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INTRODUCTION
In Unit M.2, we explored the cost of electricity in and around New Orleans, LA, during the summer. In this collaboration, we consider electrical charges during the winter.
The company that provides electricity to people in and around New Orleans, Louisiana, is called Entergy. The table below estimates the winter monthly rates charged by Entergy.
| Winter Billing Cycles (November – April) |
| Energy Charges:
|
This situation is called “Piecewise Defined.” It is piecewise because the cost changes based on the amount of electricity used; cost per kWh is not constant so the graph if this situation is not a line.
For example, a customer who used 700 kWh in a month would be charged
$0.06002 × 700 + $8.07 = $50.08
However, a customer who used 1000 kWh in a month would be charged
$0.06002 × 800 + $0.04766 × 200 + $8.07 = $65.62
Notice how this customer was charged 6.002¢ for the first 800 kWh, but was then only charged 4.766¢ per kWh for the remaining 200 kWh.
SPECIFIC OBJECTIVES
By the end of this collaboration, you should understand that
- sometimes models are comprised of more than one algebraic equation.
- setting two equations equal to each other is an algebraic method for finding points of intersection.
By the end of this collaboration, you should be able to
- create and use piecewise linear models.
- set two linear expressions equal to each other and solve the resulting linear equation.
PROBLEM SITUATION 1: ANOTHER UTILITY BILL
In Problem Situation 1 of Collaboration M.2, we examined electric bills in New Orleans, but did not consider the winter rates, which had tiered pricing. Let us now look at a tiered type of utility bill.
In order to conserve water, the Detroit area water system has a two-tiered system of charges. Customers are billed quarterly (every three months) for water usage. The cost to customers is $12.04 per thousand gallons for the first 20,000 gallons of water used, and $21.90 for each thousand gallons over 20,000. In addition there is always a flat quarterly fee of $73.00.
(1) In the table below, enter the amount of the quarterly bill for each of the water usage amounts listed.
| Quarterly Water Usage in Thousands of Gallons | Amount of Bill in Dollars |
| 15 | |
| 18 | |
| 20 | |
| 25 | |
| 28 | |
| 32 |
(2) Plot this data as a scatterplot on the graph below. Note: If completing this problem online, follow the instructions given online to create your scatterplot.
(3) Construct an algebraic model that represents the amount of the quarterly water bill (in dollars) for any amount of water used. Take a minute to try this on your own before sharing your ideas in your group. Use g for water usage and W for water bill amount.
(4) Draw the graph of the model on your scatterplot diagram. Note: If completing this problem online, follow the instructions given online.
(5) How many thousands of gallons of water were used if the quarterly water bill was $229? What if the bill was $686?
PROBLEM SITUATION 2: THE COST OF MOVING
Julio is moving from one apartment to another apartment in a neighboring town. He needs to rent a truck for one day to move his belongings. Julio uses the Internet to search for companies with trucks for rent. He finds two companies that have trucks available on the day he needs one: Rent-A-Truk and Trucks-4-Less. Rent-A-Truk advertises a rate of $40 for the day, plus 60 cents a mile for every mile over 40 miles. Trucks-4-Less advertises a rate of $25 a day plus 80 cents a mile for every mile over 25 miles.
(6) What factors will Julio use to determine which company to rent from?
(7) Fill in the blank with little or no calculation: If Julio only drives _________ miles during the day, Trucks-4-Less will end up being the better deal. Explain your reasoning.
(8) Fill in the blank with little or no calculation: If Julio drives _________ miles during the day, Rent-A-Truk will end up being the better deal. Explain your reasoning.
(9) Without doing any calculations, do you know if there will ever be a time when the two rental plans would cost the same? Consider this on your own before sharing your ideas and reasoning in your group.
(10) How much total mileage used will make the cost of both rental plans about the same? Use any method you wish to make this approximation.
(11) Now we will use algebra to find the exact mileage that would make both rental plans cost the same.
(a) Set up an algebraic model that can be used to represent the cost for using Rent-A-Truk given that one plans on driving x miles:
Total cost using Rent-A-Truk plan: \(x \le 40\): R = ______________.
Total cost using Rent-A-Truk plan: \(x > 40\): R = ______________.
(b) If a person plans on driving x miles, what would the algebraic model be to represent the cost for using Trucks-4-Less:
Total cost using Trucks-4-Less: \(x \le 25\): T = ______________.
Total cost using Trucks-4-Less: \(x > 25\): T = ______________.
(c) Julio might want to know at what mileage the cost of the Rent-A-Truk plan equals the cost of the Trucks-4-Less plan. Use your algebraic models to write one equation, using only the variable x, that represents this situation.
Hint: You can think about this situation as R = T .
Hint: Julio estimated that this situation occurs at a mileage higher than 40.
(d) Solve the equation from (c) for x and check whether the solution agrees with your answer to Question 10.
MAKING CONNECTIONS
Record the important mathematical ideas from the discussion.
