2.4.2: Exercise M.4
- Page ID
- 148559
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)(1) The following table indicates the postage rates for large envelopes and packages of different weights:
| Weight, Not Over (ounces) | Cost to Mail a Large Envelope ($) | Cost to Mail a Package ($) |
| 1 | 0.88 | 1.71 |
| 2 | 1.08 | 1.71 |
| 3 | 1.28 | 1.71 |
| 4 | 1.48 | 1.88 |
| 5 | 1.68 | 2.05 |
| 6 | 1.88 | 2.22 |
| 7 | 2.08 | 2.39 |
| 8 | 2.28 | 2.56 |
(a) Write a sentence explaining how much it costs to send a large envelope.
(b) Write a sentence explaining how much it costs to send a package.
(c) Do you think a large envelope and a package that weigh the same will ever cost the same to mail? Explain.
(d) Create a linear model that could be used to calculate the cost to mail a large envelope, given its weight in ounces. Use E to represent the cost to mail an envelope and x to represent the weight of the envelope in ounces. Write your answer in the form E = mx + b.
(e) Create a linear model that could be used to calculate the cost to mail a package, given its weight in ounces. Let P represent the price to mail a package that weighs x ounces. Write the second linear model simplified in the form P = mx + b. (Hint: This will require two linear pieces.)
(f) Find the weight at which a large envelope and a package will cost the same amount to mail, assuming your linear models still apply to higher weights. Round to the nearest tenth of an ounce.
(2) The table below shows the salary plan for new employees at a company that sells furniture. A new employee earns $25,000 in base salary and 8% commission on furniture sold after his or her first $2,000 in sales for the year.
- Complete the table by entering a new employee’s yearly salary at each level of sales.
| Sales | Yearly salary ($) |
| $0 | |
| $1500 | |
| $2000 | |
| $5000 | |
| $10,000 | |
| $50,000 | |
| $100,000 |
(b) Write an algebraic model for an employee’s salary if their sales total is $2000 or less. Use S to represent the annual salary.
(c) Write an algebraic model for an employee who sells furniture with a total sales price of x dollars and x is over $2,000. Let S be salary and x total sales price. Write your answer simplified in the form S = mx + b.
(d) How much in furniture sales would a new employee need in order to earn a salary of $40,000 for the year?
(3) According to an article in Forbes magazine, “While major soft drink companies continue to battle declining carbonated soft drink sales, another segment continues to grow…”22 That segment is bottled water. In 2010, soft drink companies sold 13.75 billion gallons of carbonated soft drinks and 8.61 billion gallons of bottled water. In 2019, the amount of carbonated soft drinks sold fell to 12.01 billion gallons, while the amount of bottled water sold increased to 14.35 billion gallons.
(a) What was the rate of change for soft drink sales from 2010 to 2019? Round to two decimal places.
(b) What was the rate of change for bottled water sales from 2010 to 2019? Round to two decimal places.
(c) Using the information from Question 3(a), and assuming that sales are linear, write a linear model that represents soft drink sales from 2010 to 2019. Use t for the number of years after 2010.
(d) Using the information from Question 3(b), and assuming that sales are linear, write a linear model that represents bottled water sales from 2010 to 2019. Let t be the number of years after 2010.
(e) Using your equations, during what year was the amount of soft drinks and bottled water sold the same?
(4) The graph below shows the increase in the number of children with autism from 2000–2020.23 Is the number of children with autism across these years best modeled by a linear equation, a piecewise model, or neither? Explain.
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22 https://petpla.net/wp-content/uploads/2020/05/BW-consumption-shift-2020-2.jpg