Section 9B Linear Modeling

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Section 9BLinear Modeling

• Pages 571-585

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Linear Modeling
9-B
LINEAR constant rate of change
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Understanding Rate of Change
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
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Understanding Rate of Change
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
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Understanding Rate of Change
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
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Understanding Rate of Change
9-B
Rate of Change Rule (p574) To calculate the
change in dependent variable from the change in
independent variable (change in dependent
(rate of x (change in independent
variable)
change)
variable)
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Understanding Linear Equations
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
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Understanding Linear Equations
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
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Understanding Linear Equations
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
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Understanding Linear Equations
9-B
General Equation for a Linear Function
(p576) dependent var. initial value
(rate of change x independent var.)
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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
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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
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Graphing Linear Equations
Example Lake Water Depth W 1200 - (9.75)(d)
h D
0 1200
1 1190.25
2 1180.5
3 1170.75
5 1151.25
10 1102.5
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Linear Modeling
9-B
LINEAR constant rate of change
(slope) straight line graph
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Understanding Slope
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.
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Understanding Slope
Example Calculate the slope of the Straightown
graph.
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Understanding Slope
Example Calculate the slope of the Water Lake
Depth graph.
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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) Clearly identify
independent and dependent variable. B) Find a
linear equation to describe the situation. C)
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) Clearly identify
independent and dependent variable. B) Find a
linear equation to describe the situation. C) 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. A) Clearly
identify independent and dependent variable. B)
Find a linear equation to describe the
situation. C) How much time can you rent for
25?
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9-B

• Homework
• Pages 582-583
• 23a-b(rate), 25a-b(rate), 28, 30, 34, 36, 38

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More Practice
39/584 Suppose your dog weighed 2.5 pounds at
birth and weighed 15 pounds after 1 year. A)
Based on these two data points, find a linear
function that describes how weight varies with
age. B) Predict your dogs weight at 5 and 10
years of age. C) Comment on the validity of the
model. D) Sketch a graph of the weight function.
41/584 A Campus Republican fundraiser offers
raffle tickets for 10 each. The prize for the
raffle is a 350 television set, which must be
purchased with proceeds from the ticket sales A)
Find a function that gives the profit/loss for
the raffle as it varies with the number of
tickets sold. B) What is the profit/loss if 40
tickets are sold? C) How many tickets must be
sold for the raffle sales to equal the cost of
the prize? D) Sketch a graph of the profit/loss
function.
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43/584 A 1200 washing machine in a laundromat
is depreciated for tax purposes at a rate of 75
per year. A) Find a function for the depreciated
value of the washing machine as it varies with
time. B) What is the depreciated value after 2
years? C) When will the depreciated value be
0. D) Sketch a graph of the function.
55/584 The cost of publishing a poster is 2000
for setting up printing equipment, plus 3 per
poster printed. A) Find a function for the cost
of publishing a poster as it varies with the
number of posters printed. B) How much does it
cost to print 2000 posters? C) How many posters
can you print for 2800 D) Sketch a graph of
the function.
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Algebraic Linear Equations
Slope Intercept Form y b mx
b is the y intercept or initial valuem is the
slope or rate of change.
More Practice/584 45, 47, 49, 51
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9-B

• Homework
• Pages 584
• 40, 44, 48, 50, 54, 56, 58