4.8: Green Savings

INTRODUCTION

This collaboration uses a personal finance context (buying a car) to lead you through a complex mathematical modeling exercise. The first half of the exercise is presented in this collaboration, and the second half is presented in Collaboration 4.9. You will be confronted with an open-ended problem, which requires you to identify missing information, make the relevant accommodations, and perform the appropriate calculations to solve the problem.

SPECIFIC OBJECTIVES

By the end of this collaboration, you should understand that

• mathematical modeling and quantitative reasoning can be used to help make personal financial decisions.

By the end of this collaboration, you should be able to

• identify missing information.
• demonstrate an ability to determine reasonable values for missing information.
• construct a linear model of a savings scenario.
• describe in words and graphs the effect changes in assumptions have on linear models.

PROBLEM SITUATION: COMPARING THE COSTS OF CARS

You have decided to investigate some pros and cons of buying a hybrid car. A hybrid car gets better gas mileage than a gas-powered car, but it usually costs more. While there are many issues to consider in this choice, at this point you want to investigate the savings at the gas pump for the hybrid vehicle and determine if these savings will offset the higher price of the car. Suppose Internet research yields the following information:

• A 2023 Honda Accord hybrid costs $31,895. This car is supposed to get 46 miles per gallon (mpg) in regular city driving and 41 mpg on the highway.²⁵

• A completely gas-powered 2023 Honda Accord costs $27,295. This car is supposed to get
  29 mpg in regular city driving and 37 mpg on the highway.

(1) What additional information is necessary to determine how much money will be spent on gas
in a year of driving one of these vehicles? Discuss in your group and write down some reasonable values for any missing information.

(2) (a) If you purchase the hybrid Accord, how much money would you save in gasoline purchases the first year? Take a minute to think about this individually before sharing your ideas and calculations in your group. Round figures to two decimal places.

(b) How long would it take for the gasoline savings to make up for the additional cost of the hybrid?

Round to the nearest year.

(3) (a) You have seen in earlier collaborations the importance of being able to model quantities both algebraically (with equations) and graphically. Determine an algebraic model (equation) for the amount of money saved in gasoline. Define all variables used in the model.

(b) Produce a graph of this model. Note: If completing this problem online, follow the instructions given online to create your graph.

(c) Explain whether the model is linear, exponential, or neither.

(4) How does the graph change if the price of gas increases?

(5) How does the graph change if the number of miles driven in any given year increases?

FURTHER APPLICATIONS

(6) Below is a graph about Tyrone’s savings account. The coordinates of the first and last points on the graph are provided.

(a) How would you best describe the situation modeled by the graph?

(b) Write an equation for the graph. Be sure to define your variables, and explain your group’s approach.

(c) If the trend continues, how long will it take for Tyrone to have $500? Explain how you found your answer.

MAKING CONNECTIONS

Record the important mathematical ideas from the discussion.

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²⁵ https://automobiles.honda.com/accord-sedan