hours in the city of Boston, how many daylight hours will there be on the 100 th day of the year?

H = 3.1 sin(0.017(100) - 1.35) + 12.12
= 3.1 sin(1.7 - 1.35) + 12.12
= 3.1 sin(0.35) + 12.12
= 1.062983203 + 12.12
= 13.18

After calculating the number of daylight hours using this trigonometric function, we can state that on the 100 th day of the year, there will be sunlight for approximately 13 hours in the city of Boston.
Real - life calculations that involve trigonometric functions
Trigonometric functions can be used to calculate various topics that are significant in the real world.
1)

Calculating Temperature
2) Calculating Daylight Hours
3) Determining population growth
4) Modeling prey vs predators
5) Calculating the angel of inclination from the sun

Calculating daylight hours using trigonometric functions
The daylight hours for a certain city in a period of days can be calculated using a trigonometric function. The daylight hours for Boston is calculated using this sine formula:

H = 3.1 sin(0.017t - 1.35) + 12.12,
where t represents the day of the year and H represents the number of daylight hours
Visual Representation

This is the graph that represents the trigonometric function used to calculate daylight hours of Boston.
Characteristics of trigonometric graph
Domain: {xER | 0 =<x =< 365 }

Range: {yER | 9.12 =< y <= 15.22 }

No x-intercept

y-intercept = (0,9.12)

Increasing Interval: [0, 15.22]

Decreasing Interval: [15.22 , 9.12]
Now, we have finally learned about three important functions used in the real world and how they are represented visually.
By: Gajane Gopalakrishnan
It is important to calculate daylight hours for various purposes, but a significant reason to determine this variable is for religious prayers. Individuals who fast and want to end their fasting after the daylight hours have finished would want to know when they can end their fast so they can cook for their family and prepare a feast. Daylight hours is also important for farmers to know. It is more of an advantage for them to harvest crops and grow plants a few days before the time period where there will be days with the most daylight hours. This way their crops can gain a rich source of sunlight.
Works Cited
Parmanand, Jagnandan. Rational Functions in the Real World. SOPHIA. SOPHIA LEARNING. June 17 2013. <http://www.sophia.org/rationalfunctions-in-the-real-world/rational-functions-in-the-real-world--3-tutorial>

Melchior. Tanganyika.Nl.Forum. Jan 16 2008. phpBB. June 17 2013. <http://tanganyika.nl/plein/viewtopic.php?f=4&t=5996>

Leslie, Martin. "Using Trig Functions to Model Daylight Hours." June 17 2013. PDF file. <http://math.arizona.edu/~mleslie/solns/lab.pdf>
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