Resources

Resources

Interpret parts of quadratic expressions: word problems

Fernando built a robot to filter air and water efficiently. He expects the robot to filter more than

512

liters of air and water while using less than

9

Joules of energy.

\newline

A

represents the number of minutes the robot filters air, and

W

represents the number of minutes the robot filters water.

\newline

21 A+14 W>512

represents the number of minutes the robot filters air and water to filter more than

512

liters of air and water.

\newline

0.36 \mathrm{~A}+0.6 \mathrm{~W}<9

represents the number of minutes the robot filters air and water while using less than

9

Joules of energy.

\newline

Does the robot meet both of Fernando's expectations by filtering air for

15

minutes and filtering water for

5

\newline

1

\newline

(A) The robot meets both of Fernando's expectations.

\newline

(B) The robot filters the expected amount of air and water, but it doesn't use the expected amount of energy.

\newline

(C) The robot uses the expected amount of energy, but it doesn't filter the expected amount of air and water.

\newline

(D) The robot doesn't meet either of Fernando's expectations.

Genghis Motors wants to build cars and motorbikes on a budget. The company plans to spend at most

\$ 75000

and use at least

320

workers to build cars and motorbikes.

\newline

8000 C+5500 M \leq 75000

represents the number of cars

C

M

the company can build by spending at most

\$ 75000

\newline

52 C+38 M \geq 320

represents the number of cars and motorbikes the company can build by using at least

320

\newline

Can the company meet both of its expectations by building

4

6

\newline

1

\newline

(A) The company meets both of its expectations.

\newline

(B) Dhe company uses the expected amount of money but not the expected number of workers.

\newline

(C) The company uses the expected number of workers but not the expected amount of money.

\newline

(D) The company doesn't meet either of its expectations.

Vic's favorite cookie recipe uses

250

grams of flour for every

170

grams of butter. Imani made cookies using

750

grams of flour for every

630

grams of butter.

\newline

What will Vic think of the amount of butter in Imani's cookies?

\newline

1

\newline

(A) Imani's cookies have too much butter.

\newline

(B) Imani's cookies don't have enough butter.

\newline

(C) Imani's cookies have the right amount of butter.

The Soup Shack usually makes tomato soup with

9

tomatoes for every

12

bowls of soup. Today, they used

6

tomatoes to make

8

\newline

How does the tomato taste in today's soup compared to the usual recipe?

\newline

1

\newline

(A) Today's soup has a weaker tomato taste.

\newline

(B) Today's soup has a stronger tomato taste.

\newline

(C) Today's soup will taste the same.

You're a manager in a company that produces rocket ships. Machine A and Machine B both produce cockpits and propulsion systems. Machine A and Machine B produce cockpits at the same rate, and they produce propulsion systems at the same rate. Machine A ran for

26

hours and produced

4

6

propulsion systems. Machine

\mathrm{B}

56

hours and produced

8

12

propulsion systems.

\newline

We use a system of linear equations in two variables.

\newline

Can we solve for a unique amount of time that it takes each machine to produce a cockpit and to produce a propulsion system?

\newline

1

\newline

(A) Yes; it takes Machine A and Machine B

2

hours to produce a cockpit and

3

hours to produce a propulsion system.

\newline

(B) Yes; it takes Machine A. and Machine B

1

hour to produce a cockpit and

4

hours to produce a propulsion system.

\newline

(C) No; the system has many solutions.

\newline

(D) No; the system has no solution.

You're a manager in a company that produces rocket ships. Machine A and Machine B both produce cockpits and propulsion systems. Machine A ran for

22

hours and produced

3

5

propulsion systems. Machine B ran for

44

hours and produced

6

10

propulsion systems.

\newline

Assume both machines produce cockpits at the same rate and both produce propulsion systems at the same rate.

\newline

Can we use a system of linear equations in two variables to solve for a unique amount of time that it takes each machine to produce a cockpit and to produce a propulsion system?

\newline

1

\newline

(A) Yes; it takes Machine A and Machine B

4

hours to produce a cockpit and

2

hours to produce a propulsion system.

\newline

(B) Yes; it takes Machine A and Machine B

2

hours to produce a cockpit and

4

hours to produce a propulsion system.

\newline

(C) No; the system has many solutions.

\newline

(D) No; the system has no solution.

A gallon of Moo Milk costs

\$ 5.12

\newline

What is the price, in dollars, of an

8

ounce glass of Moo Milk?

\newline

128

1

\newline

1

\newline

\$ 0.04

\newline

\$ 0.32

\newline

\$ 0.64

\newline

\$ 16

Nadia built a robot to filter air and water efficiently. She expects the robot to filter more than

343

liters of air and water while using less than

49

Joules of energy.

\newline

12 A+8 W>343

represents the number of minutes the robot filters air

A

W

to filter more than

343

liters of air and water.

\newline

3 A+4 W<49

represents the number of minutes the robot filters air and water while using less than

49

Joules of energy.

\newline

Does the robot meet both of Nadia's expectations by filtering air for

20

minutes and filtering water for

15

\newline

1

\newline

(A) The robot meets both of Nadia's expectations.

\newline

(B) The robot filters the expected amount of air and water, but it doesn't use the expected amount of energy.

\newline

(C) The robot uses the expected amount of energy, but it doesn't filter the expected amount of air and water.

\newline

(D) The robot doesn't meet either of Nadia's expectations.

Astrid wants to buy pumpkins and watermelons. She wants to buy more than

10

fruits, and she has a budget of

\$ 117

\newline

P+W>10

represents the number of pumpkins

P

and watermelons

W

Astrid should buy to get more than

10

\newline

5.5 P+3.5 W \leq 117

represents the number of pumpkins and watermelons she can buy on her budget of

\$ 117

\newline

Does Astrid meet both of her expectations by buying

7

3

\newline

1

\newline

(A) Astrid meets both of her expectations.

\newline

(B) Astrid buys the expected number of fruits, but she goes beyond her budget.

\newline

(C) Astrid stays within her budget, but she doesn't buy the expected number of fruits.

\newline

(D) Astrid doesn't meet either of her expectations.

Antonius the chef wants to cook several batches of lasagna and spaghetti for his restaurant. He plans to use at least

4

5

kilograms of pasta and more than

6

3

liters of sauce to cook spaghetti and lasagna.

\newline

0.3 S+0.65 L \geq 4.5

represents the number of batches of spaghetti

S

L

Antonius can cook to use at least

4

5

kilograms of pasta.

\newline

0.25 S+0.8 L>6.3

represents the number of batches of spaghetti and lasagna Antonius can cook to use more than

6

3

liters of sauce.

\newline

Does Antonius meet both of his expectations by cooking

10

batches of spaghetti and

4

batches of lasagna?

\newline

1

\newline

(A) Antonius meets both of his expectations.

\newline

(B) Antonius uses the expected amount of pasta but not the expected amount of sauce.

\newline

(C) Antonius uses the expected amount of sauce but not the expected amount of pasta.

\newline

(D) Antonius doesn't meet either of his expectations.

Sicilia randomly selected

25 \%

of the Sound Sleep email subscribers and asked them how many hours of sleep they average each night. Of the subscribers surveyed,

16

average less than

6

hours of sleep each night. Based on the data, what is the most reasonable estimate of the number of Sound Sleep email subscribers who average less than

6

hours of sleep?

\newline

1

\newline

16

\newline

48

\newline

64

\newline

400

The formula for the volume of a pyramid is

\frac{A \cdot h}{3}

A

is the area of the pyramid's base and

h

\newline

The base of the Great Pyramid of Giza is about

53

000

square meters, and its height is about

150

\newline

What calculation will give us the estimated volume of the Great Pyramid of Giza in cubic meters?

\newline

1

\newline

\frac{\frac{53,000}{100^{2}} \cdot 150}{3}

\newline

\frac{53,000 \cdot 150 \cdot 100}{3}

\newline

\frac{53,000 \cdot 150 \cdot 100^{2}}{3}

\newline

\frac{53,000 \cdot 150}{3}

V=\frac{1}{3} \ell w h

\newline

The formula gives the volume

V

of a rectangular pyramid with base length

\ell

w

h

. What is the volume, in cubic meters, of a rectangular pyramid with a base length of

5

meters, a base width of

4

meters, and a height of

3

V=A t

\newline

A confectioner is applying a sugar coating to some spherical candies. The given equation approximates

V

, the volume of coating in cubic millimeters, needed to create a

t

- millimeter thick coating on a candy with a surface area of

A

square millimeters. If

20 \pi

cubic millimeters of coating is needed to create a

0

2

-millimeter thick coat on a spherical candy, what is the uncoated candy's radius in millimeters?

\newline

(The formula for the surface area,

A

, of a sphere with radius

r

A=4 \pi r^{2}

V=\frac{s^{2} h}{3}

\newline

The formula gives the volume of a right square pyramid of height

h

and square base of side length

s

. If the side length of a right square pyramid increases by

30 \%

while the height remains constant, what happens to the volume?

\newline

1

\newline

(A) The volume increases by

30 \%

\newline

(B) The volume increases by

69 \%

\newline

(C) The volume increases by

169 \%

\newline

(D) The volume increases by

300 \%

12 g

gives the number of kilometers a car can travel using

g

liters of gasoline. How many kilometers can this car travel on

5 \frac{1}{2}

liters of gasoline?

An ice cream stand uses the expression

2+0.5 x

to determine the cost, in dollars, of an ice cream cone that has

x

scoops of ice cream. What is the cost, in dollars, of an ice cream cone with

5

scoops of ice cream?

1.08 s+1.02 b

predicts the end-of-year value of a financial portfolio, where

s

is the value of stocks in dollars and

b

is the value of bonds in dollars in the portfolio at the beginning of the year. What is the predicted end-of year value, in dollars, of a portfolio that begins the year with

200

dollars in stocks and

100

dollars in bonds?

100+20 m

gives the volume of water, in liters, in Marcel's pool after Marcel spends

m

minutes filling his pool. What is the volume of water, in liters, in Marcel's pool after he fills it for

5 \frac{1}{4}

1,000+250 x

\newline

Teresa currently has

\$ 1,000

in her bank account. The expression above models the amount of money in her account in

x

months if she deposits

\$ 250

into the account each month. Based on the expression, what is the amount, in dollars, Teresa will have in her account

6

months from now? (Ignore the

\$

symbol when entering your answer.)

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

12 h+30 w

h

is the number of hours worked and

w

is the number of wagons sold. What is the amount, in dollars, paid to an employee who works

6

hours and sells

3

4 x

gives the perimeter of a square with a side length of

x

units. What is the perimeter of a square with a side length of

\frac{5}{7}

Rossana heads the finance department of a small city. This year, the budget for the department is

1

32

million dollars. If Rossana is asked to bring the department's budget to under

1

12

million dollars next year, which of the following can be used to represent

x

, the amount of money, in millions of dollars, Rossana must cut from the department's budget?

\newline

1

\newline

1.32-x>1.12

\newline

1.32-x<1.12

\newline

x-1.32>1.12

\newline

x-1.32<1.12

In order to bring his business to the next level, Christov wants to gain at least

2

000

followers on a popular social media platform. From his own personal account, he knows that each original post gains him approximately

3

new followers and every

5

reposts gains about

1

. Which of the following inequalities represents the numbers of posts,

P

R

, Christov needs to reach his goal of gaining at least

2

000

\newline

1

\newline

3 P+0.2 R \geq 2,000

\newline

3 P+5 R \leq 2,000

\newline

1 P+5 R \geq 2,000

\newline

0.2 P+5 R \leq 2,000

The amount of water vapor,

v

(in grams), that will saturate

1

kilogram of dry air when the temperature is

x

degrees Celsius is approximated by the following equation.

\newline

v=4.04 \cdot 1.07^{x}

\newline

For each increase of

1

degree, what is the percent increase in the amount of water vapor required to saturate the air according to this model?

\newline

\%

symbol when entering your answer. For example, if the answer is

11.2 \%

11

2

A mug of warm apple cider is gradually cooling. Its temperature in degrees Celsius

\left({ }^{\circ} \mathrm{C}\right)

can be modeled with the expression

23+24 \cdot 2^{-0.014 t}

, where the variable

t

represents the time in minutes. Approximately how many minutes will it take for the temperature to reach

29^{\circ} \mathrm{C}

\newline

(Round to the nearest minute.)

h(t)=-4.9 t^{2}+9.8 t+39.2

\newline

Kaia throws a stone vertically upward from a bridge. The height, in meters, of the stone above the water

t

seconds after the throw can be modeled by the quadratic function given. How many seconds after the throw does the stone hit the water?

\newline

1

\newline

1

\newline

2

\newline

4

\newline

8

The population of Japan,

p

, in millions of people at time

t

, where negative values of

t

represent a number of years before January

1^{\text {st }}

2000

and positive values of

t

represent a number of years after January

1^{\text {st }}, 2000

is projected as:

\newline

p=128-0.012(t-9.17)^{2}

\newline

According to this projection, during which year does Japan reach its maximum population?

Target heart rate is determined by multiplying the reserve heart rate by the training intensity level,

p

0 \leq p \leq 1

, and adding the resting heart rate. An adult woman with a resting heart rate of

65

beats per minute (bpm) and a reserve heart rate of

125 \mathrm{bpm}

is told by a trainer that her target heart rate should not exceed

140 \mathrm{bpm}

. What is the maximum training intensity level,

p

, that is consistent with her trainer's advice?

Hamadi is renting a car. The rental charge is

\$ 17.50

\$ 0.16

per mile. Hamadi can spend at most

\$ 33

for the cost of the car rental. If Hamadi rents the car for one day, which of the following is one possible number of miles Hamadi can drive the rental car?

\newline

1

\newline

53

\newline

97

\newline

\mathbf{1 1 0}

\newline

\mathbf{1 5 0}

Ling needs to fit a pane of glass in an opening with a maximum diagonal length of

62

inches. The glass has a diagonal length of

61

98

inches at room temperature. For every increase of

1

\left({ }^{\circ} \mathrm{C}\right)

, the pane of glass will expand

0

00056

inch diagonally. Which of the following is closest to the maximum temperature increase from room temperature the pane of glass can experience and still fit in the opening?

\newline

1

\newline

1^{\circ} \mathrm{C}

\newline

4^{\circ} \mathrm{C}

\newline

10^{\circ} \mathrm{C}

\newline

36^{\circ} \mathrm{C}

h=16,890-3.4 t

\newline

h

, in feet, of a mountain at a time,

t

, measured in thousands of years from the present is given by the equation. What is the best interpretation of

16

890

as shown in the given equation?

\newline

1

\newline

(A) The height of the mountain is presently

16

890

\newline

(B) The height of the mountain will reach

0

16

890

years from now.

\newline

(C) The height of the mountain increases by

16

890

feet every thousand years.

\newline

(D) The height of the mountain decreases by

16

890

feet every thousand years.

w=3.1+\frac{l}{2}

\newline

w

, in ounces, of a glass soda bottle design with a side length of

l

inches is given by the equation. By how many ounces does the weight of the bottle increase for every inch of increase in the side length?

\newline

1

\newline

0

5

\newline

2

\newline

3

1

\newline

3

6

C=1.5 b+67.5

\newline

The total cost,

C

, in dollars, to produce

b

books is given by the equation. What is the meaning of

1

5

in the equation?

\newline

1

\newline

1

5

books will cost

\$ 67.50

\newline

(B) It costs an extra

\$ 1.50

for each book produced.

\newline

(C) If you produce one book, your total cost will be

\$ 1.50

\newline

(D) There is a flat fee of

\$ 1.50

to produce any number of books.

D=1,874-0.55 t

\newline

D

, in meters, between an antarctic glacier and the coast

t

days after January

1

2010

is approximated by the equation. How does the distance between the glacier and the coast change over time?

\newline

1

\newline

(A) The glacier moves

0

55

meters per day closer to the shore.

\newline

(B) The glacier moves

1

874

meters per day closer to the shore.

\newline

(C) The glacier moves

0

55

meters per day further from the shore.

\newline

(D) The glacier moves

1

874

meters per day further from the shore.

C=290+155 h

\newline

A team of painters is hired to paint the exterior of a museum. The total cost,

C

, in dollars, incurred by the museum for

h

hours of painting is given by the equation. What is the best interpretation of

155

in the equation?

\newline

1

\newline

A The total cost can be at most

\$ 155

\newline

B It takes the team

155

hours to paint the exterior.

\newline

(C) Each additional hour of painting the exterior costs

\$ 155

\newline

\$ 155

is the base cost regardless of how many hours it takes to paint the exterior.

5.5 B+4 R=28

\newline

The given equation models the cost if Amit picks

B

pounds of blueberries and

R

pounds of raspberries at a farm where blueberries cost

\$ 5.50

per pound and raspberries cost

\$ 4.00

per pound. According to the equation, how much does Amit spend in total on both types of berries?

\newline

1

\newline

\$ 9.50

\newline

\$ 22.00

\newline

\$ 28.00

\newline

\$ 56.00

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.

w=500-6.2 t

\newline

A rain barrel is being used during the dry season to water a garden with previously collected rainwater. With a hose connected to its outlet, the equation approximates amount of rainwater in the barrel,

w

, in liters, after

t

minutes of watering. What is the meaning of

6.2 t

in this equation?

\newline

1

\newline

6.2 t

minutes to empty the barrel.

\newline

6.2 t

minutes are needed to fill the barrel.

\newline

6.2 t

liters of rainwater is emptied after

t

\newline

6.2 t

liters in the barrel when filled for

t

P=47.4(G-174.5)

\newline

P

, in dollars, to an amusement park serving

G

guests over one day is given by the equation. What is the minimum number of guests that need to be served in order to make a positive profit?

\newline

1

\newline

48

\newline

\mathbf{1 7 4}

\newline

\mathbf{1 7 5}

\newline

8

272

h=11.2-0.125 d

\newline

The ceiling height,

h

, in feet, for a particular room in a house a distance of

d

feet from the west wall is given by the equation. In order for the ceiling height to decrease by

1

foot, how much does the distance from the west wall change in feet?

\newline

1

\newline

0

125

\newline

8

\newline

\mathbf{1 1 . 2}

\newline

89

6

v=14.2(12.5-s)

\newline

v

, in meters per second, of an airplane

s

seconds after landing is given by the equation. What is the best interpretation of

14.2

as shown in the equation?

\newline

1

\newline

(A) The plane takes

14.2

seconds to stop.

\newline

(B) The position of the plane changes

14.2

meters every second.

\newline

(C) The velocity changes by

14.2

meters per second every second.

\newline

(D) The velocity of the plane just as it lands is

14.2

meters per second.

Katlin wants to make

64

ounces of chocolate milk that is

12 \%

chocolate. She has light chocolate milk that is

5 \%

chocolate and heavy chocolate milk that is

21 \%

chocolate. How many ounces of light chocolate milk and heavy chocolate milk does Katlin need to combine?

\newline

1

\newline

24

40

\newline

28

36

\newline

36

28

\newline

40

24

Anthropologists can predict the height of a human based on the length of their femur. The relationship between the human's height in centimeters,

H

, and length of the femur in centimeters,

f

, can be modeled by the given equation.

\newline

H=2.72 f+42.12

\newline

One anthropologist found the femurs of two different people. One femur measured

22

centimeters and the other femur measured

23

centimeters. What is the difference in their heights?

\newline

1

\newline

1

\newline

2

72

\newline

27

2

\newline

42

12

C=40(5-s)

\newline

After Hiro's family photoshoot, the photographer sells the printed photos by the sheet. Hiro has a

\$ 200

credit from a prior session, and the equation gives the total credit left,

C

, in dollars, if he buys

s

sheets. How many sheets can Hiro purchase with his credit?

\newline

1

\newline

5

\newline

8

\newline

40

\newline

200

m=14(11-t)

\newline

m

, in grams, of a piece of marble after being in a rock tumbler for

t

minutes is given by the equation. After how many minutes in the rock tumbler does the piece of marble disintegrate by reaching a mass of

0

\newline

1

\newline

3

\newline

11

\newline

14

\newline

\mathbf{1 5 4}

B=11.5(132-n)

\newline

The monthly electricity bill,

B

, in dollars, for a museum using

n

solar panels is given by the equation. What is the best interpretation of

132

in the equation?

\newline

1

\newline

(A) The museum is currently using

132

\newline

(B) The monthly electric bill is

132

dollars without solar panels.

\newline

(C) The electric bill is

0

dollars when the museum uses

132

\newline

(D) The electric bill decreases by

132

dollars for each additional solar panel.

A tennis club is organizing group lessons. The club supplies each player with

40

new balls, which costs the club

\$ 1

each ball. Each player pays

\$ 300

for the lessons. The club must pay each instructor

\$ 1,000

for conducting the lessons, and there must be at least

1

instructor for every

6

players. Which amount of players and instructors meets these requirements and still gives the club a net profit?

\newline

1

\newline

6

2

\newline

10

3

\newline

13

2

\newline

16

3

A memory chip is being designed to hold a number of transistors and heat sinks. The transistors hold memory while the heat sinks cool the chip. There must be at least one heat sink for every

2

000

transistors to prevent overheating. Also, each transistor has an area of

2.0 \times 10^{-10} \mathrm{~mm}^{2}

(square millimeters), each heat sink has an area of

3.6 \times 10^{-6} \mathrm{~mm}^{2}

, and the total area of transistors and heat sinks must be at most

2 \mathrm{~mm}^{2}

. What is the approximate maximum number of transistors that the chip can hold according to this design?

\newline

1

\newline

2.78 \times 10^{2}

\newline

5.56 \times 10^{5}

\newline

1.0 \times 10^{9}

\newline

1.0 \times 10^{10}

A business analyst is deciding the amount of time allotted to each employee for meetings and training. He wants the sum of meeting and training time to be no more than

16

hours per month. Also, there should be at least one training hour for every two meeting hours. Finally, there should be at least

2

meeting hours per month to discuss shortterm goals. What is the difference between the maximum and minimum number of monthly training hours that could be allotted to an employee?

\newline

1

\newline

10

\newline

13

\newline

14

\newline

16

...