1. Section 1.2 Estimation, Graphs and Mathematical Models
Objectives
Using estimation techniques to arrive at an approximate answer to a problem.
Apply estimation techniques to information given by graphs.
Develop mathematical models that estimate relationships between variables.
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2. Estimation
The process of arriving at an approximate answer to a question.
Methods of Estimation
Rounding Numbers
Using Graphs
Using Mathematical Models
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3. Rounding Numbers
Look at the digit to the right of the digit where rounding is to occur.
a. If the digit to the right is 5 or greater, add 1 to the digit to be rounded. Replace all digits to the right by zeros.
b. If the digit to the right is less than 5, do not change the digit to be rounded. Replace all digits to the right by zeros.
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4. Example 1 Rounding Whole numbers
Round the U.S. population (295,734,123)
To the nearest million: 295,734,123.
The symbol ≈ means “approximately equal to”
To the nearest thousand: 295,734,123
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5. Rounding Decimal Numbers
Look at the digit to the right of the digit where rounding is to occur.
a. If the digit to the right is 5 or greater, add 1 to the digit to be rounded. Drop all digits to the right.
b. If the digit to the right is less than 5, do not change the digit to be rounded. Drop all digits to the right.
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6. Example 2 Rounding Decimal Numbers: The number π (pi)
π is an irrational ( a non-terminating, non-repeating decimal) number used to represent the ratio between the circumference and diameter of a circle. The first seven digits of π are: Round π:
To the nearest hundredth:
To the nearest thousand:
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7. Example 3: Estimation by Rounding
You purchased bread for $2.59, detergent for $2.17, a sandwich for $3.65, an apple for $0.47 and coffee for $ The total bill was $ Is this amount reasonable?
To check, round the cost of each
Item to the nearest dollar.
This bill of $18.67 seems high compared to the $15.00 estimate.
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8. Example 5: Estimation with Graphs Circle (Pie) Graphs
Dogs count as Company
According to the American Pet Association, there are 43,143,849 dog owners in the U.S.
The American Kennel Club did a survey asking this question of dog owners:
If you are at home with your dog and no other people are there, are you alone?
Use a circle to represent the whole and divide it into sectors proportional to their part of the whole.
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9. Example 5 continued
Compute the exact answer
Rounding to the nearest
whole number is 34,083,641
Estimate the Answer
Approximately 32 million dog owners
in the U.S. think dogs count as
company.
Using the circle graph, of the 43,143,849 dog owners, how many believe their dogs are company?
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10. Estimation with Graphs Bar Graphs
Bar graphs use a vertical or horizontal bar to represent each item.
The length of the bar determines the amount.
We can use bar graphs to make predictions.
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11. Bar Graphs continued Example1:
Estimate a man’s increased life expectancy, rounded to the nearest hundredth of a year, for each future birth year.
At the end of the example, “Life expectancy each year is increasing by approximately 0.17 year.”
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12. Example 6 Bar Graphs
Predict the life expectancy for a man born in 2020.
The life expectancy for a man born in 2020 is
approximately:
The life expectancy each year is increasing by approximately 0.17 years.
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13. Estimation with Graphs Line Graphs
Line graphs illustrate trends over time
Horizontal axis represents time
Vertical axis represents the amount
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14. Example 7 Find an estimate for the number of democracies in 1989.
There were approximately
64 democracies
Identify the year of 110 democracies.
There were 110 democracies in 1997.
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15. Example 8 Estimation with Mathematical Models
The process of finding formulas to describe real-world phenomena.
The graph on the next slide represents the median age of the U.S. population.
Create a mathematical model that estimates the median age, M of the U.S. population x years after 1970.
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16. Example 8 Continued Find the yearly increase:
Use variables to obtain our model:
The mathematical model estimates the median age, m,
of the U.S population x years after 1970.
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