Where is the water bill? It is a Piecewise Function!

Objectives

Students will:
(1.) Discuss piecewise functions.
(2.) Discuss real-world applications of piecewise-functions.
(3.) Solve problems involving the application of piecewise functions.
(4.) Meet one of the learning objectives of the VCCS (Virginia Community College System) standards for:
MTH 161: PreCalculus I
(Presents topics in power, polynomial, rational, exponential, and logarithmic functions, and systems of equations and inequalities.).

MTH 167: PreCalculus with Trigonometry
(Presents topics in power, polynomial, rational, exponential, and logarithmic functions, systems of equations, trigonometry, and trigonometric applications, including Law of Sines and Cosines, and an introduction to conics.).

(5.) Meet the QM (Quality Matters) and USDOE (United States Department of Education) requirements for distance education as regards the provision of RSI (Regular and Substantive Interaction).
Federal Register: Distance Education and Innovation
St. John's University: New Federal Requirements for Distance Education: Regular and Substantive Interaction (RSI)
Student – Content Interaction: Very high
Student – Student Interaction: Flexible
Student – Faculty Interaction: High

Water Rates by Calhoun County Water Authority; State of Alabama, USA

As of the 7th day of July, 2018; the water rates by the Calhoun County Water Authority is found here
We shall focus on the Residential Rates. These rates exclude taxes.

                The information is written here for you:
                First 3,000 Gallons:                $17.35 Minimum Per Month 
                Next 2,000 Gallons:                 $5.24 per 1,000 Gallons  
                Next 5,000 Gallons:                 $3.96 per 1,000 Gallons  
                All over 10,000 Gallons:            $3.10 per 1,000 Gallons 
            

Calculate the water rates for the following consumption of water. (1.) $2500$ gallons
(2.) $3700$ gallons
(3.) $5000$ gallons
(4.) $7500$ gallons
(5.) $12000$ gallons
(6.) $4692$ gallons
(7.) $6456$ gallons

Solution: $1st$ Method: Manual/Arithmetic Method
This application has four pieces.
Let $g$ = gallons of water (in gallons)
Let $r$ = rate (in dollars per gallons)
(1.) $2500$ gallons falls in the first piece.
rate for $2500$ gallons @ $$17.35$ = $$17.35$

(2.) $3700$ gallons falls in the second piece.
Before we use the second piece, we have to go through the first piece first.
rate for $3000$ gallons @ $$17.35$ = $$17.35$
$3700 - 3000 = 700$
We need to find the rate for the remaining $700$ gallons
The first piece (for $3000$ gallons) is a constant rate. It is a flat fee.
However, the second piece is $$5.24$ per $1000$ gallons
The second piece is $$5.24$ per $1000$ (not per $700$) gallons

But, we have $700$ (not $1000$ gallons remaining). What do we do?
We have to use Proportional Reasoning Method to calculate the rate per gallons (rather than using the rate per thousand gallons)
Remember: per gallon means for $1$ gallon; and per $1000$ gallons means for a thousand gallons

Let us set up the Proportional Reasoning
Let $x$ be the cost per gallon

                dollars            gallons
                5.24                      1000
                x                         1
            

This means that:
$$ \dfrac{5.24}{x} = \dfrac{1000}{1} \\[5ex] 1000 * x = 5.24 * 1 \\[3ex] 1000 * x = 5.24 \\[3ex] x = \dfrac{5.24}{1000} \\[5ex] x = 0.00524 $$ Similarly;
$$ \dfrac{3.96}{x} = \dfrac{1000}{1} \\[5ex] 1000 * x = 3.96 * 1 \\[3ex] 1000 * x = 3.96 \\[3ex] x = \dfrac{3.96}{1000} \\[5ex] x = 0.00396 $$ And;
$$ \dfrac{3.10}{x} = \dfrac{1000}{1} \\[5ex] 1000 * x = 3.1 * 1 \\[3ex] 1000 * x = 3.1 \\[3ex] x = \dfrac{3.1}{1000} \\[5ex] x = 0.0031 $$
Can we re-write the application?

                First 3,000 Gallons:                $17.35 Minimum Per Month 
                Next 2,000 Gallons:                 $0.00524 per gallon 
                Next 5,000 Gallons:                 $0.00396 per gallon
                All over 10,000 Gallons:            $0.0031 per gallon
            

Ask students if they understood how the rates per gallon were calculated.

So, back to completing the second question:
rate for $700$ gallons @ $0.00524 per gallon = $0.00524 * 700$ = $$3.668$
Please do not approximate intermediate calculations especially if it deals with money!
rate for $3700$ gallons @ = 17.35 + 3.668 = 21.018
Now, you can round your final answer to the nearest cent.
NOTE: If your professor does not want you to round, do not round.
rate for $3700$ gallons = $$21.02$

(3.) $5000$ gallons falls in the second piece.
Before we use the second piece, we have to go through the first piece first.
rate for $3000$ gallons @ $$17.35$ = $$17.35$
$5000 - 3000 = 2000$
rate for $2000$ gallons @ $0.00524 per gallon = $0.00524 * 2000$ = $$10.48$
rate for $5000$ gallons @ = 17.35 + 10.48 = 27.83
rate for $5000$ gallons = $$27.83$

(4.) $7500$ gallons falls in the third piece.
Before we use the third piece, we have to go through the first and second pieces.
rate for $3000$ gallons @ $$17.35$ = $$17.35$
$7500 - 3000 = 4500$
rate for $2000$ gallons @ $0.00524 per gallon = $0.00524 * 2000$ = $$10.48$
$4500 - 2000 = 2500$
rate for $2500$ gallons @ $0.00396 per gallon = $0.00396 * 2500$ = $$9.90$
rate for $7500$ gallons @ = 17.35 + 10.48 + 9.90 = 37.73
rate for $7500$ gallons = $$37.73$

(5.) $12000$ gallons falls in the fourth piece.
Before we use the third piece, we have to go through the first, second, and third pieces.
rate for $3000$ gallons @ $$17.35$ = $$17.35$
$12000 - 3000 = 9000$
rate for $2000$ gallons @ $0.00524 per gallon = $0.00524 * 2000$ = $$10.48$
$9000 - 2000 = 7000$
rate for $5000$ gallons @ $0.00396 per gallon = $0.00396 * 5000$ = $$19.80$
$7000 - 5000 = 2000$
rate for $2000$ gallons @ $ per gallon = $0.0031 * 2000$ = $$6.20$
rate for $12000$ gallons @ = 17.35 + 10.48 + 19.80 + 6.20 = 53.83
rate for $12000$ gallons = $$53.83$

(6.) $4692$ gallons falls in the second piece.
Before we use the second piece, we have to go through the first piece first.
rate for $3000$ gallons @ $$17.35$ = $$17.35$
$4692 - 3000 = 1692$
rate for $1692$ gallons @ $0.00524 per gallon = $0.00524 * 1692$ = $$8.86608$
rate for $4692$ gallons @ = 17.35 + 8.86608 = 26.21608
rate for $4692$ gallons = $$26.22$

(7.) $6456$ gallons falls in the third piece.
Before we use the third piece, we have to go through the first and second pieces.
rate for $3000$ gallons @ $$17.35$ = $$17.35$
$6456 - 3000 = 3456$
rate for $2000$ gallons @ $0.00524 per gallon = $0.00524 * 2000$ = $$10.48$
$3456 - 2000 = 1456$
rate for $1456$ gallons @ $0.00396 per gallon = $0.00396 * 1456$ = $$5.76576$
rate for $6456$ gallons @ = 17.35 + 10.48 + 5.76576 = 33.59576
rate for $6456$ gallons = $$33.60$

Some students may ask if it is possible to have just one function that will find the rate for any gallon(s) of water?.
Or is it possible to find the rate for gallons of water that is contained in the second piece, without having to go through the first piece?

Solution: 2nd Method: Piecewise Function/Algebraic Method
What if we have to calculate the water rates for "several" gallons of water?
Do we have to solve this manually all the time? That will be time consuming!
We can write it as a piecewise function and use each function for the number of gallons that correspond to that piece.
Besides, writing it as a piecewise function helps us to write a computer program that will find the rate for any gallon of water.
This application has four pieces.
Let $g$ = gallons of water (in gallons)
Let $r$ = rate (in dollars per gallons)
$r = f(g)$
This can be written as: $r(g)$

For the first piece;
$r(g)$ = $$17.35$

For the second piece;
We have to "finish" with the first piece first
Then, we can multiply the remaining gallons of water by $0.00524$
$r(g) = 17.35 + 0.00524(g - 3000)$
$r(g) = 17.35 + 0.00524g - 15.72$
$r(g) = 0.00524g + 1.63$

For the third piece;
We have to "finish" the second piece first
Then, we can multiply the remaining gallons of water by $0.00396$
What is the complete gallons for the second piece? - It is $5000$
In other words, $5000$ is the end point for the second piece. $5000$ is included.
Let us find the rate for that end point, $5000$
$r(g) = 0.00524(5000) + 1.63$
$r(g) = 26.2 + 1.63 = 27.83$
So, $$27.83$ is the most that can be charged for the second piece.
Any remaining gallons over $5000$ would be multiplied by $0.00396$
$r(g) = 27.83 + 0.00396(g - 5000)$
$r(g) = 27.83 + 0.00396g - 19.8$
$r(g) = 0.00396g + 8.03$

For the fourth piece;
We have to "finish" the third piece first
Then, we can multiply the remaining gallons of water by $0.0031$
What is the complete gallons for the second piece? - It is $10000$
In other words, $10000$ is the end point for the third piece. $10000$ is included.
Let us find the rate for that end point, $10000$
$r(g) = 0.00396(10000) + 8.03$ $r(g) = 39.6 + 8.03 = 47.63$
So, $$47.63$ is the most that can be charged for the third piece.
Any remaining gallons over $10000$ would be multiplied by $0.0031$
$r(g) = 47.63 + 0.0031(g - 10000)$
$r(g) = 47.63 + 0.0031g - 31$
$r(g) = 0.0031g + 16.63$

We can now write the piecewise function as:
$$ r(g) = \begin{cases} $17.35; & \quad 0 \leq g \leq 3000 \\[2ex] 0.00524g + 1.63; & \quad 3000 \lt g \leq 5000 \\[2ex] 0.00396g + 8.03; & \quad 5000 \lt g \leq 10000 \\[2ex] 0.0031g + 16.63; & \quad g \gt 10000 \end{cases} $$ Let us recalculate all the questions using the Piecewise Function method.

(1.) $2500$ gallons falls in the first piece.
The rate for the first piece is $$17.35$
$\therefore$ the rate for $2500$ gallons of water = $$17.35$

(2.) $3700$ gallons falls in the second piece.
The rate for the second piece is $0.00524g + 1.63$
For $g = 3700$, $r = 0.00524(3700) + 1.63$
$r = 19.388 + 1.63$
$r = 21.018$
$\therefore$ the rate for $3700$ gallons of water = $$21.02$

(3.) $5000$ gallons falls in the second piece.
The rate for the second piece is $0.00524g + 1.63$
For $g = 5000$, $r = 0.00524(5000) + 1.63$
$r = 26.2 + 1.63$
$r = 27.83$
$\therefore$ the rate for $5000$ gallons of water = $$27.83$

(4.) $7500$ gallons falls in the third piece.
The rate for the third piece is $0.00396g + 8.03$
For $g = 7500$, $r = 0.00396(7500) + 8.03$
$r = 29.7 + 8.03$
$r = 37.73$
$\therefore$ the rate for $7500$ gallons of water = $$37.73$

(5.) $12000$ gallons falls in the fourth piece.
The rate for the fourth piece is $0.0031g + 16.63$
For $g = 12000$, $r = 0.0031(12000) + 16.63$
$r = 37.2 + 16.63$
$r = 53.83$
$\therefore$ the rate for $12000$ gallons of water = $$53.83$

(6.) $4692$ gallons falls in the second piece.
The rate for the second piece is $0.00524g + 1.63$
For $g = 4692$, $r = 0.00524(4692) + 1.63$
$r = 24.58608 + 1.63$
$r = 26.21608$
$\therefore$ the rate for $4692$ gallons of water = $$26.22$

(7.) $6456$ gallons falls in the third piece.
The rate for the third piece is $0.00396g + 8.03$
For $g = 6456$, $r = 0.00396(6456) + 8.03$
$r = 25.56576 + 8.03$
$r = 33.59576$
$\therefore$ the rate for $6456$ gallons of water = $$33.60$

Which of the two methods do you prefer?
What are your reasons?
What are the pros and cons that you see for each method?
Do you have any other method for solving Piecewise Function applications?

General Project Requirements

(1.) You may work as a group (peer tutor to teach/correct one another).
However, this is an individual project.
In other words, you will submit your own project.

(2.) No two projects should be the same.
(a.) You may not do any of my examples even if the rates of the company have changed. Use my examples as guides to completing your own.
(b.) You may not do any of the projects of my previous students. (Student Project samples are provided below. Use them as guides to completing your own.)

(3.) Research any real-world application of piecewise function.
(a.) All information used for this project should be verifiable on the direct website of the company/organization.
(b.) Textbook examples are NOT allowed.
(c.) Find any application that has at least 2 pieces, and has at least a function in each piece that includes the independent variable.

(4.) Write the complete address of the direct page of the website where you found the application.
(a.) *If the "direct" web address is too long, please shorten it by pasting the "complete web address" into www.tinyurl.com
* *This is only for traditional students (onsite) students*
*Generate a short address and write that address "as is".*
(b.) For online students, please copy and paste the link as is.
(c.) Please set the link to open in a new window.
(d.) Required:Please submit the name of the company, the direct web page that has the application, and the objectives, in the Projects: Company Names and Websites page in the Canvas course. I shall review and respond.

(5.) As a student, you have access to Microsoft Office suite of apps.
(a.) You can download and install these apps on your laptop/desktop. Please contact the IT/Tech support in your college for assistance if you do not know.
In that regard, the project is to be typed using Microsoft Word (not Google Docs).
As at the time of this writing (05/05/2023), the reason for using Microsoft Word over Google Docs is because Microsoft Word has the opening brace for piecewise functions but Google Docs does not.
The file name for the Microsoft Office Word project should be saved as: firstName–lastName–project
Use only hyphens between your first name and your last name; and between your last name and the word, project.
No spaces.

(b.) For all English terms (entire project): use Times New Roman; font size of 14; line spacing of 1.5


(c.) For all Math terms: symbols, variables, numbers, formulas, expressions, equations and fractions among others, please use the Math Equation Editor.
(i.) Set the font to Cambria Math; font size of 14; and align accordingly
(ii.) Insert a space after each each equation as applicable. Just make a good work that is organized and spacious.
(iii.) Align the functions in each piece of the piecewise function accordingly.







(d.) Include page numbers. You may include at the top of the pages or at the bottom of the pages but not both.


(6.) Mr. C (SamDom For Peace) wants you to do this real-world project very well.
Hence, he highly recommends that you submit a draft so he can give you feedback.
It is important that you submit your draft to me via email or in the Projects: Drafts page in the Canvas course.
If you do not mind other students reviewing the feedback I provide for your draft, then please submit your draft in the Projects: Drafts page in the Canvas course.
If you mind other students reviewing my feedback for your draft, please submit your draft to my school email.
Please follow the due dates listed in the course syllabus.

When you submit your draft, I shall review and provide feedback.
When everything is fine (after you make changes as applicable based on my feedback), please submit your work in the appropriate area in the Canvas course.
Only projects submitted in the Canvas course are graded.
Draft projects are not graded. In other words, projects submitted via email and/or in the Projects: Drafts page are not graded because they are drafts. Submitting drafts is highly recommended. If your professor gives you an opportunity to submit a draft, please use that opportunity.
Submitting drafts is not required. It is highly recommended because I want to give you the opportunity to do your project very well and make an excellent grade in it.
The actual project (not the draft) should be submitted in the appropriate area (Assignments page: Piecewise Function Project) in the Canvas course.

(7.) Research Skills: Cite your source properly. Use APA, MLA, or Chicago Manual of Style. Indicate the style you used.

(8.) Writing Skills: Write or type the main application entirely.

(9.) Mathematical Skills: Arithmetic This is the Arithmetic method.
Use random numbers to test the real-world application manually for each piece.
Show all work including intermediate calculations/values.
Write down your results.

(10.) Mathematical Skills: Algebra: This is the Algebraic Method (Piecewise Function Method).
Write a piecewise function for that application.
Test the piecewise function with the same random numbers that you used for the Arithmetic method.
Please NOTE:
(a.) The intermediate results should be the same for both methods.
(b.) The final results should be the same for both methods.
If either of the results are not the same, there are issues. Please fix them.

These are the required information.
You may or may not not use a table format.
Please be creative.

Required Information

Name:	Your name
Date:	The date
Instructor:	Samuel Chukwuemeka
Project:	Water Bill: Residential Rates
Company:	Calhoun County Water Authority (https://calhouncwa.com/rates.html) (Name of the company and the direct Website of company that has the piecewise function)
Objectives:	(1.) Calculate the water bill of the residents of Calhoun County in the State of Alabama within each range of specific water usage manually using Arithmetic method. (2.) Write a piecewise function of the residential rates. (3.) Recalculate the same water bill of the residents of Calhoun County in the State of Alabama within each range of specific water usage algebraically using the Piecewise Function method.
Information:	Write the direct information from the direct website
Arithmetic Method:	Test each piece manually
Piecewise Function:	Write the piecewise function of the information
Piecewise Function Method:	Test the same piece algebraically
Citation:	Indicate the type of citation format. Cite your source(s) accordingly.

Students Projects

First Sample: Omar: Blue Ridge Rural Water Company: Water Rates


Second Sample: Michelle: City of Akron, Ohio: Water Rates


Third Sample: John: City of Akron, Ohio: Water Rates


The teacher should guide each student to the successful completion of the project.
Let students know you are willing to help.

Checklist

Checklist for Piecewise Function Project: Water Bill
(1.) Project is a piecewise function that has at least 2 pieces, and has at least a function in each piece that includes the independent variable.
(2.) Title of project.
(3.) Name of Student.
(4.) Name of Instructor.
(5.) Objectives of project.
(6.) Name of company.
(7.) Direct website of company.
(8.) Information on website written as is
(9.) Arithmetic Method:
(a.) Numbers in each piece, including 0 are tested.
(b.) All work including conversions are shown.
(c.) Intermediate calculations and values (before rounding) are shown.
(d.) Final results (rounded) are written and unit (currency) is included.
(10.) Piecewise Function:
(a.) Variables are defined.
(b.) All work is shown to determine the algebraic function for each piece.
(c.) Piecewise function is developed and written well using beginning brace.
(d.) The function in each piece and the corresponding domain is written well using any of the approved formats including the use of semicolon or comma.
(e.) The functions in the piecewise function are aligned properly.
(11.) Algebraic Method (Piecewise Function Method):
(a.) The same numbers including 0 tested using the Arithmetic Method, are also tested using the Algebraic Method.
Testing each number implies that each number within the domain of each piece in the piecewise function is tested in the function for that piece.
(b.) Intermediate calculations and values (before rounding) are shown. Intermediate values must be the same as the Arithmetic Method.
(c.) Final results (rounded) are the same as the Arithmetic Method and the unit is included.
(12.) References:
(a.) The reference style is specified.
(b.) Sources are cited according to the specified reference style.
(13.) Page numbers are included in every page.
(14.) Times New Roman font, font size of 14, and line spacing of 1.5 is used for the project.
(15.) Cambria Math font, font size of 14, and the Math Equation Editor is used for all Math terms including numbers, arithmetic operations, symbols, variables, fractions, formulas, expressions, and equations among others.
(16.) Correct file name.

References

Chukwuemeka, S.D (2016, April 30). Samuel Chukwuemeka Tutorials - Math, Science, and Technology. Retrieved from https://precalculus.appspot.com/

Calhoun County Water Association. (n.d.). Calhoun County Water Association. Retrieved from http://www.calhouncwa.com/rates.htm