Section 1.3 Problem Solving

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 1.3

Problem Solving


What You Will Learn

Problem-solving techniques

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Polya’s Procedure

George Polya (1887-1985), a mathematician who was educated in Europe and taught at Stanford, developed a general procedure for solving problems.

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Guidelines for Problem Solving1. Understand the Problem.2. Devise a Plan to Solve

the Problem.3. Carry Out the Plan.4. Check the Results.

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1. Understand the Problem.

• Read the problem carefully at least twice.

• Try to make a sketch of the problem. Label the information given.

• Make a list of the given facts that are pertinent to the problem.

• Determine if you have sufficient information to solve the problem.

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2. Devise a Plan to Solve the Problem.

• Can you relate this problem to a previous problem that you’ve worked before?

• Can you express the problem in terms of an algebraic equation?

• Look for patterns or relationships.

• Can you express the problem more simply?

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2. Devise a Plan to Solve the Problem.

• Can you substitute smaller or simpler numbers to make the problem more understandable?

• Will listing the information in a table help in solving the problem?

• Can you make an educated guess at the solution? You can work backward to determine the correct procedure.

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3. Carrying Out the Plan.

• Use the plan you devised in step 2 to solve the problem.

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4. Check the Results.• Ask yourself, “Does the answer make sense?” and “Is it reasonable?”• If the answer is not reasonable, recheck your method for solving the problem and your calculations.

• Can you check the solution using the original statement?

• Is there an alternative method to arrive at the same conclusion?

• Can the results of this problem be used to solve other problems?

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Example 2: Shuttle Revenue

Airport Express shuttle service provides service from San Antonio International Airport to downtown hotels, approximately 10 miles away. One particular shuttle makes 16 round trips per day, carrying 5 passengers per trip. The fare each way is $18. What are the receipts from one day’s operation for this particular shuttle?

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SolutionList all the information:Distance from airport to downtown hotels ≈ 10 miles* Number of round trips per day = 16* Number of passengers per trip = 5* Fare each way = $18We need only the information with a *.


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SolutionRound trip costs 2 $18 = $36The total receipts for one day is the product of the number of round trips, the number of passengers, and the cost per round trip

= 16 5 $36 = $2880


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Example 3: Retirement

It is never too early to start planning for retirement. U.S. Census Bureau data indicate that at age 65 the average woman will live another 20 years and the average man will live another 17.2 years. The data also indicate that about 33% of the average person’s retirement income will come from Social Security.

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When discussing retirement planning, many investment firms and financial planners use the graph in the figure on the next slide which shows how long a typical retiree’s assets (or “nest egg”) will last based on the percentage of the assets withdrawn each year.

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a)If a typical retiree has retirement assets of $500,000, how much can he or she withdraw annually if he or she wishes the assets to last 21 years?

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Solutiona) From the graph, we can see that for assets to last 21 years, about 6% of the assets can be withdrawn annually. The amount that can be withdrawn isAmount = 6% of assetsAmount = 0.06(500,000) = $30,000Thus, about $30,000 can be withdrawn annually.


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b)How much should a retiree have in assets if he or she wishes to withdraw $25,000 annually and wishes his or her assets to last 18 years?

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Solutionb) From the graph, we can see that for assets to last 18 years, about 7% of the assets can be withdrawn annually. That amount has to equal $25,000:7% of assets = $25,0000.07 assets = $25,000assets = 25,000 ÷ 0.07 = 357,142.86The retiree needs $357,142.86


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Example 4: Determining a Tip

The cost of Freddie Rojo’s meal before tax is $28.00.

a) If a 6 ½ % sales tax is added to his bill, determine the total cost of the meal including tax.

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SolutionChange 6 ½ to a decimal: 0.065Sales tax = 6 ½ % of mealSales tax = 0.065(28.00) = 1.82The total bill = cost of meal + sales taxTotal bill = 28.00 + 1.82 = $29.82The bill including sales tax is $29.82


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b) If Freddie wants to leave a 10% tip on the pretax cost of the meal, how much should he leave?

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SolutionTo find 10% of any number, we multiply the number by 0.1010% of pretax cost = 0.10(28.00)

= 2.80A simple way to find 10% of any number is to move the decimal point in the number one place to the left:

28.00 2.80


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c) If he wants to leave a 15% tip on the pretax cost of the meal, how much should he leave?

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SolutionTo find 15% of $28.00, multiply:15% of $28.00 – 0.15(28.00) = 4.20

Or to find 15%, find 10% and add it to ½ that amount:

$2.80 + = $2.80 + $1.40 = $4.20


$2.80

2

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Example 6: Spraying Weed KillerThe instructions on the Ortho Weed-B-Gon lawn weed killer indicate that to cover 1000 square feet (ft2) of lawn, 20 teaspoons (tsp) of the weed killer should be mixed in 5 gallons (gal) of water. Ron Haines wishes to spray his lawn with the weed killer using his pressurized sprayer.a) How much weed killer should be mixed with 8 gal of water to get a solution of the proper strength?

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SolutionUse: 20 teaspoons of weed killer is to be mixed with 5 gal water

Example 6: Spraying Weed Killer

Note: teaspoons and gallons are placed in the same relative positions.Replace the ? with an x.

20 tsp

5 gal water

? tsp

8 gal

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Solution


Ron must mix 32 tsp of weed killer with 8 gal of water.(32 tsp = 10 2/3 tablespoons = 2/3 cup)

20

5

x

820 8 5x

160 5x

x 160

532

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Example 6: Spraying Weed KillerThe instructions on the Ortho Weed-

B-Gon lawn weed killer indicate that to cover 1000 square feet (ft2) of lawn, 20 teaspoons (tsp) of the weed killer should be mixed in 5 gallons (gal) of water. Ron Haines wishes to spray his lawn with the weed killer using his pressurized sprayer.b) How much weed killer is needed to cover an area of 2820 ft2 of lawn?

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SolutionUse: 1000 ft2 requires 20 teaspoons of weed killer


Areas may be on top or bottom as long as they are in the same relative position.Replace the ? with an x.

1000 sq ft

20 tsp

2820 sq ft

? tsp

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Solution


Thus, 56.4 tsp are needed.

1000

20

2820

x1000 x 20 2820

1000x 56,400

x 56,400

100056.4

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Section 1.3 Problem Solving