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Section 1.2 – Estimation, Graphs, and Mathematical Models
Objective #1:
Applying estimation techniques to approximate answers.
Estimating means to do the rounding before we do the problem. If we
are given a place value, we round each number to that place value and
then perform the operations.
Estimate the following:
Ex. 1
1175 + 326 + 79 + 5
Solution:
First, round each number to the nearest hundred and then add:
1175
→
1200
→
1200
326
→
300
→
300
79
→
100
→
100
+5
→
+0
→
+0
1600
The answer is 1600.
Ex. 2
In the year 2000, the US population was 282,158,336 people.
By the year 2010, it had increased by 28,074,527 people. For the
year 2050, the US population is projected to be 439,010,253 people.
(Source: www.census.gov)
a)
Round each figure to the nearest million and estimate the US
population in 2010.
b)
Round each figure to the nearest million and estimate the
projected change in the US population from 2000 to 2050.
Solution:
a)
282,158,336
→
282,000,000
+ 28,074,527
→ + 28,000,000
310,000,000
The US population was 310 million people in 2010.
b)
439,010,253
→
439,000,000
– 282,158,336
→ – 282,000,000
157,000,000
The US population is expected to grow by 157 million people
from 2000 to 2050.
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Ex. 3
Juanita pays $3.10 per day to ride the bus back and forth to
work Monday through Friday. If a monthly bus pass is $40, is it
cheaper for her to buy a monthly bus pass?
Solution:
We can estimate how much she pays per month without a bus pass.
For one week, she pays $3.10 per day for five days:
≈
$3 5days
•
day week
=
$15
week
There are approximately four weeks in a given month:
≈
$15 4 weeks
•
week month
= $60 per month
€ it is cheaper
€
€Thus,
for her to buy a monthly pass.
Objective #2: Reading and Interpreting Circle Graphs (Pie Charts)
€
€
Circle graphs use sectors to represent the data for different categories.
Usually, the sector is a percentage of the whole circle graph. Consider the
following example.
Use the graph below to answer the following questions:
Ex. 4 The circle graph below shows the percentage of different types of
tickets issued by a local police department during a particular year.
Other
Not stopping
at a stop sign
5%
8%
Illegally
parked
35%
12%
Speeding
19%
Driving under
the influence
a)
21%
Running a red
light
What was the most common ticket issued?
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b)
If 19,375 tickets were issued last year, estimate the number of
tickets issued for running a red light?
c)
If 182 tickets were issued for driving under the influence last
month, estimate the total number of tickets issued last month.
Solution:
a) The most common ticket issued was for speeding.
b) We need to find 21% of 19,375:
Running a red light tickets is 21% of 19,375.
= 0.21•19,375 ≈ 0.20•20,000 = 4,000
So, approximately 4,000 tickets were issued for running a red light
last year.
c) 182 is 19% of the total tickets: 182 = 0.19•T
Estimate: 200 ≈ 0.20T or 200/0.20 ≈ T which yields 1000
So, approximately 1,000 tickets were issued last month.
Objective #3:
Reading and Interpreting Bar Graphs
Bar graphs use bars to represent the data for different categories.
Sometimes a bar graph is drawn horizontal and sometimes vertically. A bar
graph may appear as a single, double, or triple bar graph.
Use the graph below to answer the following questions:
Ex. 5
Number of new houses sold in the US per month
Source: (http://www.census.gov/construction/nrs/)
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a)
b)
c)
d)
Which month had the lowest number of new homes sold?
Which month had the highest number of new homes sold?
Approximately how many new homes were sold in Jun, 2013?
What is the difference in the number of new homes sold in Jun, 2013
and Nov, 2012?
Solution:
a)
The ninth bar is the lowest. So, Jul, 2013 had the lowest.
b)
The twelfth bar is the highest. So, Oct, 2013 had the highest.
c)
Approximately 450,000 homes.
d)
≈ 450,000 – 400,000 = 50,000. So, ≈ 50,000 more new
homes were sold in Jun, 2013.
Use the graph below to answer the following questions:
Average cost per year of going to a fouryear institution in 2013-2014 dollars
Ex. 6
$40,000
$35,000
$30,000
$25,000
$20,000
$15,000
$10,000
$5,000
$0
– Four-Year Private College/University
– Four-Year Public College/University
Source: U.S. Department of Education, National Center for Education
Statistics
a)
b)
c)
Estimate the change in cost per year in attending a four-year public
college/university from 2007-08 to 2012–13.
If the cost continues to change at that rate, estimate the cost to
attend a four-year public college/university in 2026–27.
Find a mathematical model to find the cost, x years after 2012-13.
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Solution:
a)
The cost in 2007–08 was approximately $15,000 and the cost
in 2012–13 was approximately $17,500. Thus, the increase in
cost was $17500 – $15000 = $2500 over that five year period.
This mean that the cost increased by ≈ $2500/5 years ≈ $500
per year.
b)
2026–27 is 14 years later than 2012-13. So, the cost is
estimated to be $500(14) + $17,500 = $24,500.
c)
Replace 14 by x in part b) to get a model: c = $500x + $17,500.
Objective #4:
Reading and Interpreting Line Graphs
Line graphs use points connected by lines to represent the data for
different categories. It may appear as a single or double line graph.
Use the graph below to answer the following questions:
Ex. 7
Texas
Florida
Source: www.census.gov
a)
b)
Approximately what year is Texas' population projected to reach 30
million people?
If, in 2010, 8% of the US population lived in Texas, what was the
US population in 2010?
Solution:
a)
The line crosses 30 million between 2020 and 2025 , but it is a
little closer to 2025. So, it is projected to reach 30 million in
2023.
b)
In 2010, Texas' population is about 25 million. So, we are
looking for 25 million is 8% of the total population:
25 = 0.08•total or 25/0.08 = 312.5 million.
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Use the graph below to answer the following questions:
Ex. 8
Median income of the Middle Class
adjusted for inflation
$59,000
$58,000
$57,000
$56,000
$55,000
$54,000
$53,000
$52,000
$51,000
$50,000
Source:
US Census
Bureau
a)
Estimate the median income in 2009.
b)
Between what two years did the median income decrease the
greatest?
For what year(s) was the median income below $53,000?
If the median income continues to decrease at the same rate as it did
from 2000 to 2014, find a mathematical model for finding the median
income x years after 2000 and use the model to find the median
income in 2018.
Solution:
a) The median income was approximately $55,000.
b) The greatest decrease occurred between 2007 and 2008.
c) The median income was below $53,000 for the years 2011 and
2012.
d) The median income in 2000 was ≈ $57,750 and the median
income in 2014 was ≈ $53,750, so, the median income was
changing by ($53,750 – $57,750)/(2014 – 2000)
= – $4000/14 years ≈ – $286 per year
So, our model would be median inc. = mx + b = – 286x + 57,750
Since 2018 is 18 years after 2000, then
median inc. = – 286(18) + 57,750 = $52,602 ≈ $52,500.
c)
d)