3.2: The Cost of Driving part 1 / The Cost of Driving part 2
- Page ID
- 148739
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The Cost of Driving, Part 1
INTRODUCTION
Greg owns a rental property in Beaufort, SC. His mortgage, insurance, and HOA fees total up to $1,400 per month. In addition, he’s expecting the following maintenance expenses over the next year:
- Furnace replacement: $5,500
- Gutters replacement: $2,500
- Landscaping: $2,000
What’s the minimum monthly rent Greg should be charging his tenants for the next year, just to break even? As a group, discuss your strategies for solving this problem.
SPECIFIC OBJECTIVES
By the end of this collaboration, you should understand that
- units can be used in dimensional analysis to set up calculations.
- precision should be based on several factors, including the size of the numbers used and the precision of the original values. Rounding can produce large differences in results.
By the end of this collaboration, you should be able to
- solve a complex problem with multiple pieces of information and steps.
- use dimensional analysis.
- investigate how changing certain values affects the result of a calculation.
PROBLEM SITUATION: COST OF DRIVING
Jenna’s job requires her to travel. She owns a 2020 Toyota 4Runner, but she also has the option to rent a car for her travel. In either case, her employer will reimburse her for the mileage using the rate set by the Internal Revenue Service. In 2023, this rate was set at 65.5 cents per mile.9 Over the next two lessons you will explore the question of whether it would be better for Jenna to drive her own car or to rent a car.
(1) What do you need to know to calculate the cost of Jenna driving her own car?
(2) What do you need to know to calculate the cost of Jenna renting a car?
Skill Building
This section introduces skills that will help you with the problem situation.
(3) Gas mileage is rated for either city driving or highway driving. Most of Jenna’s travel will take place on the highway. For one trip, she drives 150 miles and the price of gas is $3.50/gallon. Her 4Runner gets 18 miles/gallon. If Jenna rents a car, she can request a small, fuel-efficient model such as the Hyundai Elantra, which gets 40 miles/gallon.
(a) Use your estimation skills to compare the cost of gas for the two vehicles. Which one costs more? Explain your answer.
(b) Now, calculate the cost of gas for each vehicle. Think about the discussion on units in Preparation 3.2. Try this on your own first before sharing your ideas in your group. Round to two decimal places.
4Runner:
Elantra:
(4) Using the information below, calculate Jenna’s total cost of driving a rental car for a round trip.
- Price of gas: $3.50/gallon
- Length of trip (one way): 193 miles
- Gas mileage of rental car: 40 miles/gallon
- Price of the rental car: $118.98 plus 15.3% tax (Gas is not included in the rental price and the car must be returned to the rental agency with a full tank.)
Work through the following series of calculations to find the total cost of Jenna driving a rental car for a round trip. Enter your answers rounded to two decimal places when necessary.
Round trip distance =
Amount of gas needed for round trip =
Price of gas for round trip =
Total price of rental car =
Total cost for round trip =
(5) Using the information below, calculate the total cost of Jenna driving her own car for a round trip.
- Price of gas: $3.50 per gallon
- Length of trip (one way): 193 miles
- Gas mileage of Jenna’s car: 18 miles per gallon
Maintenance costs for Jenna’s car:
- General maintenance (oil and fluid changes): $45 every 3,000 miles
- Tires: Tires for Jenna’s car cost $960; they are supposed to be replaced every 50,000 miles
- Repairs: The website Edmunds.com estimates repairs on a three-year-old 2020 4Runner will be approximately $426 per year; this is based on driving 15,000 miles.10
Work through this problem in your group. Round to two decimal places in your calculations.
MAKING CONNECTIONS
Record the important mathematical ideas from the discussion.
The Cost of Driving, Part 2
INTRODUCTION
This is the formula that Jenna could use to figure out the total cost of renting a Hyundai Elantra:
\[R = \dfrac{g\times m}{40} + 137.18\nonumber \]
Discuss in your group what you think happens to the value of R as g and m change.
R = total cost of renting car
g = price of gas
m = total miles
SPECIFIC OBJECTIVES
By the end of this collaboration, you should understand that
- units can be used in dimensional analysis to set up calculations.
- precision should be based on several factors, including the size of the numbers used and the precision of the original values. Rounding can produce large differences in results.
By the end of this collaboration, you should be able to
- solve a complex problem with multiple pieces of information and steps.
- use dimensional analysis.
- investigate how changing certain values affects the result of a calculation.
PROBLEM SITUATION: COMPARING COSTS
Let us return to the problem situation from the previous collaboration (3.2, Part 1). Recall that Jenna’s job requires her to travel. She owns a 2020 Toyota 4Runner, but she also has the option to rent a car for her travel. In either case, her employer will reimburse her for the mileage using the rate set by the Internal Revenue Service. In 2023, this rate was set at 65.5 cents per mile.11
(1) Calculate how much profit Jenna makes after she pays her expenses in the situation where she rents a car, and in the situation where she drives her own car. The information for both situations was used in Question 4 and 5 in Part 1. It is displayed again below. Enter your calculations in the table below. Round to two decimal places in your calculations.
|
Info from Q4, 3.2 part 1 |
Info from Q5, 3.2 part 1 |
|
Maintenance costs for Jenna’s car:
|
|
Total cost of driving a rental car for a round trip = $170.96 |
Total cost of Jenna driving her own car for a round trip = $99.22 |
|
Total Cost to Jenna |
Cost per mile |
Jenna’s Profit After Expenses |
|
|
Jenna’s own car |
(a) |
(c) |
(e) |
|
Rental car |
(b) |
(d) |
(f) |
Trip Length and Gas prices
Jenna’s job requires her to make trips of various lengths. The price of gas tends to change over time. In mathematics, we call parameters that change variables. In the following questions you will explore Jenna’s expenses of renting versus the cost of using her own car for making a single trip by varying the gas price and trip’s length.
(2) In the table below, work in your group to enter:
- Jenna’s cost of a round trip for each given gas price and trip length if Jenna decides to rent a car.
- Since her trips vary in length, it is useful for Jenna to know the cost per mile of renting a car for each trip.
Round all your calculations to two decimal places. Note: Complete the first row with the calculated costs from Question 1.
|
Gas Price |
Round Trip Length |
Cost for Jenna Renting a Car |
Cost for Jenna Renting a Car (per mile) |
|
$3.50 |
386 miles |
From Q1 |
From Q1 |
|
$5.00 |
386 miles |
(a) |
(b) |
|
$3.50 |
772 miles |
(c) |
(d) |
|
$5.00 |
772 miles |
(e) |
(f) |
(3) (a) In the table below, work in your group to enter:
- Jenna’s cost of a round trip for each given gas price and trip length if Jenna drives her own car. Show your calculations in the table below.
- Since her trips vary in length, it is useful for Jenna to know the cost per mile of driving her own car for each trip. Write your results in the table below.
Round all your calculations to two decimal places. Note: Complete the first row with the calculated costs from Question 1.
|
Gas Price |
Round Trip Length |
Cost for Jenna Driving her Own Car |
Cost for Jenna Driving her Own Car (per mile) |
|
$3.50 |
386 miles |
From Q1 |
From Q1 |
|
$5.00 |
386 miles |
(a) |
(b) |
|
$3.50 |
772 miles |
(c) |
(d) |
|
$5.00 |
772 miles |
(e) |
(f) |
(b) In calculating the cost for each trip that Jenna makes, we perform similar calculations. Create a mathematical equation for the cost per mile of Jenna driving her own car. An equation is a statement of equality, meaning that it tells you that two expressions are equal to each other. Take a minute to think this through individually before sharing your ideas in your group. Round to four decimal places where necessary.
(c) Use the equation from (b) to calculate how much it would cost for Jenna to drive 1,000 miles if the price of gas is $4.00. Take a minute to think this through individually before sharing your ideas in your group.
(4) Using the work you have done in Questions 2 and 3 as a group, write a note to Jenna explaining how she can decide if it is better to drive her own car or get a rental. Your explanation should include information about what factors affect the cost of driving and why.
MAKING CONNECTIONS
Record the important mathematical ideas from the discussion.
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