8.6: Linear Functions
- Page ID
- 193499
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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 Rectangular Coordinate Systems and Graphs
1) \(x\)-intercept: 3; \(y\)-intercept: -4
3) \(y=\frac{5}{3}x+4\)
5) \(\sqrt{72}\approx8.4853\)
7) \(2 \sqrt{96130}\approx620.0968\)
9) \((2,11.5)\)
Linear Functions
1) Terry starts at an elevation of 3000 feet and descends 70 feet per second.
3) 3 miles per hour
5) \(d(t)=100-10t\)
7) Yes.
9) No.
11) No.
13) No.
15) Increasing.
17) Decreasing.
19) Decreasing.
21) Increasing.
23) Decreasing.
25) \(3\)
27) \(-13\)
29) \(45\)
31) \(f(x)=-12x+72\)
33) \(y=2x+3\)
35) \(y=-13x+223\)
37) \(y=45x+4\)
39) \(-54\)
41) \(y=23x+1\)
43) \(y=-2x+3\)
45) \(y=3\)
47) Linear, \(g(x)=-3x+5\)
49) Linear, \(f(x)=5x-5\)
51) Linear, \(g(x)=-252x+6\)
53) Linear, \(f(x)=10x-24\)
55) \(f(x)=-58x+17.3\)
59) a. \(a=11900\); \(b=1000.1\) b. \(q(p)=1000p-100\)
63) \(x=-163\)
65) \(x=a\)
67) \(y=dc-ax-adc-a\)
69) $45 per training session.
71) The rate of change is 0.1. For every additional minute talked, the monthly charge increases by $0.1 or 10 cents. The initial value is 24. When there are no minutes talked, initially the charge is $24.
73) The slope is -400. This means for every year between 1960 and 1989, the population dropped by 400 per year in the city.
75) c.
Linear Functions and Mathematical Models: Modeling with Linear Functions
1) Determine the independent variable. This is the variable upon which the output depends.
3) To determine the initial value, find the output when the input is equal to zero.
5) 6 square units
7) 20.012 square units
9) 2,300
11) 64,170
13) \(P(t)=75000+2500t\)
15) (–30, 0) Thirty years before the start of this model, the town had no citizens. (0, 75,000) Initially, the town had a population of 75,000.
17) Ten years after the model began.
19) \(W(t)=0.5t+7.5\)
21) (-15,0): The \(x\)-intercept is not a plausible set of data for this model because it means the baby weighed 0 pounds 15 months prior to birth); (0, 7.5): The baby weighed 7.5 pounds at birth.
23) At age 5.8 months.
25) \(C(t)=12025-205t\)
27) (58.7, 0): In roughly 59 years, the number of people inflicted with the common cold would be 0); (0, 12,025): Initially there were 12,025 people afflicted by the common cold.
29) 2064
31) \(y=-2t+180\)
33) In 2070, the company’s profit will be zero.
35) \(y=30t-300\)
37) (10, 0) In 1990, the profit earned zero profit.
39) Hawaii
41) During the year 1933
43) $105,620
45) a) 696 people; b) 4 years; c) 174 people per year; d) 305 people; e) \(P(t)=305+174t\); f) 2,219 people
47) a) \(C(x)=0.15x+10\); b) The flat monthly fee is $10 and there is an additional $0.15 fee for each additional minute used; c) $113.05
49) a) \(P(t)=190t+4360\); b) 6,640 moose
51) a) \(R(t)=16-2.1t\); b) 5.5 billion cubic feet; c) During the year 2017
53) More than 133 minutes
55) More than $42,857.14 worth of jewelry
57) $66,666.67
Linear Functions and Mathematical Models: Applications
1) \(y=25x+1200\)
3) \(y=20x+350\)
5) \(y=80x+240\,000\)
7) \(y=\frac{2}{5}x\); \(68\)
9) \(y=7x-338\); \(138\)
11) \(F=\frac{9}{5}C+32\); \(77\)
13) \(y=0.375x+29.8\); \(42.925\) million people in 2025
15) \(y=25x+1200\); 14,400 students in 2010
17) \(y=0.18x+10\); the cost is $\(82\) for a home using \(400\) kWh of electricity per month
19) a) \(y=12x+110\,000\); b) \(y\) = $230,000; c) \(x\) = $7,500
21) a) \(y=3x+1000\); b) when $100 is spent on ads, 1,300 cups of coffee are sold
Intersection of Straight Lines
1) \(x=3\), \(y=13\)
3) \(x\) = $11.50, \(y\) = 16,500 items
5) a) Plan I costs $87; Plan II costs $99. Plan I is better. b) \(x\) = 150 miles; both plans cost $61.50.
7) Supply curve: \(y=400x-1200\)
9) \(x\) = 4,000 cookies; cost = revenue = $3,200
11) \(x\) = 8,000 pairs of socks; cost = revenue = $36,000
13) a) cost function \(y=10x+700\); b) fixed cost = $700; c) \(x\) = 140 pounds; d) revenue = cost = $2,100
Chapter Review
1) Yes
3) Increasing.
5) \(y=-3x+26\)
7) 3
9) \(y=2x-2\)
11) Not linear.
13) parallel
15) \((-9, 0)\); \((0, -7)\)
17) Line 1: \(m=-2\); Line 2: \(m=-2\); Parallel
19) \(y=-0.2x+21\)
23) 250.
25) 118,000.
27) \(y=-300x+11500\)
29) a) 800; b) 100 students per year; c) \(P(t)=100t+1700\)
31) 18,500
33) $91,625
35) Extrapolation.
39) Midway through 2024.
41) \(y=-1.294x+49.412\); \(r=-0.974\)
43) Early in 2022
45) 7,660