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CHAPTER 1

Problem Solving andCritical Thinking

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1.2

Estimation, Graphs and MathematicalModels

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Objectives

1. Use estimation techniques to arrive at anapproximate answer to a problem.

2. Apply estimation techniques to informationgiven by graphs.

3. Develop mathematical models that estimaterelationships between variables.

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Estimation

The process of arriving at an approximateanswer to a question.

Methods of Estimation

Rounding Numbers

Using Graphs

Using Mathematical Models

4

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Rounding Whole Numbers

5

The symbol means is approximatelyequal to

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Example 1: Rounding A Whole Number

Round the World population 6,751,593,103 To the nearest million:

To the nearest thousand:

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Example 2: Rounding The Decimal Part of a Number

The first seven digits of are: 3.141592.Round : To the nearest hundredth: 3.141592

To the nearest thousand: 3.141592.

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Example 3: Estimation by Rounding

You purchased bread for $2.59,detergent for $2.17, a sandwichfor $3.65, an apple for $0.47 andcoffee for $5.79. The total billwas $18.67. Is this amount

reasonable? To check, round the cost of eachitem to the nearest dollar.

This bill of $18.67 seems highcompared to the $15.00

estimate.

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Example 5: Applying Estimation Techniques to aCircle Graph

Use a circle to representthe whole and divide itinto sectors proportionalto their part of the whole.

According to the U.S.Census Bureau, in 2008,there were 220,948,915Americans 20 years andolder. Assuming the circle

graph is representative ofthis age group,

a. Determine a calculation thatshows the number ofAmericans 20 years and

older who define old age by adecline in physical ability.

b. Use rounding to find areasonable estimate.

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Example 5 continued

b. Estimate the Answer

Approximately 88 million Americans 20years and older define old age by adecline in physical ability.

a. Compute the exact answer

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Estimation with Graphs: Bar Graphs

Bar graphs use a vertical or horizontal bar torepresent each item.

The length of the bar determines the amount.

We can use bar graphs to make predictions.

E l 6 A l i E ti ti d I d ti

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Example 6: Applying Estimation and InductiveReasoning to Data in a Bar Graph

a. Estimate a mans increased

life expectancy, rounded to thenearest hundredth of a year, foreach future birth year.

For each subsequent birth year, a mans lifeexpectancy is increasing by approximately 0.17year.

75.1 65.60.17

2006 1950

E l 6 A l i E ti ti d I d ti

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b. Estimate the life expectancyof a man born in 2020.

An American man born in 2020 will have a lifeexpectancy of approximately 77.5 years.

65.6 0.17 70

65.6 11.977.5

Example 6: Applying Estimation and InductiveReasoning to Data in a Bar Graph continued

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Estimation with Graphs: Line Graphs

Line graphs illustrate

trends over time

Horizontal axis

represents time

Vertical axis

represents theage

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Example 7: Using a Line Graph

Find an estimate forthe percentage ofseniors who smokedcigarettes in 1995.

In 1995,approximately

34% of seniorssmoked cigarettes.

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Example 7: Using a Line Graph continued

In which five-yearperiod did thepercentage ofseniors who usedsmoked cigarettesdecrease at thegreatest rate?

The percentage of seniors smoking cigarettesdecreased at the greatest rate in the five year periodfrom 2000 through 2005.

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Estimation with Mathematical Models

The process of finding formulas to describe real-world phenomena.

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Example 8: Modeling the Cost of Attending aPublic College

a. Estimate the yearly increase in tuition and

fees.

The average yearly increase is about $358.

6585 3362

2009 20003223

9

358

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Example 8: continued

b. Obtain a mathematical

model that estimates theaverage cost of tuitionand fees, for the school

year ending x yearsafter 2000.

3362 358T x

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Example 8 continued

c. Use the mathematical model to project the

average cost of tuition and fees for theschool year ending in 2014.

The model projects that the average cost oftuition and fees for the school year endingin 2014 will be $8374.

3362 358

3362 358(14)

3362 5012

8374

T x

T