Transcript Section 3B Putting Numbers in Perspective

Section 9B
Linear Modeling
Pages 571-585
9-B
Linear Modeling
LINEAR
constant rate of change
9-B
Understanding Rate of Change
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
9-B
Understanding Rate of Change
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
9-B
Understanding Rate of Change
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
9-B
Understanding Linear Equations
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
9-B
Understanding Linear Equations
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
9-B
Understanding Linear Equations
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
9-B
Understanding Linear Equations
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
Graphing Linear Equations
Example - Straightown: P = 10000 + 500y
y
P
0
10,000
1
10,500
2
11,000
3
11,500
5
12,500
10
15,000
population
Growth of Straightown
17000
16000
15000
14000
13000
12000
11000
10000
9000
8000
12, 16000
10, 15000
5, 12500
3, 11500
2, 11000
1, 10500
0, 10000
0
5
10
years
15
Graphing Linear Equations
Example – Rain Depth: D = 1/2 + (1)(h)
h
D
0
1/2
1
3/2
2
5/2
3
7/2
5
11/2
10
21/2
rain depth (inches)
Rain Gauge Depth
12
11
10
9
8
7
6
5
4
3
2
1
0
10, 10.5
5, 5.5
3, 3.5
2, 2.5
1, 1.5
0, 0.5
0
2
4
6
hours
8
10
Graphing Linear Equations
Example – Lake Water Depth: W = 1200 - (9.75)(d)
Water Lake Depth
D
0
1200
1
1190.25
2
1180.5
3
1170.75
5
1151.25
10
1102.5
lake depth (inches)
h
1200
1190
1180
1170
1160
1150
1140
1130
1120
1110
1100
0, 1200
1, 1190.25
2, 1180.5
3, 1170.75
5, 1151.25
10, 1102.5
0
1
2
3
4
5
days
6
7
8
9
10
9-B
Linear Modeling
LINEAR
constant rate of change (slope)
straight line graph
Understanding Slope
We define slope of a straight line by:
change in dependent var iable
slope 
change in independent var iable
( y2  y1 )
slope 
( x2  x1 )
rise
slope 
run
where (x1,y1) and (x2,y2) are any two points
on the graph of the straight line.
slope  rate of change
Understanding Slope
Example: Calculate the slope of the Straightown graph.
population
Growth of Straightown
17000
16000
15000
14000
13000
12000
11000
10000
9000
8000
12, 16000
10, 15000
5, 12500
3, 11500
2, 11000
1, 10500
0, 10000
0
5
10
years
15
Understanding Slope
Example: Calculate the slope of the Water Lake Depth graph.
lake depth (inches)
Water Lake Depth
1200
1190
1180
1170
1160
1150
1140
1130
1120
1110
1100
0, 1200
1, 1190.25
2, 1180.5
3, 1170.75
5, 1151.25
10, 1102.5
0
1
2
3
4
5
days
6
7
8
9
10
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
9-B
Homework:
Pages 582-583
34, 36, 38, 54, 56, 58
formulas, answers and graphs for each problem.
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
9-B
Homework:
Pages 584
# 40, 44, 48, 50, 54, 56, 58