Resources

Resources

Evaluate two-variable equations: word problems

Mr. Green uses a linear function to estimate how installing solar panels on the roof of his house will reduce his electric bill.

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• Mr. Green does not have any solar panels yet, and his electric bill is

133

\$

per month on average.

\newline

100

square feet of solar panels he installs, Mr. Green's monthly electric bill will decrease by

13.50

\$

\newline

How many square feet of solar panels must Mr. Green install to reduce his monthly electric bill to

25

\$

\newline

\_\_\_

Christine uses a GPS fitness device to track her steps while she walks to school and then around the track. The graph shows the relationship between the number of laps around the track she completes and the total number of steps she takes.

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\newline

How many steps does Christine take to get from her home to the track?

\newline

\_\_\_

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Complete the equation to represent the relationship between the total number of steps Christine takes,

S

, and the number of laps she completes,

n

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S = \_\_\_ n + \_\_\_

A buoy floating in the ocean is bobbing in simple harmonic motion with period

8

seconds and amplitude

4\,\text{ft}

. Its displacement

d

from sea level at time

t=0

0\,\text{ft}

and initially it moves downward. (Note that downward is the negative direction.)

\newline

Give the equation modeling the displacement

d

as a function of time

t

\newline

d=

\square

A small college with

1,200

total students has a student government of

40

members. From its members, the student government will elect a president, vice president, secretary, and treasurer. No single member can hold more than

1

4

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The permutation formula

nPr

can be used to find the number of unique ways the student government can arrange its members into these positions.

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What are the appropriate values of

n

r

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n=

\square

\newline

r=

\square

300-1.5 p

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To promote his new business, Claudio made

300

tamales to give to pedestrians. The expression above describes the number of tamales remaining after passing out samples to

p

pedestrians. If Claudio passed out samples to

160

pedestrians, how many tamales would remain?

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\square

The following data points represent the number of points scored last basketball game by each player on the Dragons basketball team.

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5,8,11,7,?

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If the mean of the data set is

8

points, find the missing number of points.

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\square

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Carly is cutting a piece of paper stock based on the width of a photo. The expression above describes the length of the paper stock, in inches, for a photo that is

w

inches wide. If the photo is

8.5

inches wide, what is the length of the paper stock in inches?

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\square

A school district wants to conduct a sample survey to determine the average number of sports played by high school seniors in the district. Which of the following survey methods is most likely to produce valid results?

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1

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(A) The district surveys

500

randomly selected high school seniors in the district who play baseball.

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(B) The district surveys every tenth student to enter the football stadium at the district championship game.

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(C) The district surveys

500

randomly selected high school seniors in the district.

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(D) The district selects one high school in the district and surveys all of its students.

A college newspaper wants to determine whether current students like a movie that was filmed on campus

10

years ago. Which of the following sampling methods is most appropriate to estimate the proportion of all current students who like the movie?

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1

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(A) Creating an online poll on which current students can express their opinions and then using the first

50

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50

current students at random

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50

current students who have seen the movie at random

\newline

50

alumni who attended the college when the movie was filmed at random

Maria and Jose have a large tomato garden. On day

1

when the tomatoes first ripened, they picked

3

tomatoes. Each day thereafter, as more tomatoes ripened, they doubled the number of tomatoes they picked. Let

T

be the number of tomatoes picked on day

d

. Which of the following best explains the relationship between

d

T

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1

\newline

(A) The relationship is exponential because

T

6

d

d=1

d=2

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(B) The relationship is linear because

T

increases by a factor of

2

d

1

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(C) The relationship is linear because

T

3

d

d=0

d=1

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(D) The relationship is exponential because

T

increases by a factor of

2

d

1

A block of ice weighing

100

kilograms begins to melt. Each hour the block of ice loses

5 \%

of its weight. Let

W

be the weight, in kilograms, of the block of ice after melting for

t

hours. Which of the following best explains the relationship between

t

W

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1

\newline

(A) The relationship is exponential because

W

is multiplied by a factor of

0

95

t

1

\newline

(B) The relationship is linear because

W

is multiplied by a factor of

5 \%

t

1

\newline

(C) The relationship is linear because

W

5

t

t=0

t=1

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(D) The relationship is exponential because

W

9

75

t

t=0

t=2

600

liters of milk is leaking. Each week it loses

60

liters of milk. Let

M

be the number of liters of milk remaining in the tank after leaking for

t

weeks. Which of the following best explains the relationship between

t

M

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1

\newline

(A) The relationship is linear because

M

always decreases as values of

t

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(B) The relationship is exponential because

M

60

t

1

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(C) The relationship is exponential because

M

is multiplied by a factor of

0

6

t

1

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(D) The relationship is linear because

M

60

t

1

An iceberg floating in the North Pacific Ocean weighs

80

metric tons. Each month it loses

5

metric tons of weight. Let

W

be the weight in metric tons of the iceberg after

t

months have passed. Which of the following best explains the relationship between

t

W

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1

\newline

(A) The relationship is exponential because

W

is multiplied by a factor of

30

t

1

\newline

(B) The relationship is exponential because

W

is multiplied by a factor of

0

3

t

1

\newline

C) The relationship is linear because

W

5

t

1

\newline

(D) The relationship is linear because

W

always decreases as values of

t

-14

is a radioactive isotope used to determine the age of samples of organic matter. The amount of carbon

-14

in a sample decreases once the organism is no longer alive. The half-life of carbon

-14

is approximately

5730

years. This means that every

5730

years, the amount of carbon

-14

in an organism that is no longer living will be halved. Which of the following functions,

C

, models the fraction of carbon

-14

remaining in a sample after

t

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1

\newline

C(t)=2^{\frac{t}{5730}}

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C(t)=\left(\frac{1}{2}\right)^{\frac{t}{5730}}

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C(t)=\left(\frac{1}{2}\right)^{5730 t}

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C(t)=2^{5730 t}

P(t)=(0.82)^{t}

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A local department of transportation has determined that on a particular bus route, the probability of waiting more than

t

minutes for a bus at a certain intersection is modeled by the function shown. What is the best description of the meaning of

0

82

in this context?

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1

\newline

(A) The probability that a person will have to wait more than

1

minute for a bus is

82 \%

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82 \%

of the buses are on the bus route at any given time.

\newline

82 \%

of the buses on the route are late.

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(D) The probability that a person will have to wait less than

1

minute for a bus is

82 \%

C

gives the cost, in dollars, to shred

w

pounds of confidential documents of a company.

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What is the best interpretation for the following statement?

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C^{\prime}(500)=80

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1

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(A) The cost to shred documents, when the weight of the documents is

500

pounds, is increasing at a rate of

\$ 80

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(B) The average cost to shred documents is

\frac{80}{500}

dollars per pound.

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(C) The cost to shred

500

pounds of documents is

\$ 80

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(D) The cost to shred documents, when the weight of the documents is

500

pounds, is increasing at a rate of

\$ 80

Carlos has taken an initial dose of a prescription medication. The function

M

gives the amount of medication, in milligrams, in Carlos's bloodstream after

t

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What is the best interpretation for the following statement?

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M^{\prime}(5)=-1

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1

\newline

5

hours, Carlos's weight is decreasing by

1

milligram per hour.

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5

hours, the amount of medication is decreasing by

1

milligram per minute.

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5

hours, the amount of medication is decreasing by

1

milligram per hour.

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5

hours, Carlos is missing

1

milligram of medication in his bloodstream.

Dina is conducting a study on the amount of electricity used per person in the country of Qatar. She finds that from the year

2000

2008

, the amount of electricity used per person grew by an average of

6 \%

from one year to the next.

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E(t)

be the amount of electricity used per person in kilowatt-hours and

t

be the number of years after

2000

. If the amount of electricity used per person in

2000

was approximately

12

300

kilowatt-hours, which of the following mathematical models will give Dina how much electricity was used per person between

2000

2008

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1

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E(t)=12,300 \cdot(1.06)^{t}

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E(t)=12,300 \cdot(0.06)^{t}

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E(t)=12,300+(1.06) \cdot t

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E(t)=12,300+(0.06) \cdot t

I=0.5(18+m)

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A landlord determines that the total amount of income,

I

, in thousands of dollars generated by renting out a home is given by the equation. Negative values of

m

represent the time

m

months before January

1

2010

and positive values of

m

represent the time

m

months after January

1

2010

. The landlord had an income of

0

when the home was bought and only began receiving income from rent afterward. When did the landlord buy the home?

\newline

1

\newline

(A) The landlord bought the house

18

months before January

1

2010

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(B) The landlord bought the house

9

months before January

1

2010

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(C) The landlord bought the house

9

months after January

1

2010

\newline

(D) The landlord bought the house

18

months after January

1

2010

C=20+0.10 G

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A grid hosting service charges a flat amount for up to a certain number of graphics processing units used per month and charges for each additional graphics processing unit used beyond that. The monthly charge in dollars,

C

, is shown, where

G

is the number of graphics processing units over

1

000

used that month. What is the meaning of the number

20

in this equation?

\newline

1

\newline

(A) The hosting service charges

\$ 20

per graphics processing unit.

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(B) The hosting service charges a total of

\$ 20

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(C) The hosting service charges

\$ 20

100

graphics processing units used in a month.

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(D) The hosting service charges

\$ 20

1

000

graphics processing units used in a month.

C=0.4 m+68

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Lúcia hires a truck driver to deliver the food for her catering company. For a delivery that is

m

miles away, the total cost,

C

, in dollars, is given by the equation. What is the meaning of the

0.4 \mathrm{~m}

in this equation?

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1

\newline

0.4 \mathrm{~m}

miles in each delivery.

\newline

0.4 m

is a flat fee charged for every delivery.

\newline

0.4 \mathrm{~m}

dollars per mile to deliver the food.

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(D) A delivery that is

m

miles away raises the total delivery cost by

0.4 \mathrm{~m}

C=24 f+88 e

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The cost in collars,

C

, from an airplane flight given that the average cost of a first class seat is

f

dollars and the average cost of an economy seat is

e

dollars is given by the equation if all seats are sold. What is the best interpretation of

88 e

as shown in the equation?

\newline

1

\newline

88 e

economy seats on the flight.

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(B) The total cost due to economy seats is

88 e

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(C) The increase in total cost for each economy seat sold is

88 e

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(D) The increase in total cost is

88 e

dollars when economy seat prices increase by

1

t=\frac{V}{1.1}+0.5

\newline

A cylindrical ceramic mug with a volume of

V

cups is filled with water and placed in a microwave oven of constant power. It takes approximately

t

minutes for the water to begin boiling as given by the equation. What does

1

1

mean in the equation?

\newline

1

\newline

(A) The volume of the ceramic mug can be a maximum of

1

1

\newline

1

1

additional cups of water take

1

more minute to microwave and boil.

\newline

(C) The time it takes to microwave and boil

1

additional cup of water is

1

1

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(D) The minimum amount of time to microwave and boil any amount of water is

1

1

V=220 L+451 H

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The sample variance,

V

, in the expected total premiums of

L

lowrisk beneficiaries and

H

high-risk beneficiaries based on an underwriter's premium allocation is given by the equation. The values of

L

H

are independent. What is the best interpretation of

451 \mathrm{H}

as shown in the equation?

\newline

1

\newline

451 H

high-risk beneficiaries.

\newline

(B) The total variance due to high-risk beneficiaries is

451 \mathrm{H}

\newline

(C) The variance in premiums for a high-risk beneficiary is

451 \mathrm{H}

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(D) The expected total premiums for high-risk beneficiaries is

451 \mathrm{H}

S=120(5-h)

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Simone arrived at her cabin after a big snow storm and realized that she would need to shovel some snow before she could go inside. The amount of snow,

S

, in cubic feet, that is left to shovel after shoveling for

h

hours is given by the equation. What does the

5

mean in the equation?

\newline

1

\newline

(A) It will take Simone

5

hours to shovel all of the snow.

\newline

5

hours, there will be

120

cubic feet of snow left to shovel.

\newline

5

cubic feet of snow to shovel when Simone arrived at the cabin.

\newline

5

cubic feet of snow left to shovel after shoveling for

120

Which of the following savings account plans would yield exponential growth of the money in the account?

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1

\newline

2 \%

of the initial value of the account is added to the value of the account each year.

\newline

1.5 \%

of the current value of the account is added to the value of the account each year.

\newline

\$ 500

signing bonus is added when the account is first opened, and

0.5 \%

of the initial value of the account is added to the value of the account each year.

\newline

\mathrm{A} \$ 100

customer loyalty bonus is added to the account each year.

The ages of the members of four teams are summarized below. Answer the questions about them.

\newline

Team A: The range of ages is

23

and the mean age is

43

\newline

Team B: The range of ages is

27

and the mean age is

44

\newline

Team C: The range of ages is

21

and the mean age is

41

\newline

Team D: The range of ages is

20

and the mean age is

46

\newline

(a) Based on the information above, which team's ages have the most variability?

\newline

\newline

\newline

\newline

\newline

(b) Based on the information above, which team has the oldest members on average?

\newline

\newline

\newline

\newline

Farmer Jim uses

5

-strand wire fencing around his paddocks.

\newline

His fences have

3

barbed wire strands and

2

plain wire strands, as shown in the diagram below

\newline

Barbed wire costs

\$ b

per metre. Plain wire costs

\$ w

\newline

Which of these expressions gives the total cost of the wire needed for a fence of length

F

Lois and Clark own a company that sells wagons. The amount they pay each of their sales employees (in dollars) is given by the expression

12h + 30w

h

is the number of hours worked and

w

is the number of wagons sold. What is the amount paid to an employee that works

6

hours and sells

3

The average monthly temperature, in degrees Fahrenheit, for a town in central Indiana can be modeled by the sinusoidal function

T(m)=25.7 \sin \left(\frac{\pi}{6}(m-4)\right)+61.2

1 \leq m \leq 12

months. Based on the model, which of the following is true?

\newline

(A) The maximum average monthly temperature is

61

2

degrees Fahrenheit.

\newline

(B) The maximum average monthly temperature occurs at

m=1

\newline

(C) The minimum average monthly temperature is

35

5

degrees Fahrenheit.

\newline

(D) The minimum average monthly temperature occurs at

m=7

1

. A candle is placed at a distance of

12 \mathrm{~cm}

from of a concave mirror with a focal length of

8 \mathrm{~cm}

. The candle is

5 \mathrm{~cm}

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A. Where is the image located?

\newline

B. What is the height of the image?

Suppose that the amount of time it takes to build a highway varies directly with the length of the highway and inversely with the number of workers. Suppose also that it takes

300

22

24

miles of highway. How long will it take

225

workers to build

27

miles of highway?

\newline

\square

A utility service company has a fleet of cars and trucks. The gas tank of each car has a volume of

x

gallons, and the gas tank of each truck has a volume of

y

gallons. If it takes

z

gallons of gas to fill the empty gas tanks of the entire fleet, which equation shows the possible number of cars,

c

, and number of trucks,

t

, in the fleet?

At the Silver City Middle School field day, a favorite event is the obstacle course. Sophie was happy that she finished it in

62

seconds. Most of the

26

students in her class took over

60

seconds. The median time was

67

seconds, and the interquartile range was about

14

seconds. The fastest student in the class finished in

54

\newline

What is a typical obstacle course time of a student in Sophie's class?

\newline

14

\newline

60

\newline

62

\newline

67

12 g

gives the number of kilometers a car can travel using

g

liters of gasoline.

\newline

How far can this car travel on

5 \frac{1}{2}

liters of gasoline?

\newline

\square

\mathrm{km}

20 l+25 g-10

gives the number of dollars that Josie makes from mowing

l

lawns and raking

g

\newline

How many dollars does Josie make from raking

4

gardens and mowing

3

\newline

\square

15

\$ 90

and sells them for

\$ 405

p

represents the total profit in dollars and cents when the store buys any number of sweaters,

s

, write a proportional equation for

p

s

that matches the context.

\newline

p=\_\_\_\_\_

The amount of Cu

-67

in the blood of a patient may be modelled by the following equation.

m(t)=m_0\left(\frac{1}{2}\right)^{\frac{t}{h}}

m(0)=m_0

is the initial amount of Cu

-67

, that is, the amount present at the time of injection. Here

t

represents time measured in days from the time of injection and

h=2.6

days is the half-life. Sketch the graph of

m(t)

over a period of just over five half-lives from the time of injection (at time

t=0

m_0=8

mg. Find the ratio

\frac{m(t)}{m_0}

predicted by the model

10.4

days after injection. This is the fraction of the Cu

-67

content remaining. Plot the corresponding point on your graph. When the mass of Cu

-67

0.2

mg the radiation can no longer be detected. How long after the injection must the patient wait until the radiation can no longer be detected?

A charity organization had to sell a few tickets to their fundraiser just to cover necessary production costs. After selling

10

tickets, they were still at a net loss of

\$ 800

(due to the production costs). They sold each ticket for

\$ 70

\newline

y

represent the net profit (in dollars) when they have sold

x

\newline

Complete the equation for the relationship between the net profit and number of tickets sold.

\newline

y=\square

For a drink recipe, the amount of carrot juice is proportional to the amount of guava juice. This equation represents the proportional relationship between the number of quarts of carrot juice

c

and guava juice

g

3c=4g

How many quarts of carrot juice would be used with

1

quart of guava juice?

Which situation would yield the highest amount in the account?

\newline

\$ 200

in an account the earns

5 \%

simple interest for two years.

\newline

\$ 400

in an account the earns

3 \%

interest compounded annually

\newline

\$ 200

in an account the earns

5 \%

interest compounded annually for two years.

\newline

\$ 400

in an account the earns

3 \%

simple interest for two years.

Over summer break, Emmett needs to read a book for his English class. The function

f(x)

gives the total number of chapters he has read by day

x

of summer break.

\newline

f(12) = 24

\newline

\newline

12

days of summer break, Emmett has read

24

\newline

24

days of summer break, Emmett has read

12

Marvin sells fresh, homemade pasta at the farmers' market. For every

x

grams of flour he uses, he can make

f(x)

\newline

f(500) = 6

\newline

\newline

500

grams of flour, Marvin can make

6

\newline

6

grams of flour, Marvin can make

500

Jaxon was out at a restaurant for dinner when the bill came. He wanted to leave a

13 \%

tip. If dinner cost

d

dollars and cents, which expression represents the cost of dinner plus tip?

\newline

0.13 d

\newline

1.013 d

\newline

1.13 d

\newline

13 d

Rajani was out at a restaurant for dinner when the bill came. She wanted to leave a

28 \%

tip. If dinner cost

d

dollars and cents, which expression represents the cost of dinner plus tip?

\newline

1.028 d

\newline

1.28 d

\newline

0.28 d

\newline

28 d

Parker was out at a restaurant for dinner when the bill came. He wanted to leave a

35 \%

tip. If dinner cost

d

dollars and cents, which expression represents the cost of dinner plus tip?

\newline

1.35 d

\newline

35 d

\newline

1.035 d

\newline

0.35 d

Maya was out at a restaurant for dinner when the bill came. She wanted to leave a

33 \%

tip. If dinner cost

d

dollars and cents, which expression does NOT represent the cost of dinner plus tip?

\newline

0.33 d

\newline

1 d+0.33 d

\newline

(1+0.33) d

\newline

d+0.33 d

f

folders that cost

\$ 0.90

m

markers that cost

\$ 0.35

each. Sales tax is

3.5 \%

. Which expression represents the total amount Xin paid, including tax?

\newline

1.035(f+m)

\newline

0.035(0.9 f+0.35 m)+(0.9 f+0.35 m)

\newline

0.9(0.035 f)+0.35(0.035 m)

\newline

1.035(0.9+0.35)

Ian used a discount coupon for

5 \%

off at his favorite clothing store. He bought

4

pairs of shoes for

x

dollars each and

2

y

dollars each. Which expression represents the total amount Ian paid after the coupon was applied?

\newline

(4 x+2 y)-0.05(4 x+2 y)

\newline

0.05(4 x+2 y)+(4 x+2 y)

\newline

(4 x+2 y)-0.95(4 x+2 y)

\newline

95(4 x+2 y)

F=1.35 P+1.50 A

\newline

Milo wishes to purchase a bouquet consisting of

P

A

alstroemeria. If

F

is the total cost, in dollars, of a bouquet of these two types of flowers, what is the meaning of

1.50

in the equation?

\newline

1

\newline

\$1.50

is the cost of one peony.

\newline

\$1.50

A

is the cost of one alstroemeria.

\newline

\$1.50

is the cost of one alstroemeria.

\newline

\$1.50

A

is the cost of one alstroemeria.

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