8.8: Quantway Core 3.2 Workforce (Environment) - Student Lesson
- Page ID
- 148831
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)INTRODUCTION
Greg owns a rental property in Beaufort, SC. His mortgage, insurance, and HOA fees total up to $1,100 per month. In addition, he’s expecting the following maintenance expenses over the next year:
- Furnace replacement: $5,000
- Upgrading solar panels: $2,000
- Landscaping: $1,500
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.
Completing the CaS Chart
You may want to use the Comprehension and Synthesis (CaS) Chart in this collaboration. You may have used a CaS Chart in a previous collaboration. Remember, using the CaS Chart helps you have a deeper understanding of the problem situation. CaS Charts will help you to identify the main issue(s) that need(s) to be resolved and will help you to recognize the quantitative information that is available in the problem situation that can help you to solve the problem. You will use CaS Charts in some of the Quantway collaborations to “unpack” problem situations and support problem-solving.
As you read the problem situation, complete the CaS Chart. You may wish to return to these steps as you complete the CaS Chart. Read through the problem situation, Problem Situation: Solar Energy in Homes. As you are reading:
- Complete Column A. Hint: What issues in the problem situation do you need to understand in order to solve the problem? Is there contextual information that you need to know in order to understand the problem situation?
- Complete Column B. Hint: What quantitative information is provided in the problem situation that will help you solve the problem? Hint: Quantitative information is often a number, but also could be a number word, like “two”.
- Complete Column C. Hint: It is not necessary to solve the problem or use calculations right now. In this column, brainstorm ways you might address the issues presented in the problem situation (Column A) using the quantitative information in Column B. There are no wrong answers.
|
Column A |
Column B |
Column C |
|
What is/are the main issue(s) in this problem situation? |
What is the key quantitative information you need to solve the issue(s) in the problem situation? |
Describe in writing how the information in Column B will help you address the issue(s) in Column A later in the lesson. |
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.
SPECIFIC LANGUAGE AND LITERACY OBJECTIVES
By the end of this collaboration, you should be able to
- read and comprehend the problem situation about solar energy in homes.
- complete a Comprehension and Synthesis (CaS) Chart with contextual and quantitative information from the problem situation about solar energy in homes.
- demonstrate an understanding of mathematics by writing complete and correct responses to questions.
- demonstrate the ability to interpret, analyze, and synthesize answers about solar energy efficiency in homes.
- use appropriate quantitative and environmental science vocabulary to discuss mathematics in this collaboration.
PROBLEM SITUATION: SOLAR ENERGY IN HOMES
You are a solar energy specialist at QW Solar Now. QW Solar Now is a company that specializes in helping homeowners plan for and install solar panels in their homes. The Pressman family is interested in installing a solar energy system in their New Jersey home. You are at their home to provide an estimate of potential costs and benefits of installing solar panels.
You first provide a brief background to Tali and David Pressman about the environmental benefits of solar energy. You explain that solar panels collect, store, and convert energy from the sun into electricity. Since solar is renewable energy, they do not have to worry about running out of solar energy. Solar energy is also good for the environment. Carbon dioxide (CO2) and other greenhouse gases are not released in the air when using solar energy. Burning fossil fuels such as gas and coal releases CO2 and greenhouse gases. CO2 and greenhouse gases contribute to global warming. The Pressmans are excited about helping the environment. They also care about costs.
You know that installing solar panels can be expensive. But, over time, solar panels save customers money. You want to help the Pressmans understand that solar panels will save them money in the long run. The most common system for homes is called a 5 kilowatt (5 kW) solar energy system. It costs about $20,000 to install. However, eventually the homeowners will save money on electricity bills because collecting and using solar energy is free.
Tali and David could even make a profit from installing solar panels. Sometimes solar panels collect more solar energy than a family needs to power their home. If homeowners collect more solar energy than they need, they can sell the excess energy back to the electric company.
Your main task today is to help the Pressman family learn the financial benefits of installing solar panels. To understand potential benefits for the Pressmans, you need to know how much they pay for electricity and the average amount of electricity they use per month.
You analyze the Pressmans’ electric bill from the previous year. You discover that they are paying 16¢ per kWh of electricity. On average, they use 903 kWh per month.
(1) Think about this problem situation and what you learned in Collaboration 2.3 about alternative energy. Do you think solar energy will be a popular source of energy in homes in the future? Why or why not? Write your answer in 1-2 complete sentences. Don’t forget to use the quantitative information you wrote in Column B of the CaS Chart to support your thinking.
(2) In order for Tali and David Pressman to better understand their own energy use, they must first understand how energy is measured.
First, you explain to the Pressmans that a watt (W) is a unit of power. Power is the rate at which energy is produced or consumed. For example, a 60W light bulb uses energy at a higher rate than a 25W light bulb.
A watt-hour (Wh) is a unit of energy. For example, if you leave a 25W (watt) light bulb on for one hour, it will use 25 Wh (watt-hours of energy). If you leave the same 25W light bulb on for four hours, it will use 100 Wh of energy. Power companies, because they are often measuring large amounts of energy, measure the amount of energy produced or consumed in kilowatt-hours (kWh). 1 kilowatt-hour equals 1,000 watt-hours.
Suppose the Pressmans want to power the six lights in their kitchen for a full 24-hour day. Each light uses a 60W bulb. How many kilowatt-hours of energy would be required?
(3) Approximately how much do the Pressmans pay for electricity annually? (Hint: Use information from the CaS Chart and problem situation. You need to know the cost of electricity and how much the Pressmans use.) Use dimensional analysis to solve this problem. Show your work. Round your answer to the nearest cent.
(4) Tali and David Pressman are considering installing a 5 kW solar energy system for their home. This is the typical solar energy system installed in a single-family residential home. A 5 kW solar energy system generates about 5000 kWh of energy per year. How much would the Pressmans save in electricity costs in one year using a 5 kW solar energy system?
(5) How many 5 kW solar energy systems would the Pressmans need to install to get more energy than they need (an energy surplus)?
(6) New Jersey pays 7.5¢ per kilowatt-hour to buy back surplus electricity from home solar energy systems. Using your answer to Question 4, calculate the total annual revenue the Pressmans would earn from installing enough solar energy systems to generate surplus energy.
(7) A federal tax credit reduces the amount of income tax an individual or family pays the government each year. A federal tax credit of 30% is offered on all home-installed solar energy systems.
(a) If the Pressmans purchased one 5 kW solar energy system, how many years will pass, assuming the cost of energy remains the same, until the money saved on electricity is greater than the initial cost of the solar energy system? Recall from the problem situation, the most common system for homes is called a 5 kilowatt (5 kW) solar energy system. It costs about $20,000 to install.
(b) If the Pressmans purchased three 5 kW solar energy systems, how long until the money saved on electricity combined with the profit from the energy generated is greater than the initial cost of the solar energy system?
(8) Think back to Question 1: "Do you think solar energy will be a popular source of energy in homes in the future? Why or why not?"
Has your opinion about the future of solar energy changed? Why or why not? Discuss in your group. Use the quantitative information from your answers to Questions 3–7 to support your thinking in 1–2 sentences.
FURTHER APPLICATIONS
(9) Write an email message to David and Tali Pressman explaining which you think is the better investment. Make a recommendation to the Pressmans, and explain your reasoning. Be sure to use quantitative information from the previous questions.
Writing Tip!
It can be a challenge to use numbers to back up your statement. However, doing this makes your statement much stronger. When writing this recommendation email, you should be sure to include costs associated with initial investment, the cost of electricity, and profit generation.
(10) Most current solar energy systems are only guaranteed to last for 25 years. How does this affect the benefits of buying a solar energy system?
MAKING CONNECTIONS
Record the important mathematical ideas from the discussion.