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Section 9BSection 9BLinear ModelingLinear Modeling
Pages 571-585Pages 571-585
Linear ModelingLinear Modeling
9-B
LINEAR
constant rate of change
Understanding Rate of Understanding Rate of ChangeChange
9-B
Example: The population of Straightown increases at a rate of 500 people per year. How much will the
population grow in 2 years? 10 years?
The population of Straightown varies with respect to time (year) with a rate of change of 500 people per year.
P = f(y)
In 2 years, the population will change by:
(500 people/year ) x 2 years = 1000 people
Understanding Rate of Understanding Rate of ChangeChange
9-B
Example/571: During a rainstorm, the rain depth reading in a rain gauge increases by 1 inch each hour.
How much will the depth change in 30 minutes?
The rain depth varies with respect to time (hour) with a rate of change of 1 inch per hour.
D = f(h)
In 30 minutes, the rain depth will change by:
(1 inch/hour ) x (1/2 hour) = (1/2) inch
Understanding Rate of Understanding Rate of ChangeChange
9-B
Example 27/583: The water depth in a lake decreases at a rate of 1.5 inches per day because of evaporation. How much does the water depth change in 6.5 days? in 12.5 days?
The water depth varies with respect to time (days) with a rate of change of -1.5 inches per day.
W = f(d)
In 6.5 days, the water depth will change by:
(-1.5 inches/day ) x (6.5 days) = -9.75 inches
Understanding Linear Understanding Linear EquationsEquations
9-B
Example: The population of Straightown is 10,000 and increasing at a rate of 500 people per year. What will the population be in 2 years?
The population of Straightown varies with respect to time (years) with an initial value of 10,000 and a rate of change of 500 people per year.
P = f(y)P = 10000 + 500y
P = 10000 + (500)(2) = 11000 people
Understanding Linear Understanding Linear EquationsEquations
9-B
Example: The rain depth at the beginning of a storm is ½ inch and is increasing at a rate of 1 inch per hour? What is the depth in the gauge after 3 hours?
The rain depth varies with respect to time (hours) with an initial value of ½ inch and a rate of change of 1 inch per hour.
D = f(h)P = 1/2+ (1)(h)
P = 1/2 + (1)(3) = 7/2 inches or 3.5 inches
Understanding Linear Understanding Linear EquationsEquations
9-B
Example 27*/583: The water depth in a lake is 100 feet and decreases at a rate of 1.5 inches per day because of evaporation? What is the water depth after 6.5 days?
The water depth varies with respect to time (days) with an initial value of 100 feet (1200 inches) and a rate of change of 1.5 inches per day.
W = f(d)P = 1200-(1.5)(d)
P = 1200-(1.5)(6.5) = 1200 – 9.75 = 1190.25 inches
Understanding Linear Understanding Linear EquationsEquations
9-B
General Equation for a Linear Function (p576):
dependent var. = initial value + (rate of change x independent var.)
NOTE: rate of change = dependent variable per independent variable
y P
00 10,00010,000
11 10,50010,500
22 11,00011,000
33 11,50011,500
55 12,50012,500
1010 15,00015,000
Growth of Straightown
12, 16000
5, 12500
1, 10500
0, 10000
10, 15000
3, 115002, 11000
80009000
1000011000120001300014000150001600017000
0 5 10 15
years
po
pu
lati
on
Graphing Linear Graphing Linear EquationsEquations
Example - Straightown: P = 10000 + 500y
h D
00 1/21/2
11 3/23/2
22 5/25/2
33 7/27/2
55 11/211/2
1010 21/221/2
Graphing Linear Graphing Linear EquationsEquations
Example – Rain Depth: D = 1/2 + (1)(h)
Rain Gauge Depth
5, 5.5
1, 1.5
0, 0.5
10, 10.5
3, 3.5
2, 2.5
0123456789
101112
0 2 4 6 8 10
hours
rain
de
pth
(in
ch
es
)
h D
00 12001200
11 1190.251190.25
22 1180.51180.5
33 1170.751170.75
55 1151.251151.25
1010 1102.51102.5
Graphing Linear Graphing Linear EquationsEquations
Example – Lake Water Depth: W = 1200 - (9.75)(d)
Water Lake Depth
5, 1151.25
1, 1190.250, 1200
10, 1102.5
3, 1170.75
2, 1180.5
11001110112011301140115011601170118011901200
0 1 2 3 4 5 6 7 8 9 10
days
lak
e d
ep
th (
inc
he
s)
Linear ModelingLinear Modeling
9-B
LINEAR
constant rate of change (slope)
straight line graph
var
var
change in iabldependent
independen
eslope
change in t iable
2 1
2 1
( )
( )
y yslope
x x
We define slope of a straight line by:
where (x1,y1) and (x2,y2) are any two points on the graph of the straight line.
riseslope
run
Understanding SlopeUnderstanding Slope
slope rate of change
Growth of Straightown
12, 16000
5, 12500
1, 10500
0, 10000
10, 15000
3, 115002, 11000
80009000
1000011000120001300014000150001600017000
0 5 10 15
years
po
pu
lati
on
Understanding SlopeUnderstanding SlopeExample: Calculate the slope of the Straightown graph.
Understanding SlopeUnderstanding SlopeExample: Calculate the slope of the Water Lake Depth graph.
Water Lake Depth
5, 1151.25
1, 1190.250, 1200
10, 1102.5
3, 1170.75
2, 1180.5
11001110112011301140115011601170118011901200
0 1 2 3 4 5 6 7 8 9 10
days
lak
e d
ep
th (
inc
he
s)
More Practice
33/583 The price of a particular model car is $15,000 today and rises with time at a constant rate of $1200 per year.
A) Find a linear equation to describe the situation.B) How much will a new car cost in 2.5 years.
35/583 A snowplow has a maximum speed of 40 miles per hour on a dry highway. Its maximum speed decreases by 1.1 miles per hour for every inch of snow on the highway.
A) Find a linear equation to describe the situation.B) At what snow depth will the plow be unable to move?
37/583 You can rent time on computers at the local copy center for $8 setup charge and an additional $1.50 for every 5 minutes.
B) Find a linear equation to describe the situation.C) How much time can you rent for $25?
53, 55, 57/584
Homework:Homework:
Pages 582-583Pages 582-583
34, 36, 38, 54, 56, 5834, 36, 38, 54, 56, 58
formulas, answers and graphs for each formulas, answers and graphs for each problem.problem.
9-B
Algebraic Linear Equations
Slope Intercept Form
y = b + mx
b is the y intercept or initial value
m is the slope or rate of change.
More Practice/584: 45, 47, 49, 51
Homework:Homework:
Pages 584Pages 584
# 40, 44, 48, 50, 54, 56, 58 # 40, 44, 48, 50, 54, 56, 58
9-B
