RICHARD N. AUFMANN - Discovering Mathematics A Quantitative Reasonin...

Complete Solutions Manual

Discovering Mathematics A Quantitative Reasoning Approach

RICHARD N. AUFMANN

Palomar College

Prepared by Christi Verity

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1 © 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Chapter 1: Introduction to

Problem Solving

T

HINK ABOUT IT 1.1

• Inductive

• Specific

• One example is 5, which is an odd number.

EXERCISE SET 1.1

• The numbers are the squares of consecutive

integers. 8 2 = 64.

• Subtract 1 less than the integer subtracted

from the previous integer.

3.13 .15 Add 2 to the numerator and denominator.

4.6 .7 Add 1 to the numerator and denominator.

• –13. Use the pattern of adding 5, then

subtracting 10 to obtain the next pair of numbers.

• Subtract 8 from 5, add 12 to –3, subtract 16

from 9, add 20 to –7, etc., increasing the amount by 4 each time while alternating between adding and subtracting.

• Each image has a smaller square inside a larger

square. The smaller square moves to a new position in a counterclockwise direction. The next figure is shown.

• Each image has a smaller square inside a larger

square. The smaller square moves to a new position in a counterclockwise direction. The next figure is shown.

• Each image has a smaller square and a circle

inside a larger square. The smaller square moves to a new position in a corner in a counterclockwise direction. The circle moves to a new position in a counterclockwise direction. The next figure is shown.

• Each image has a smaller square and a circle

inside a larger square. The smaller square moves to a new position in a corner in a counterclockwise direction. The circle moves to a new position in a clockwise direction. The next figure is shown.

• Each figure is a circle with a polygon,

alternating positions inside and outside. The first figure is a triangle (3 sides) with a circle inside. The second is a circle with a square (4 sides) inside. The next figure is shown.

• Each figure is a polygon within a polygon,

alternating blue and yellow interiors. The first figure is a 7-sided figure inside an 8-sided figure. The second figure is a 6-sided figure inside a 7-sided figure. The next figure is shown.

• The amount is decreasing by $1000 per month.

Thus, the amount in year 6 will be $5000.

• a. Greater since the temperature is increasing.
• No. Fall and winter would come and the

temperature would decrease.

• More than. The increase in average annual

salary increases from $7763 to $8927 to $10,266. Thus the increase from 20 years to 25 years will be more than $10,266, giving an average annual salary for a teacher having 25 years of experience of greater than $78,705 + $10,266 = $88,971, which is greater than

$88,000.

• Fewer. From the bar graph, it appears that as

the price of cell phones increases, the number of cell phones sold decreases. Assuming the same trend, if the price of the cell phone is $700, the company will sell fewer than 333,000 cell phones.

• For every 10 seconds, the distance is increasing

by 300 feet. Therefore, after 70 seconds, the athlete will run a distance of 2100 feet.

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2 Chapter 1: Problem Solving

© 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

• For every 2 hours, the distance is decreasing by

100 miles. Therefore, after 6 hours, the trip will have 200 miles remaining.

• From the table, as the depth increases, the

temperature decreases. Since the last entry of the table is 40 m and 11°C, then for 45 m, the temperature should be less than 11°C.

• Each year the car depreciates by a smaller

amount: $5336, $5191, $2930, and $2878. The

value of the car at year 5 is $11,743. After year 6, the value of the car will be less than $11,000.

• From the table, the distance (in feet) is equal to

the time (in seconds) squared. So at 7 seconds the distance is 49 feet, and at 8 seconds the distance is 64 feet.

• The pattern from 0 seconds to 7 seconds,

repeats again starting at 8 seconds. So at 14 seconds, the distance will be the same as at 6 seconds, which is 0 inches.

• Answers will vary. For instance,

11 1 .42 8 ´=

• Answers will vary. For instance, 5 – (–3) = 8.

• A diamond shape can have four equal sides and

is not a square.

• Answers will vary. For instance, 20.

• Whales are mammals and do not have legs.

• A penguin is a bird that does not fly.

• a. Deductive
• Inductive

• a. Inductive
• Deductive

• Place 4 coins in the left balance pan, 4 coins in

the right balance pan, and set 4 coins aside.There are three possibilities. The right side goes down, the left side goes down, or the scale balances (the heavier coin is set aside).Take the 4 coins that include the heavier coin and place 2 on the left balance pan and 2 on the right balance pan.From the 2 coins on the heavier pan, place one coin on each balance pan to determine the heavier coin.

• Label the coins 1, 2, 3, … 8. Set coins 5, 6, 7,

and 8 aside.

Weighing 1: Place coins 1 and 2 on one

balance and coins 3 and 4 on the other.If the scale is balanced, the heavier or lighter coin is 5, 6, 7, or 8. Coins 1, 2, 3, or 4 are of equal weight.If the scale is unbalanced, the heavier or lighter coin is 1, 2, 3, or 4. Coins 5, 6, 7, or 8 are of equal weight.

Weighing 2: From the four coins that contain

the heavier or lighter coin, choose two coins, placing one coin on each balance, and set the other two coins aside. Suppose that coins 5, 6, 7, or 8 are not of equal weight. Choose coins 5 and 6 and place one on each side of the balance.If the scale is balanced, then coins 5 and 6 are of equal weight and the coin that is heavier or lighter is either coin 7 or 8.If the scale is unbalanced, then coins 7 and 8 are of equal weight and the coin that is heavier or lighter is either coin 5 or 6.

Weighing 3: Suppose the coin that is heavier

or lighter is either coin 7 or 8. Place coin 7 on one balance and coin 1 (or any one of coins 2 to 6, since they are of equal weight), on the other balance.If the scale is balanced, then coin 8 is the coin that is heavier or lighter.If the scale is unbalanced, then coin 7 is the coin that is heavier or lighter.

• Label each stack of coins as 1, 2, 3, … 10.

From each stack select the same amount of coins as the labels, so select 1 coin from stack 1, 2 coins from stack 2, and so on to 10 coins from stack 10. Weigh the selected coins. Since the counterfeit coins each weigh 0.1 g more, the multiple of 0.1 g over the weight expected will determine which stack contains the counterfeit coins.

• Label each stack of coins as 0, 1, 2, 3, … 10.

Note that there are 11 stacks. From each stack select the same amount of coins as the labels, so select 0 coins from stack 0, 1 coin from stack 1,

• coins from stack 2, and so on to 10 coins from

stack 10. Weigh the selected coins. Since the counterfeit coins each weigh 0.1 g more, the multiple of 0.1g over the weight expected will determine which stack contains the counterfeit coins. Note that if the weight is not more than expected, the counterfeit coins are in stack 0.

35.Util Auto Tech Oil

AX1 X1 X3

TX1 X1 X3

MX2X3X3

JX2 X2X2

   

Maria: the utility stock; Jose: the automotive stock; Anita: the technology stock; Tony: the oil stock

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Chapter 1: Problem Solving 3

© 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

36.Soup Entree Salad Dessert

CX2 X3 X2

SX2 X3 X2

OX1 X2 X2

GX2X2X2

    Changs: entrée; Steinbergs: salad; Ontkeans:

dessert; Gonzaleses: soup

37.Coin Stamp Comic Baseball

AX3 X3 X4

CX2 X4 X1

PX3X3X2

SX2 X3 X3

    Atlanta: stamps; Chicago: baseball cards; Philadelphia: coins; San Diego: comic books

• a. Yes. Change the color of Iowa to yellow

and Oklahoma to blue.

b. No. One possible explanation: Oklahoma,

Arkansas, and Louisiana must each have a different color than the color of Texas and they cannot all be the same color. Thus, the map cannot be colored using only two colors.

THINK ABOUT IT 1.2

• Estimate

• Estimate

• Exact

• Estimate

• Exact

• Exact

2.6

5.32 10 0.00000532

- ´= The answer is b, greater than 0 but less than 1.

• True

• False;
• 1

10 0.001

1000- ==

• Yes

EXERCISE SET 1.2

1.1487 miles 1500 miles 1500 miles 500 miles per day

• days

» =

2.11 children 10 children; 3 slices 4 slices

• slices

10 children 40 slices

• child

40 slices

• pizzas
• slices/pizza

»» ´= =

• pizzas is conservative; 5 pizzas should be

plenty.

3. 2(25) 30 80+=

So, 25°C ≈ 80°F.

• 238 square feet 240 square feet

$0.28 $0.30

240 $0.30 $72

» » ´= The cost will be approximately $72.

• 400 20 8000 square feet´=

• Since 1 gallon of paint covers 350 square feet,

and the room has 400 square feet of wall space, including windows and doors, you should purchase 2 gallons of paint.

• Answers will vary depending on the grid.

Using a 3 by 5 grid, we count 30 cars. Multiply 30 by the number of sections, 15, to get 450.So our estimate is 450 cars.

• Answers will vary depending on the grid.

Using a 3 by 5 grid, we count 28 flowers.Multiply 24 by the number of sections, 15, to get 360. So our estimate is 360 flowers.

• Answers will vary depending on the grid.

Using a 3 by 5 grid, we count 18 cadets.Multiply 15 by the number of sections, 15, to get 225. So our estimate is 225 cadets.

• Answers will vary depending on the grid.

Using a 4 by 5 grid, we count 9 marchers.Multiply 9 by the number of sections, 20, to get

• So our estimate is 180 marchers.

11.6, 400,000 square feet 1,600,000 people

• square feet/(1 person)

= or 1.6 million people

12.

68.3 square miles 68 square miles 10,000 people 68 sq mi 680,000 people

• sq mi

» ´= Approximately 680,000 people

• If a 10-foot pole casts a shadow of 18 feet, and

a telephone pole has a shadow of 72 feet, then the telephone pole will be more than half as high as its shadow is long. The telephone pole has height

10 72 40 feet 18 ´=

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