Where is the light 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

Residential Service Rates by Georgia Power; State of Georgia, USA

As of the 30th day of December, 2018; the power rates for the Residential Service during the Summer Period (June - September) by Georgia Power is found in the Georgia Power Bill Calculator of the Georgia Public Service Commission.
The information is here
It is great to know that the Georgia Public Service Commission developed a Power Bill calculator.
We shall check our work with their calculator.
And of course...we shall also check our work with my calculator.

                The information is written here for you:
                Basic Service:          $10.00 
                Tier                    Usage                   Cost per KWh
                1st tier                up to 650KWh            $0.056582
                2nd tier                next 350KWh             $0.093983
                3rd tier                over 1000KWh            $0.097273 
            

Calculate the power costs for the following consumption of power. (1.) $0$ KWh
(2.) $300$ KWh
(3.) $700$ KWh
(4.) $1000$ KWh
(5.) $1200$ KWh

Solution: $1st$ Method: Manual/Arithmetic Method
This application has three pieces.
The basic service fee is not a piece. It must be paid whether one consumed any power or not.
Let $p$ = power (in KWh)
Let $r$ = cost (in dollars per KWh)
(1.) $0$ KWh falls in the first piece.
Basic service fee = $$10.00$
cost for $0$ KWh @ $$0.056582$ per KWh = $0 * 0.056582 = 0$
$10.00 + 0.00 = 10.00$
cost for $0$ KWh = $$10.00$

(2.) $300$ KWh falls in the first piece.
Basic service fee = $$10.00$
cost for $300$ KWh @ $$0.056582$ per KWh = $300 * 0.056582 = 16.9746$
$10.00 + 16.9746 = 26.9746$
cost for $300$ KWh = $$26.97$

(3.) $700$ KWh falls in the second piece.
Basic service fee = $$10.00$
Before we use the second piece, we have to go through the first piece first.
First Piece for $650$ KWh
cost for $650$ KWh @ $$0.056582$ per KWh = $650 * 0.056582 = 36.7783$
$700 - 650 = 50$
We need to find the cost for the remaining $50$ KWh
That takes us to the second piece.
Second Piece for $50$ KWh
cost for $50$ KWh @ $$0.093983$ per KWh = $50 * 0.093983 = 4.69915$
$10.00 + 36.7783 + 4.69915 = 51.47745$
cost for $700$ KWh = $$51.48$

(4.) $1000$ KWh falls in the second piece.
Basic service fee = $$10.00$
Before we use the second piece, we have to go through the first piece first.
First Piece for $650$ KWh
cost for $650$ KWh @ $$0.056582$ per KWh = $650 * 0.056582 = 36.7783$
$1000 - 650 = 350$
We need to find the cost for the remaining $350$ KWh
That takes us to the second piece.
Second Piece for $350$ KWh
cost for $350$ KWh @ $$0.093983$ per KWh = $350 * 0.093983 = 32.89405$
$10.00 + 36.7783 + 32.89405 = 79.67235$
cost for $1000$ KWh = $$79.67$

(5.) $1200$ KWh falls in the third piece.
Basic service fee = $$10.00$
Before we use the third piece, we have to go through the first piece and also through the second piece.
First Piece for $650$ KWh
cost for $650$ KWh @ $$0.056582$ per KWh = $650 * 0.056582 = 36.7783$
$1000 - 650 = 350$
We need to find the cost for $350$ KWh
That takes us to the second piece.
Second Piece for $350$ KWh
cost for $350$ KWh @ $$0.093983$ per KWh = $350 * 0.093983 = 32.89405$
$1200 - 1000 = 200$
We need to find the cost for the remaining $200$ KWh
That takes us to the third piece.
Third Piece for $200$ KWh
cost for $200$ KWh @ $$0.097273$ per KWh = $200 * 0.097273 = 19.4546$
$10.00 + 36.7783 + 32.89405 + 19.4546 = 99.12695$
cost for $1200$ KWh = $$99.13$

Some students may ask if it is possible to have just one function that will find the cost for any consumption of power.
Or is it possible to find the cost for the consumption of power that falls in the second piece, without having to go through the first piece?
Those are really interesting questions!
That is one of the reasons for studying piecewise functions ☺☺☺

Please specify the importance of not rounding intermediate calculations.
Please specify the importance of rounding only the final answer to two decimal place (because it is dollars and cents).


Solution: 2nd Method: Piecewise Function/Algebraic Method
What if we have to calculate the power rates for "several" consumption of power?
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 consumption of power 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 consumption of power.
This application has three pieces.
Let $p$ = power consumed(in KWh)
Let $c$ = cost per KWh or power consumed (in dollars)
$c = f(p)$
This can be written as: $c(p)$

For the first piece;
Basic service fee = $$10.00$
cost for $p$ KWh @ $$0.056582$ per KWh = $p * 0.056582 = 0.056582p$
$10 + 0.056582p = 0.056582p + 10$
$c(p) = 0.056582p + 10$

For the second piece;
Basic service fee = $$10.00$
We have to "finish" with the first piece first
cost for $650$ KWh @ $$0.056582$ per KWh = $650 * 0.056582 = 36.7783$
$10 + 36.7783 = 46.7783$
Then, we can multiply the remaining consumption of power by $0.093983$
$c(p) = 46.7783 + 0.093983(p - 650)$
$c(p) = 46.7783 + 0.093983p - 61.08895$
$c(p) = 0.093983p - 14.31065$

For the third piece;
Basic service fee = $$10.00$
We have to "finish" with the first piece first
cost for $650$ KWh @ $$0.056582$ per KWh = $650 * 0.056582 = 36.7783$
$10 + 36.7783 = 46.7783$
Then we have to finish with the second piece next
$1000 - 650 = 350$
cost for $350$ KWh @ $$0.093983$ per KWh = $350 * 0.093983 = 32.89405$
$46.7783 + 32.89405 = 79.67235$
Then, we can multiply the remaining consumption of power by $0.097273$
$c(p) = 79.67235 + 0.097273(p - 1000)$
$c(p) = 79.67235 + 0.097273p - 97.273$
$c(p) = 0.097273p - 17.60065$

We can now write the piecewise function as:
$$ c(p) = \begin{cases} 0.056582p + 10; & \quad 0 \leq p \leq 650 \\[3ex] 0.093983p - 14.31065; & \quad 650 \lt p \leq 1000 \\[3ex] 0.097273p - 17.60065; & \quad p \gt 1000 \end{cases} $$ Let us recalculate all the questions using the Piecewise Function method.

(1.) 0 KWh falls in the first piece.

$ c(p) = 0.056582p + 10 \\[3ex] c(0) = 0.056582(0) + 10 \\[3ex] = 0 + 10 \\[3ex] = 10 \\[3ex] $ cost for $0$ KWh = $$10.00$

(2.) 300 KWh falls in the first piece.

$ c(p) = 0.056582p + 10 \\[3ex] c(300) = 0.056582(300) + 10 \\[3ex] = 16.9746 + 10 \\[3ex] = 26.9746 \\[3ex] = 26.97 \\[3ex] $ cost for $300$ KWh = $$26.97$

(3.) 700 KWh falls in the second piece.

$ c(p) = 0.093983p - 14.31065 \\[3ex] c(700) = 0.093983(700) - 14.31065 \\[3ex] = 65.7881 - 14.31065 \\[3ex] = 51.47745 \\[3ex] = 51.48 \\[3ex] $ cost for $700$ KWh = $$51.48$

(4.) 1000 KWh falls in the second piece.

$ c(p) = 0.093983p - 14.31065 \\[3ex] c(1000) = 0.093983(1000) - 14.31065 \\[3ex] = 93.983 - 14.31065 \\[3ex] = 79.67235 \\[3ex] = 79.67 \\[3ex] $ cost for $1000$ KWh = $$79.67$

(5.) 1200 KWh falls in the third piece.

$ c(p) = 0.097273p - 17.60065 \\[3ex] c(1200) = 0.097273(1200) - 17.60065 \\[3ex] = 116.7276 - 17.60065 \\[3ex] = 99.12695 \\[3ex] = 99.13 \\[3ex] $ cost for $1200$ KWh = $$99.13$

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?



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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 project.

(b.) You may not do any of the projects of my previous students.
The Student Project samples are provided below. Use them as guides to completing your own project.

(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.) As a student, you have free access to Microsoft Office suite of apps.
(a.) Please download the desktop apps of Microsoft Office on your desktop/laptop (Windows and/or Mac only).
Do not use a chromebook.
Do not use a tablet/iPad.
Do not use a smartphone.
Do not use the web app/sharepoint access of Microsoft Office.
(Please contact the IT/Tech Support for assistance if you do not know how to download the desktop app.)
In that regard, the project is to be typed using the desktop version/app of Microsoft Office Word only.

(b.) The file name for the Microsoft Office Word project should be saved as: firstNamelastNameproject
Use only hyphens between your first name and your last name; and between your last name and the word, project.
No spaces.

(c.) For all English terms/work (entire project): use Times New Roman; font size of 14; line spacing of 1.5.
Further, please make sure you have appropriate spacing between each heading and/or section as applicable.
Your work should be well-formatted and visually appealing. first step

(d.) For all Math terms/work: symbols, variables, numbers, formulas, expressions, equations and fractions among others, the Math Equation Editor is required.
(i.) The font is set to Cambria Math by default (set it to that font if it is not); font size of 14, and align accordingly (preferably left-aligned).
(ii.) To ensure appropriate spacing between your Math work, use a line spacing of 2.0.
Alternatively, you may use line spacing of 1.5 but insert a space after each equation as applicable.
Your work should be well-formatted, organized, well-spaced (not compact), and visually appealing.

(iii.) Align the functions in each piece of the piecewise function accordingly.

second step

third step

fourth step

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

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

(6.) Writing Skills: Write the complete address of the direct page of the website where you found the application.
Write the main application completely.
Write all the information accoridngly as noted in the Example Guide and the Checklist.

(7.) (a.) Mathematical Skills: Show all work.
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.

(b.) Algebra: This is the Algebraic Method (Piecewise Function Method).
Define your variables accordingly.
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:
(i.) The intermediate results should be the same for both methods.
(ii.) 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. Use whatever format that is visually appealing.
Please be creative.

(8.) 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.

(a.) First: (Required): Please submit the required information in the Projects: Company Names and Websites page in the Discussions forum on the Canvas course.
An example was given in the forum.
The name of the company that has the application should be written.
The complete web address of the web page that has the application should be written.
The objectives should be written.
Please be very specific. In other words, I do not have to click on anything to see the application.
Once I visit the webpage that you provided, I must see the application directly on that page.
I shall review and respond.

(b.) Second: (Highly Recommended): When the company name and website is approved, please submit your draft.
Draft projects are not graded because they are drafts. They are only for feedback.
If your professor gives you an opportunity to submit a draft, please use that opportunity.
Submitting drafts is highly recommended. 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.

Please turn in your draft in the Discussions page → Projects: Drafts forum in the Canvas course (if you would like your colleagues to read my comments and avoid any mistakes that you made).
You may also send it to me via email (if you do not want your colleagues to see my comments and learn from the comments).
I shall review and provide feedback.
Then, review my feedback and make changes as necessary.
Keep working with me until I give you the green light to turn in your actual project. This must be done before the final due date to turn in the actual project.

When everything is fine (after you make changes as applicable based on my feedback), please submit your work in the appropriate area: Assignments page → Piecewise Functions Project in the Canvas course.
Only the projects submitted in the appropriate place in the Canvas course are graded.

(9.) All work must be turned in by the final due date to receive credit.
Please note the due dates listed in the course syllabus for the submission of the draft and the actual project. In the course syllabus, we have the:
(a.) Initial due date for the Project Draft: Please turn in your draft.

(b.) Initial due date for the Project: If your draft is not ready for submission, keep working with me. Make changes based on my feedback and keep working with me.
If you prefer not to turn in a draft, please review all the resources provided for you and do your project well and submit.


(c.) Final due date for the Project Draft: This is necessary if you want a written feedback for your draft.
After this date, written feedback would not be provided for your draft. However, verbal feedback would still be provided during Office Hours/Student Engagement Hours/Live Sessions.

(d.) Final due date for the Project: All work must be turned in by this date to receive credit.
After this date, no work may be accepted.

Required Information
Name: Your name
Date: The date
Instructor: Samuel Chukwuemeka
Project: Power Bill: Residential Rates
Company: Georgia Public Service Commission (for Georgia Power)
(http://www.psc.state.ga.us/calc/electric/GPcalc.asp)
(Name of the company and the direct Website of company that has the piecewise function)
Objectives: (1.) Calculate the winter/summer power bill of the residents of the State of Georgia within each range of specific power usage manually using Arithmetic method.
(2.) Write a piecewise function of the residential rates.
(3.) Recalculate the same power bill of the residents of the State of Georgia within each range of specific power 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.


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Students Projects: Math Part

First Sample: Karan: Harrisonburg Electric Commission: Power Rates


Second Sample: Kayla: Shenandoah Valley Electric Cooperative: Power Rates


Third Sample: Josiah: Union Electric Company: Power Rates


Fourth Sample: Ortiz: Southern Rivers Energy: Power bill: Residential Electric Service Rate


Fifth Sample: Jeremiah: Dominion Energy South Carolina: Power Bill: Winter Residential Rates


Sixth Sample: Danielle: Virginia Electric and Power Company: Residential Power Electricity Supply (ES) Service Charges


Seventh Sample: Anthony: First Energy Corp Electric Company: Power Rates


Eighth Sample: Alex: Virginia Electric and Power Company: Power Rates


Ninth Sample: Sekani: Alabama Power: Summer: Power Rates


Tenth Sample: Avry: Mississippi Power: Summer: Single Phase: Power Rates


Students Projects: Programming (Computer Science) Part

First Sample: Brigid: Alabama Power: Winter: Billing Months (October through May)
The Math part is not shown here.
However, the Computer Science (Programming) part done by the same student is shown.
The program is a Python file. Please see the instructions regarding how to run it.

                Winter (October through May)
                Base Charge: $14.50 
                12.2330¢ per KWh for the first 750 KWh,
                           plus
                11.0330¢ per KWh for all over 750 KWh.
            

Run the Python Program: 1st Approach (Strongly Preferred/Highly Recommended Approach)
(1.) Copy the code from the text file below.



(2.) Visit the Google Colab website: https://colab.research.google.com/
(3.) Log into your Google account.
(4.) Click the Insert menu tab
(5.) Click the Code cell option
(6.) Paste the code you copied.
(7.) Run the program by clicking the Play button.

Run the Python Program: 2nd Approach
(1.) Download the Python file: Brigid-AlabamaPowerWinter on your computer.
No worries, it is a safe file.
(2.) Visit the website: https://www.online-python.com/
(3.) Click the Open File from Disk menu tab
(4.) Open the file from your Download folder.
(5.) Run the program.

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


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Checklist

Checklist for Piecewise Function Project: Power 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.



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References

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

Georgia Public Service Commission. (n.d.). Georgia Public Service Commission. Retrieved from http://www.psc.state.ga.us/calc/electric/GPcalc.asp



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