5.4: Corequisite- Linear Models Practice
- Page ID
- 148620
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)SPECIFIC OBJECTIVES
By the end of this lesson, you should understand that
- linear models are appropriate when the situation has a constant increase/decrease.
- slope is the rate of change.
- the rate of change (slope) has units in context.
- different representations of a linear model can be used interchangeably.
By the end of this lesson, you should be able to
- label units on variables used in a linear model.
- make a linear model when given data or information in context.
- make a graphical representation of a linear model.
- make a table of values based on a linear relationship.
- identify and interpret the vertical intercept in context.
PROBLEM SITUATION: DATA PLANS
Data plans for cell phone plans can be purchased in two different ways. One is an unlimited data plan where the customer pays a set monthly fee for unlimited data. The other is a per-gigabyte (GB) pricing service where the customer pays a set monthly fee plus a specified amount for each GB. The GBs are not prorated (that means it doesn’t matter if the customer uses 0.001 GB or 1 GB, they must still pay for the entire 1 GB). The cost of the phone itself and other fees are the same amount under each plan.
You are shopping for a data plan and need to decide which option costs less. (Note: The descriptions of these options are examples of verbal representations of mathematical relationships.)
- Per-GB Pricing: There is a monthly fee of $20 plus $10 per GB.
- Unlimited Data: The unlimited plan costs $60 per month.
(1) Which plan do you think is less expensive and why?
(2) (a) Use the table below to build a linear model for the Per-GB Pricing data plan. First, create equations for 3, 4, 5, and 10 GBs used, and then create the linear model for g GBs used.
Per-GB Pricing (P)
GBs Used (g) | Cost | Equation |
0 | 20 | P = 20 |
1 | 20 + 1(10) | P = 20 + 1(10) |
2 | 20 + 1(10) + 1(10) | P = 20 + 2(10) |
3 | P = | |
4 | P = | |
5 | P = | |
10 | P = | |
g |
20 + t times |
P = |
(b) Use the table below to build a linear model for the Unlimited data plan. First, create equations for 3, 4, 5, and 10 GBs used, and then create the linear model for g GBs used.
Unlimited Data (U)
GBs Used (g) | Cost | Equation |
0 | 60 | U = 60 |
1 | 60 | U = |
2 | U = | |
10 | U = | |
g | U = |
(c) Use your models to fill in the following table.
GBs Used (g) | Monthly Cost of Per-GB Plan (P) | Monthly Cost of Unlimited Data (U) |
0 | ||
1 | ||
2 | ||
3 | ||
4 | ||
5 | ||
6 | ||
7 | ||
8 |
(d) Using the table from Q2(c), under what conditions is P less expensive?
(e) Using the table from Q2(c), under what conditions is U less expensive?
(3) The following questions are about “Problem Situation: Data Plans.”
(a) What are the rates of change of each of the two models? What do they mean in the context of the problem situation? (Include units.)
(b) What are the vertical intercepts of each of the two models? What do they mean in the context of the problem situation?
(c) Do the two models have horizontal intercepts? If so, what are they and what do they mean in the context of the problem situation?
(4) Explain how you can tell from a graph if a linear model has a negative slope, a positive slope, or a slope of zero. Include sketches of each type of graph.