MainMain
point:
Solve
perimeter,
value,
interest,
and mixture
problems.
point:
Solve
perimeter,
value,
mixture
problems.
154
CHAPTER
2 interest,
Lectureand
Notes,
Detailed
Comments, and Additional Explorations
Math098 Worksheet (6.5) OBJECTIVE
1
OBJECTIVE 1
SECTION 6.5 LECTURE NOTES
Five-Step
Problem-Solving
Method
Five-Step
Problem-Solving
Method
Objectives
To
solve
somesome
problems
in which
we want
to find
two two
quantities,
it is ituseful
to perform
the following
five five
To solve
problems
in which
we want
to find
quantities,
is useful
to perform
the following
steps:
1.steps:
Know a five-step problem-solving method.
2.
Use 1:
a system each
of two
linear equations
or a linear
to solve
perimeter,
value,
interest,
and
• Step
variable.
For each
quantity
that function
we are
to find,
we usually
define
a variable
• StepDefine
1: Define each
variable.
For each
quantity
that
wetrying
are trying
to find,
we usually
define
a variable
mixture
problems.
to betothat
unknown
quantity.
be that
unknown
quantity.
• Step
2: Write
a system
of two
equations.
We find
a system
of two
equations
by using
the variables
• Step
2: Write
a system
of two
equations.
We find
a system
of two
equations
by using
the variables
fromfrom
step step
1. We
can
usually
write
each
equation
either
by
translating
the
information
stated
in the
1. We canvalue,
usually
write and
eachmixture
equation
either by translating the information
stated
in the
Main point: Solve perimeter,
interest,
problems.
problem
into
mathematics
or
by
making
a
substitution
into
a
formula.
problem into mathematics or by making a substitution into a formula.
OBJECTIVE
1 the system. We solve the system of equations from step 2.
• Step 3: Solve
• Step 3: Solve the system. We solve the system of equations from step 2.
• Step
4: Describe
eacheach
result.
We use
complete
sentence
to describe
the quantities
we found.
• Step
4: Describe
result.
We ause
a complete
sentence
to describe
the quantities
we found.
Five-Step
Problem-Solving
Method
• Step
5: Check.
We reread
the problem
and and
check
that that
the quantities
we found
agreeagree
withwith
the given
• Step
5: Check.
We reread
the problem
check
the quantities
we found
the given
To solve
some problems in which we want to find two quantities, it is useful to perform the following five
information.
information.
steps:
each variable. For each quantity that we are trying to find, we usually define a variable
• Step 1: Define
OBJECTIVE
2unknown
Recall
that that
the formula
of the
P ofPa of
rectangle
withwith
length
L and
width
W isW is
OBJECTIVE
2 Recall
the formula
of perimeter
the perimeter
a rectangle
length
L and
width
to
be
that
quantity.
(
P =
2
L +
2
W) P Perimeter =P2L
+
2W
(Section
4.6).
= 2L + 2W (Section 4.6).
• Step 2: Write a system of two equations. We find a system of two equations by using the variables
1. For
aFor
golden
rectangle,
length
is equal
to equation
about
1.62
timestimes
width.
If an
wants
to design
1. We
canthe
usually
write
either
bythe
translating
theIfarchitect
information
stated
in design
thethe the
1. from
astep
golden
rectangle,
the length
iseach
equal
to about
1.62
the width.
an architect
wants
to
baseproblem
of
a
building
to
be
a
golden
rectangle,
what
are
the
dimensions
of
the
base
if
the
perimeter
is
to
be
into
mathematics
or
by
making
a
substitution
into
a
formula.
base of a building to be a golden rectangle, what are the dimensions of the base if the perimeter is to800
be 800
feet?feet?
• Step 3: Solve the system. We solve the system of equations from step 2.
2. A2.1
landscaper
plansplans
to dig
rectangular
garden
for which
the length
is toisbeto3be
feet
less less
thanthan
twicetwice
the width.
to aresult.
dig
a rectangular
garden
forsentence
which
the
length
3 feet
the width.
Step . A landscaper
• Step
4: Describe
each
We use a complete
to describe
the quantities
we found.
If theIflandscaper
has
66
feet
of
fencing
to
enclose
the
garden,
what
should
be
the
dimensions
of
the
garden?
the
landscaper
has
66
feet
of
fencing
to
enclose
the
garden,
what
should
be
the
dimensions
of
the
garden?
• Step 5: Check. We reread the problem and check that the quantities we found agree with the given
Step 2are
. 7 nickels worth 35 cents? We find the total value of the nickels by multiplying the value of one nickel
WhyWhy
information.
are 7 nickels worth 35 cents? We find the total value of the nickels by multiplying the value of one nickel
times
times
the number
of nickels.
the number
of nickels.
Step 3. OBJECTIVE
2 Recall that the formulaTotal-Value
of the perimeter
P of a rectangle with length L and width W is
Formula
Total-Value
Formula
P Step = 2L4+. 2W (Section 4.6).
eacheach
havehave
valuevalue
v, then
theirtheir
totaltotal
valuevalue
T is Tgiven
by by
If n Ifobjects
n objects
v, then
is given
1. For a golden rectangle, the length is equal to about 1.62 times the width. If an architect wants to design the
base of a building to be a golden rectangle, what
are
the
dimensions of the base if the perimeter is to be 800
T =
Tvn
= vn
feet?
2. A landscaper plans to dig a rectangular garden for which the length is to be 3 feet less than twice the width.
c enclose
2015
Pearson
Education,
Inc. Inc.
If the landscaper has 66 feet Copyright
of fencing
to
thePearson
garden,
what should
be the dimensions of the garden?
c 2015
Copyright
Education,
Why are 7 nickels worth 35 cents? We find the total value of the nickels by multiplying the value of one nickel
Step times
the1. number of nickels.
Step 2. Total-Value Formula
If n objects
Step 3. each have value v, then their total value T is given by
T = vn
Step 4. Copyright c 2015 Pearson Education, Inc.
Lecture
Notes,
Detailed(Comments,
Additional
Explorations
Value Problems T = vn nand
: object, v: value T: total value) 155
3. A 9000-seat amphitheater will sell tickets at $15 and $22 for a Jack Johnson concert. How many tickets
should be sold at each price for a sellout performance to generate a total revenue of $156,000?
4. An auditorium has 900 balcony seats and 3500 main-level seats. If tickets for balcony seats will cost $20
Step 1. than tickets for main-level seats, what should the prices be for each type of ticket so that the total
less
revenue from a sell-out performance will be $149,200?
Step . 5. A 26000-seat
amphitheater will sell tickets at $20 and $35 for a PJ Harvey concert. Let x and y be the
number of tickets that will sell for $20 and $35, respectively. Assume that the show will sell out.
Step 3. a. Let R = f (x) be the total revenue (in dollars) from selling the $20 and $35 tickets. Find an equation
of f .
Step 4. b. Use a graphing calculator to draw a graph of f for 0  x  6000. What is the slope? What does it
mean in this situation?
Lecture
Notes, Detailed Comments, and Additional Explorations
155
c. How many of each ticket must be sold for the revenue to be $135,000?
Money
deposited
in an account
is called
theatprincipal.
9000-seat
amphitheater
will sell
tickets
$15 and $22 for a Jack Johnson concert. How many tickets
3.• A
should
be
sold
at
each
price
for
a
sellout
performance
to generate a total revenue of $156,000?
• A person invests money in hopes of later getting back the
principal plus additional money called the interest.
4. An auditorium has 900 balcony seats and 3500 main-level seats. If tickets for balcony seats will cost $20
• TheNotes,
annual
simpleComments,
interest rate is the
percentage
of the principal that equals the interest earned per year.
Lecture
Detailed
Additional
Explorations
155
less than tickets
for main-level and
seats,
what should
the prices be for each type of ticket so that the total
a sell-out
will beShe
$149,200?
6. revenue
A personfrom
plans
to investperformance
a total of $7500.
will invest in both an account at 5% annual interest and an
account
at 12%
annual interest.
Howtickets
much at
should
she invest
in each account
soconcert.
that the total
interest
in one
3. A
A
9000-seat
amphitheater
will
sell
$15
Johnson
How
manyy tickets
5.
6000-seat amphitheater will sell tickets at $20and
and$22
$35for
fora aJack
PJ Harvey
concert. Let
x and
be the
year
will
be
$515?
should
be
sold
at
each
price
for
a
sellout
performance
to
generate
a
total
revenue
of
$156,000?
Step 1. of tickets that will sell for $20 and $35, respectively. Assume that the show will sell out.
number
4.
7. An
A person
plans to invest
twice asseats
much
money
in
an account
at 3%
interest
as in an account
at $20
7%
balcony
and
main-level
seats.
If annual
tickets
balcony
cost
a.auditorium
Let R = fhas
(x)900
be the
total revenue
(in3500
dollars)
from selling
the
$20 andfor$35
tickets.seats
Findwill
an equation
annual
interest.
How
much
should
he
invest
in
each
account
to
earn
a
total
of
$416
in
one
year?
Step 2. than
less
of f .tickets for main-level seats, what should the prices be for each type of ticket so that the total
from a sell-out performance will be $149,200?
8. revenue
A b.
person
to invest
a total to
of draw
$8000a graph
in an account
2%xand
an account
Let x What
and y does
be the
Use plans
a graphing
calculator
of f for at
0
 6000.
What at
is 5%.
the slope?
it
Step 36000-seat
. mean
money
(in dollars)
invested
in
the
account
at
2%
and
the
account
at
5%,
respectively.
5. A
amphitheater
will
sell
tickets
at
$20
and
$35
for
a
PJ
Harvey
concert.
Let
x
and
y
be
the
in this situation?
number
of ticketsofthat
will sell for
$20
$35,
respectively.
that the show will sell out.
c. How
beand
sold
for the
revenue
toAssume
beinvesting
$135,000?
a.
Let I many
= f (x) each
be theticket
totalmust
interest
(in dollars)
earned from
the $8000 for one year. Find an
Step 4
. equation
f . be the total revenue (in dollars) from selling the $20 and $35 tickets. Find an equation
a. Let
R = of
f (x)
deposited in an account is called the principal.
• Money
b. of
Usef .a graphing calculator to find the values f (0), f (2000), f (4000), f (6000), and f (8000). What do
• A person
money
in hopesto
ofdraw
later getting
additional
calledWhat
the interest.
theyinvests
in
thiscalculator
situation?
b. Use
amean
graphing
a graphback
of f the
forprincipal
0  x plus
6000.
What ismoney
the slope?
does it
mean
in
this
situation?
c. annual
Describe
the various
possible
interest earnings
from investing
the $8000
in the earned
accounts
one
• The
simple
interest
rate is total
the percentage
of the principal
that equals
the interest
perforyear.
c. How
year. many of each ticket must be sold for the revenue to be $135,000?
6. A person
plans to invest a total of $7500. She will invest in both an account at 5% annual interest and an
Interest Problems d. How
much should be invested in each account to earn a total of $220 in one year?
• account
Money
deposited
an interest.
account
thee
principal.
atsimple 12% annual
How
much
should
invest in each
account
soathat
the total
in one
How much iinnterest will is
a called
person arn bshe
y investing $2500 in an ccount at 6interest
% annual interest for one year? year
will
be
$515?
• A person invests money in hopes of later getting back the principal plus additional money called the interest.
the meaning of a 30% lime-juice solution.
7.Discuss
person
plans
to invest
twice
as is
much
money in an
at 3%that
annual
interest
as in an
account
7%
• A
The
annual
simple
interest
rate
the percentage
of account
the principal
equals
the interest
earned
per at
year.
annual interest. How much should he invest in each account to earn a total of $416 in one year?
6.
9.
8.
A
person
to each
investofa atotal
$7500. She
will invest
both
an account
at 5%must
annual
interesttoand
an
How
manyplans
quarts
16%ofantifreeze
solution
and a in
24%
antifreeze
solution
be mixed
make
A
personatplans
to invest
a totalHow
of $8000
inshould
an account
at 2%
and an
account
at 5%.
Let xinterest
and y be
the
account
12%
annual
interest.
much
she
invest
in
each
account
so
that
the
total
in
one
8 quarts of a 18% antifreeze solution?
money
(inbedollars)
year will
$515? invested in the account at 2% and the account at 5%, respectively.
10. How many quarts each of a 8% acid solution and a 20% acid solution must be mixed to make 9 quarts of a a.
Let I = f to
(x) be the total interest (in dollars) earned from investing the $8000 for one year. Find an
7. A
person
12%
acid plans
solution?invest twice as much money in an account at 3% annual interest as in an account at 7%
equation
of
f . much should he invest in each account to earn a total of $416 in one year?
annual
Step 1. interest. How
11. A b.
chemist
needs
6
ounces
of a 20%
alcohol
solution
but fhas
only af24%
alcohol
solution.
manyWhat
ounces
Use a graphing
calculator
to find
the values
f (0),
(2000),
(4000),
f (6000),
and How
f (8000).
do
8. A person
plans
to
invest
a
total
of
$8000
in
an
account
at
2%
and
an
account
at
5%.
Let
x
and
y
be the
each of
the
24%
solution
and
water
should
she
mix
to
make
the
desired
6
ounces
of
20%
alcohol
solution?
they mean in this situation?
Step 2. (in dollars) invested in the account at 2% and the account at 5%, respectively.
money
c. Describe the various possible total interest earnings from investing the $8000 in the accounts for one
a.
Let
= 7,
f (x)
be 17,
the 27,
total31,
interest
from investing the $8000 for one year. Find an
year.I 1,
SHORT
9, 13,
35, 45,(in
49,dollars)
53, 57,earned
61
Step 3. HW
equation
of
f
.
d. How much should be invested in each account to earn a total of $220 in one year?
b. Use
a graphing
find
values
f (0),
(2000),
f (4000),
f (6000),
(8000).
MEDIUM
HW
1, 7, 9,calculator
11, 13, 17,to21,
23,the25,
27, 31,
35, f39,
43, 45,
49, 51, 53,
55, 57,and
61,f65,
67 What do
Step 4. they mean in this situation?
the meaning of a 30% lime-juice solution.
Discuss
c. Describe the various possible
totalcinterest
earningsEducation,
from investing
Copyright
2015 Pearson
Inc. the $8000 in the accounts for one
year.
9. How
quartsshould
each of
16% antifreeze
solution to
and
a 24%
antifreeze
must be mixed to make
d. many
How much
beainvested
in each account
earn
a total
of $220 solution
in one year?
8 quarts of a 18% antifreeze solution?
10.Discuss
How many
quarts each
of a 8%
acid solution
and a 20% acid solution must be mixed to make 9 quarts of a
the meaning
of a 30%
lime-juice
solution.
6. A person plans to invest a total of $7500. She will invest in both an account at 5% annual interest and an
account at 12% annual interest. How much should she invest in each account so that the total interest in one
year will be $515?
7. A person plans to invest twice as much money in an account at 3% annual interest as in an account at 7%
annual interest. How much should he invest in each account to earn a total of $416 in one year?
8. A person plans to invest a total of $8000 in an account at 2% and an account at 5%. Let x and y be the
money
Step 1. (in dollars) invested in the account at 2% and the account at 5%, respectively.
a. Let I = f (x) be the total interest (in dollars) earned from investing the $8000 for one year. Find an
Step 2. equation of f .
b. Use a graphing calculator to find the values f (0), f (2000), f (4000), f (6000), and f (8000). What do
Step 3. they mean in this situation?
c. Describe the various possible total interest earnings from investing the $8000 in the accounts for one
Step 4. year.
d. How much should be invested in each account to earn a total of $220 in one year?
Discuss the meaning of a 30% lime-juice solution.
9. How many quarts each of a 16% antifreeze solution and a 24% antifreeze solution must be mixed to make
8 quarts of a 18% antifreeze solution?
10. How many quarts each of a 8% acid solution and a 20% acid solution must be mixed to make 9 quarts of a
12% acid solution?
11. A chemist needs 6 ounces of a 20% alcohol solution but has only a 24% alcohol solution. How many ounces
each of the 24% solution and water should she mix to make the desired 6 ounces of 20% alcohol solution?
SHORT HW 1, 7, 9, 13, 17, 27, 31, 35, 45, 49, 53, 57, 61
MEDIUM HW 1, 7, 9, 11, 13, 17, 21, 23, 25, 27, 31, 35, 39, 43, 45, 49, 51, 53, 55, 57, 61, 65, 67
Copyright c 2015 Pearson Education, Inc.