P.8 Modeling with Equations Notes - Honors PreCalculus and ...

false.

statement

is false,

the necessary

to

false.IfIf the

the statement

is false,

make make

the necessary

change(s) tochange(s)

If the width

of a rectangle

islength

represented by x and the length

173. If the width173.

of a rectangle

is represented

by x and the

produce aa true

produce

truestatement.

statement.

is represented byisx represented

+ 200, write a by

simplified

algebraic

x + 200,

write a simplified algebraic

2

163. The equation 12x - 32 = 25

to 2x - 3 = 5.

that models the rectangle¡¯s perimeter.

163.

The equation 12x - 322 is= equivalent

to 2x - 3 expression

= 5.

25 is equivalent

expression that models the rectangle¡¯s perimeter.

-134-hr

o n P.8

Modeling

with Equations

Equations

with

P.8S e c t i Modeling

S e c t i o Page

n P.8

102

14-11-2008

Objective 18:05

Objective

equations to solve

! Use

0P_001-134-hr

14-11-2008

18:05

problems.

Modeling with Equations

How Long It Takes to Earn $1000

Page 102

How Long It Takes to Earn $1000

! Use equations to solve

problems.

Chapter P Prerequisites: Fundamental Concepts of Algebra

Howard Stern

Dr. Phil McGraw

02 Chapter P Prerequisites: Fundamental Concepts

of Algebra

Radio host

Television host

24 sec.

2 min. 24 sec.

Brad Pitt

Actor

4 min. 48 sec.

Problem

Solving with Equations

to solve problems.

Problem Solving with Equations

!equations

Use equations to solve problems.

Kobe Bryant

Basketball player

5 min. 30 sec.

Stern

Dr. Phil McGraw

Brad Pittof a real-world

Kobe Bryant

WeWe

have

seen that Howard

a model

is a mathematical

representation

have seen that aRadio

model

representation

of a real-world

hostis a mathematical

Television host

Actor

Basketball player

situation.

In

this

section,

we

will

be

solving

problems

that

are

presented

24 sec.

min. 24 sec.

4 min.are

48 sec.

30 sec.

situation. In this section,

we will be 2solving

problems that

presented 5inmin.in

English.

This

means

that

we

must

obtain

models

by

translating

from

the ordinary

English. This means that we must obtain models by translating from the ordinary

Chief

executiveinto

G.P.

Highof

school

teacher equations.

Janitor

language

English

the

language

ofalgebraic

algebraic

equations.

To translate,

languageofof

English

into Doctor,

the language

To translate,

U.S. average

U.S. average

U.S. average

U.S. average

however,

wewemust

understand

the

English

prose

and

be

familiar

with

the forms

however,

must

understand

the

English

prose

and

be

familiar

with

the

forms

of of

2 hr. 55 min.

13 hr. 5 min.

43 hours

103 hours

algebraic

language.Here

Here are

are some

wewe

willwill

follow

in solving

wordword

algebraic

language.

some general

generalsteps

steps

follow

in solving

problems:

problems:

Source: Time

tudy Tip

Tip

When solving word problems,

solving word

problems,

articularly

problems

involving

rly

problems

involving

eometric figures, drawing

a picture

c the

figures,

drawing

a picture

situation

is often

helpful.

situation

is often

abel

drawinghelpful.

and, where

x on your

ppropriate,

label other

of the

on your drawing

and,parts

where

awing

in terms

of parts

x.

ate,

label

other

of the

in terms of x.

I

n this section,

you¡¯ll

see examples and

exercises

onschool

how much

money

Chief

executive

Doctor,

G.P. focused

High

teacher

Janitor

solving U.S. average

problems. As you

become

familiar with13this

to solve a wide

2 hr.

55 min.

hr.strategy,

5 min. you will learn

43 hours

103 hours

Step 1 Read

the problem carefully. Attempt to state the problem in your own

problems.

Step 1 variety

Readofthe

problem carefully. Attempt to state the problem in your own

Strategy

forSolving

Solving

Word

Problems

Americans

earn.

situations

illustrate

a step-by-step U.S.

strategy

for

U.S.These

average

U.S. average

average

Strategy

for

Word

Problems

words and state what the problem is looking for. Let x (or any variable)

words

and one

state

what

the problem

is inlooking

for. Let x (or any variable)

represent

of the

unknown

quantities

the problem.

Source:

Time

represent

one

of

the

unknown

quantities

in

the

problem.

Step 2 If necessary, write expressions for any other unknown quantities in the

Step

2 If in

necessary,

write

expressions

forexamples

any otherand

unknown

quantities

thehow much money

problem

terms

of x.

n this

section,

you¡¯ll see

exercises

focusedinon

problem

terms

of

x.

Step 3 inWrite

an

equation

in

x

that

models

the

verbal

conditions

of

the

problem.

Americans earn. These situations illustrate a step-by-step strategy for solving

Step

3 4 Write

an

x that

models

the verbal

conditions

of the

Step

Solve

theequation

equation

answer

thefamiliar

problem¡¯s

question.

problems.

Asinand

you

become

with

this strategy,

youproblem.

will learn to solve a wide

variety

of problems.

Step

Check

solution

in the

original

of the

problem, not in the

Step

4 5Solve

thethe

equation

and

answer

thewording

problem¡¯s

question.

I

equation obtained from the words.

Step

5 Check the solution in the original wording of the problem, not in the

equation obtained from the words.

EXAMPLE 1

Celebrity Earnings

Celebrity

Earnings

EXAMPLE

1 published

Forbes magazine

a list

of the highest paid TV celebrities between June

2006 and June 2007. The results are shown in Figure P.14.

Forbes magazine published a list of the highest paid TV celebrities between June

Highest

Paid

TV Celebrities

between

2006 and

June 2007

2006 and June 2007.

The

results

are shown

inJune

Figure

P.14.

440

Earnings (millions of dollars)

400

Highest Paid TV Celebrities between June 2006 and June 2007

360

440

320

Figure P.14

Source: Forbes

Earnings (millions of dollars)

400

280

360

240

320200

280160

240120

200 80

$40 million

160 40

120 0

80

40

Oprah

Winfrey

Jerry

Seinfeld

Simon

Cowell

Celebrity

David

Letterman

$40 million

$32 million

Donald

Trump

$32 million

0

Figure P.14

Source: Forbes

Jerrynobody came

Simon close toDavid

Oprahindicate that

The bar heights

Oprah, who Donald

earned over four

Letterman

Trump $15 million

times moreWinfrey

than any of Seinfeld

the other TV Cowell

stars. Although

Seinfeld earned

Celebrity Cowell¡¯s by $215 million. Combined,

more than Cowell, Oprah¡¯s earnings exceeded

these three celebrities earned $365 million. How much did each of them earn?

The bar heights indicate that nobody came close to Oprah, who earned over four

Solution

times

more than any of the other TV stars. Although Seinfeld earned $15 million

Step

1 Let

x represent

oneearnings

of the unknown

quantities.

We

about

more

than

Cowell,

Oprah¡¯s

exceeded

Cowell¡¯s

byknow

$215something

million. Combined,

Seinfeld¡¯s

earnings

and

Oprah¡¯s

earnings:

Seinfeld

earned

$15

million

more

than

these three celebrities earned $365 million. How much did each of them earn?

Cowell, and Oprah¡¯s earnings exceeded Cowell¡¯s by $215 million. We will let

Rectangle

Circle

A lw

P 2l 2w

A pr2

C 2p-r

100x

s

s

n if ?

a right

The

hen p.

true

e, the

eorem

ice, x,

t: If

nd not

a right

200

Length

0 x

a is

oning

value

olving

often

pecific

and

write the quadratic

equation

Triangle

Trapezoidin105

Section P.8

Modeling

with Equations

general

form.

A q bh

A

q h(a b)

2

Factor wording

out 2, theof

GCF.

+

24002

= 0 original

Step 5 Check the21x

proposed

solution

in the

the problem. The

104

Chapter

P

Prerequisites:

Fundamental

Concepts

of

Algebra

problem states that21x

all -men

(100%,

represented

by

100

using

the model)107

will

Section

Modeling

with Equations

Factor

the trinomial.

2021x + 1202

= 0 P.8

b

consider the objective essential or very important. Does this occur if we increase the

SectionFreshmen,

P.8 Modeling

109

Life Objectives of

1969¨C2006 with Equations

w

Set each

variable factor equal

to 0.

x - 52%,

20 = by

0 0.6

or

x r+year

120College

= 80

0 years,

1969

percentage,

each

for

our

EXAMPLE

2 Modeling Attitudes of

h proposed solution?

Step 4 Solve the equation and answerWomen

the question.

h

Men

College Freshmen

x = 20

x = - 120 Solve for x.

100%

Solution l

52 + 0.61802 = 52 +

48

=

100

88%

b

Researchers

have surveyed college freshmen

a

82%

a. Step 1 Let x90%

represent one of the unknown quantities.

We will let

2(x+200)+2x=1040

This

is the

equation thatevery

models

thesince 1969. Figure P.15 shows that

The path cannot 80%

have

a negative width.

Because

is geometrically

impossible,

120

75%

year

72%

Thus,

This

using

shown

in vertical

Figure

P.15,

first-year college

men

willsome life goals have changed

xof=the

the

ramp¡¯s

distance.

70%

attitudes

about

problem¡¯

we

useverifies

The

widthtrends

path

should

be 20

feet.alls conditions.

x = 20.that

60%

consider

the objective

essential

very P.18.

important 80 years after 1969.dramatically. Figure P.15 shows that the

52%

is illustrated

in or

Figure

Rectangular

47%of theproperty.

StepThe

5 situation

Check the

solution

problem.

Hasof 2006 was more interested in

Applywording

the

distributive

2x + Circular

400

+ 2x in

= the

1040original

46%

50%proposed

freshmen class

Solid

Cylinder

Sphere

Cone

40%

2 Represent

other

unknown

quantities

in

terms

of

There

are

no

other

x.

the Step

landscape

architect

doubled

the

garden¡¯s

area

with

the

20-foot-wide

path?

Thethan the freshmen of 1969 had

making

money

Combine3 like terms: 2x + 2x = 4x.

30% 4x +2 400 = 1040

2h

30%

Vtimes

pr h

Vsquare

dp r feet. Becausebeen.

V80 ap

r2x

The dimen

In

1969,

of first-year college men

quantities,

so

we

can

skip

this

step.

areaunknown

ofVthelwh

garden

is 80

feet

60

feet,

or

4800

and

+

that theSubtract

freshmen

of considered

2006

less52%very

Check Point20% Figure P.15 shows

4x = 640

400 class

from both

sides.was ¡°being

well

off

financially¡±

length is u

the

and

width of the

expanded

rectangle,

60 interested

+Step

x represent

3 Write

anlength

equation

in

models

thethan

conditions.

Figureof

P.18

shows

in 10%

developing

a philosophy

the freshmen

1969

had

r x that

x = 160of life

Divide both sides by 4. essential or very important. For the period from

the 80

right

triangle

formed

by

the

the wall,

and theessential

ground.

We

can

# 1969

Step 5 C

been.

In+ 1969,

the

considered

or very

1969 through

2006, this percentage increased by

2x = 88%

80 that

+ of

2 is

20

=women

1202006

feet

is ramp,

the expanded

rectangle¡¯s

length.

2006

1969 this objective

Thus,

approximately

0.6 each year. If this trend

find

the

ramp¡¯s

vertical

distance,

using

the

Pythagorean

Theorem.

x,

r

perimeter

well

¡°Developing

a meaningful

important. Sinceh then, ¡°Being

this very

percentage

has

decreased

by

approximately

1.1

h

h

philosophy ofrectangle¡¯s

life¡±

60year.

+ xIf =this

60 +

20 offcontinues,

=x80

expanded

width.

continues,

by which year will all male freshmen

width

=financially¡±

= feet

160.isbythe

each

trend

which

year

will

only

33%

of

female

w

2

considerr ¡°being very well off financially¡±

Objective

(leg)2

plus

(leg)Life

equals

(hypotenuse)2

consider

¡°developing

meaningful

philosophy

of life¡±

essential

orimportant?

length

= x +isa200

=

160

+ 200

= 360.

Thefreshmen

area oflthe

expanded

rectangle

120

feet

times

80 feet,

or 9600

square

feet.orThis

essential

very

Figure P.15

Because t

very important?

is

double

the

area

of

the

garden,

4800

square

feet,

as

specified

by

the

problem¡¯s

Source:

John Macionis,

Sociology

,

Twelfth

2

2 360 feet. (The 360-foot

The dimensions

of an American

are

x22008 +football

120field

=160 feet

122by

Solution

Edition, Prentice Hall,

dimension

conditions.

length is usually described as 120 yards.) Step 1 Let x represent one of the unknown quantities. We are interested in the

Step 4 Solve the equation and answer the question. The quadratic equation

year

when

all male

freshmen,

or 100% of the The

men, will consider this life objective

onoriginal

a Digital

Camera

EXAMPLE

3122proposed

2A Price Reduction

Step

solution

in the

of the

Check P

can be solved

most

efficiently

bywording

the

square

rootproblem.

property.

x25 + Check

1202 =the

essential

or very

important.

Let

A

rectangular

garden

measures

16

feet

by

12

feet.

A

path

Check

Point

perimeter

of the

field

thethe

dimensions

found isP = 2l + 2w,ofin

We will

befootball

using

the

formula

perimeterthat

of awe

rectangle,

the wid

2

2 using2for

This

isentire

the

equation

resulting

from

the

120

122

Your

local

computer

store

is=having

athat

terrific

sale

on

digital

cameras.

After

40%

uniform

width

isxThe

to +be

added

so

as to

surround

the

garden.

The

landscape

our

next

example.

formula

states

a rectangle¡¯s

perimeter

is the

sum

ofa twice

dimensi

21360 you

feet2

+wants

21160a feet2

=camera

720

feet

+$276.

320

feet

=an

1040

x =Theorem.

the

number

of320

years

after

1969 when all

price

reduction,

purchase

digital

forPythagorean

What

was

thefeet.

camera¡¯s

price

artist

doing

work

garden

and

path

to cover

area

of

square

its

length

andthe

twice

its

width. the

2

male

freshmen

will

consider

before

the

reduction?

120 and 122.is 1040 feet, our ¡°being very

xshould

+ 14,400

= 14,884

feet. How

the

path

be?

Because

thewide

problem¡¯s

wording

tells us that Square

the perimeter

We w

well off financially¡± essential or very

2

dimensions

are correct. x = 484

Isolate x2 by subtracting 14,400 from

Solution

The

formu

important.

The solution

next problem

relies on knowing

the Pythagorean

Theorem.

Finding

the Dimensions

of an

American

Football

Field

EXAMPLE

4 to our

both

sides.

x represent

Steptheorem

1 Let relates

one of the

unknown

quantities.

Wetriangle,

will let a triangle with

The

the

lengths

of

the

three

sides

of

a

right

The length

a rectangular

basketball

court root

is 44property.

feet more than

Check

Point

xthe

=4original

2484

or of

= 2484camera

Applyprior

the square

xmeasuring

=of

price

ofxthe

digital

the

reduction.

one

angle

90¡ã. The

side

opposite

the

90¡ã

angle

isto288

called

the

hypotenuse.

Step

Represent

other

unknown

quantities

terms of x. There are no other

The

length

an

American

football

field

is 2200

feet

more

than

thewhat

width.

If inits

the

EXAM

the

width.

If

the

perimeter

of

the

basketball

court

is

feet,

are

x

=

22

x

=

22

Simplify.

unknown

quantities

to

find,

so

we

can

skip

this

step.

The

other

sides

are

called

legs.

The

legs

form

the

two

sides

of

the

right

angle.

Step

2 Represent

other

unknown

quantities

in terms of x. There are no other

perimeter

of the field

is 1040

feet, what

are its dimensions?

dimensions?

Step

3 step.

Write an equation in x that models the conditions.

A rectang

unknown

to find,

we canvertical

skip

this

Becausequantities

thesoramp¡¯s

distance, this measurement must be

x represents

80 2x

to be add

The

Pythagorean

Theorem

Solution

positive.

We

reject

Thus,

the

ramp¡¯s

vertical

distance

is

22

inches.

22.

Step We

3 will

Write

that

models

the conditions.

camera¡¯s

original

A = lw,The

usean

theequation

formula in

forxthe

area

of a rectangle,

in our

next example.

landscape

The 1969

increased

0.6B

each year

100% of the

price

the

40%

the

reduced

price,

The

sum

ofCheck

the

squares

of the

lengths

of

the

legsoriginal

of$276.

a its

x represent

Step

1minus

one

of is

the

unknown

quantities.

We

know

something

about

The

formula

states

that

aproposed

rectangle¡¯s

area

is the

product

of

length

and

Step

5Let

thereduction

solution

in the

wording

thewidth.

equals male

percentage

byof its

forproblem.

x years

x freshmen.

x

addition o

right

triangle

the

square

ofthat

the

length

ofthe

theWe

the

length;

the equals

length

is

200

feet more

than

the

width.

letthe ground form a

The

problem¡¯s

wording

implies

the

ramp,

wall,will

and

the reduction

52of +

0.6x

=

100

hypotenuse.

right triangle. This

can be checked

the converse

the Pythagorean

(40% ofusing

the

the reduced

60 feet

Original

60 x

Solution

a

c

Solving

asides

Problem

Involving

Landscape

EXAMPLE

5a triangle

x

=

the

width.

price)

minus

aoriginal

b,

If the Iflegs

have

lengths

and

the

$276.c is Design

price has

Theorem:

of

lengths

where

the

length

of

the

a,and

b, isand

c,price,

Leg

80 feet

Hypotenuse

Step 1 L

P-BLTZMC0P_001-134-hr 14-11-2008 18:05 Page 109

hypotenuse

hasand

length

longest side,

if a2c,+then

the 4triangle

is aequation

right triangle.

Let¡¯s

check

b2 = c2, then Step

Solve

the

anduniform

answer

the

question.

x.

2 Represent

unknown

quantities

in

terms

of

Because

the

length

is

AStep

rectangular

gardenother

measures

80

feet

by

60

feet.

A

large

path

of

width

is

x

= triangle

276 with the ramp¡¯s length

that a vertical distance

inches0.4x

forms a right

2 x of

2 22

+width,

bshorter

= we

c2. add

aboth

morealong

than the

200and

to the

to represent

thegarden.

length.

Thus,

to200

befeet

added

sides

onewidth

longer

sideA of 2 the

The

2

2

C

b 120= 100

of 4122Solve

inches

its horizontal

distance

of 120 inches. Is 22

= 122 ? This is the equation that models

52 ++0.6x

Step

theand

equation

andwork

answer

the question.

P.17 The garden¡¯s area is

landscape

designer

doing the

wants

to double the garden¡¯sLeg

area with the theFigure

The situat

problem¡¯s conditions.

Simplifying the arithmetic,

we

obtain

the

true

statement

Thus,

14,884

=

14,884.

x

+

200

=

the

length.

to be doubled by adding the path.

- 0.4x

276 the

Thispath

is thebe?

equation that models the problem¡¯

s

addition of this path.xHow

wide= should

rectangula

a vertical distance of 22 inches formsconditions.

a right triangle. 52 - 52 + 0.6x = 100 - 52 Subtract 52 from both sides.

longer side

Figure P.16 illustrates an American football field and its dimensions.

Solution

b. Every vertical rise of 0.6x

1 inch= requires

a horizontal

runx of

the

like terms:

276 Combine

- 12

0.4xinches.

= 1x0.6x

-Because

0.4x

=

0.6x.

Simplify.

= 48

Step 3 Write

an equation

in x that

the

Because the

perimeter

Step 2 R

hasx arepresent

vertical

distance

ofunknown

22models

inches,

it conditions.

requires

a will

horizontal

distance

of

Stepramp

1 Let

one

of the

quantities.

We

let

0.6x

276

Solution

Using

the

Pythagorean

Theorem

EXAMPLE

6

of 22(12)

the field

is

1040

feet,

0.6x

48

added alon

Divide

both

sides

by

0.6.

=

inches, or 264 inches.

horizontal distance is only 120 inches,

= so this Divide both sides by 0.6.

a. Step 1 Let x re

0.6x = The

0.6 width

the

of

the

path.

0.6

0.6

ramp does not satisfy construction laws for access ramps for the disabled.

a. A wheelchair ramp

122theinches460.

has athehorizontal distance of

Twicewith

the a length oftwice

The situation is illustrated

in

Figureplus

P.17. The

figure isshowsperimeter.

the original 80-by-60

x = 80 foot Simplify.

The situation is i

width

120 inches. What is length

the ramp¡¯s

vertical

distance?

Simplify:

0.6 ! 276.0

x

= of

460

A radio

tower

iswidth

supported

by two

wires

are each

Check Point

6 and

x added

rectangular

garden

the path

along

boththat

shorter

sides130

andyards

one

Step 2 Represe

Using

current

trends,

byaccess

80 years

aftertall

1969,

or in 2049,x all122

male

b.long

Construction

laws

areground

very specific

when

it comes

to

ramps

foris the

in. freshmen will

and

attached

to

the

50

yards

from

the

base

of

the

tower.

How

the

unknown quanti

longer

side.

The digital camera¡¯s2(x+200)

price before the

$460.

+reduction

2x was

=

1040.

consider

¡°being

well off financially¡±

you agree

Step 3 W

disabled. Every vertical rise of 1 inch

requires

a very

horizontal

run of 12essential

inches.or very important. (Do

tower?

Step 3 Write a

12

0

in

with

this

projection

thatx.extends

soproblem.

farthe

into path

the future?

Are there unexpected. events

Step

25 Represent

other

unknown

quantities

in

terms

of

Because

is

StepDoes

Check

the

proposed

solution

in

the

original

wording

of

the

The

this ramp satisfy the requirement?

the rectangle

right trianglm

that might cause model breakdown to occur?)

added

along both

shorter sides

and

one the

longer

side,

Figure should

P.17 shows

that

Figure P.18

find x, the ramp

priceIn

before

the

reduction,

$460,

minus

40%

reduction

equal

the

reduced

our final example, the conditions are modeled by a rational equation.

price given in the original wording, $276:

(

80 + 2x = the length of the new, expanded rectangle

460 - Dividing

of 460the

= 460

- of

0.414602

= 460 - 184 = 276.

Cost

a Yacht

EXAMPLE

760 +40%

x = the width

of the

new,

expanded rectangle.

This verifies that the digital camera¡¯s price before the reduction was $460.

Step 4 Solve th

Step

3 Write

an agrees

equation

in x that

models

conditions.

The area

of the

x2 + 1202 = 122

A

group

of friends

to share

the cost

of a the

$50,000

yacht equally.

Before

rectangle is

be doubled

the addition

of group

the path.

x

purchase

made,

one

moreabyperson

joins

the

andpurchase

enters the

agreement.

Asfor

a

After

30%

price

reduction,

you

a new

computer

Check must

Point

result,

each

person¡¯s

share

is

reduced

by

$2500.

How

many

people

were

in

the

$840. What was the The

computer¡¯s

price beforemust

the reduction?

area, or length times width,

twice

the area of

x2 +

original group?

of the new, expanded rectangle

be

that of

the garden.

Percentage Calling Objective

¡°Essential¡± or ¡°Very Important¡±

e

2

5

3

Solution

Step 1

2 ! 80We!will60let

Let x represent(80+2x)(60+x)

one of the unknown=

quantities.

x = the number of people in the original group.

Study Tip

The Pythagorean Theorem is an if ?

then statement: If a triangle is a right

triangle, then a2 + b2 = c2. The

x = 2

x = 22

Because x repre

positive. We rejec

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