Resources

Resources

Interpret parts of quadratic expressions: word problems

Antoine stands on a balcony and throws a ball to his dog, who is at ground level. The ball's height (in meters above the ground),

x

seconds after Antoine threw it, is modeled by:

\newline

h(x) = -2x^2 + 4x + 16

\newline

How many seconds after being thrown will the ball hit the ground?

Katya is a ranger at a nature reserve in Siberia, Russia, where she studies the changes in the reserve's bear population over time. The relationship between the elapsed time

t

, in years, since the beginning of the study and the bear population

P

, on the reserve is modeled by the following function. In how many years will the reserve's bear population be

P

? Round your answer, if necessary, to the nearest hundredth.

t

An architect has created a scale drawing for a residential townhouse in the shape of a rectangular prism. In the drawing, the internal width of the townhouse is labeled as

\text{feet}

, the internal length as

\text{feet}

, and the internal height as

\text{feet}

. The local building department tells the architect that if built, the building would be too tall according to local zoning laws and that its height must be reduced by

\text{feet}

. If the architect creates a second scale drawing where the townhouse's height is reduced by

\text{feet}

, what will be its new proposed internal volume in cubic feet?

2024

, Roman Matveevich plans to take out a loan for business development for three years in the amount of

1,600,000

rubles. Conditions for its return such: Each January debt increases by

25\%

compared to the end work of the year; -- from February to April of each year it is necessary to pay one fee - press part of the debt; - payments in

2025

2026

should be equal: -- by May

2027

, the debt must be repaid in full. It is known that all funds after full repayment of the loan it will be equal to

2,637.5

thousand rubles. How many thousand rubles is the payment Roman Matveevich in

2026

Three years ago, the residents of Planet

X

found a black hole leading to another galaxy. Since then, they have been leaving Planet

X

120

residents per year.

\newline

The following equation describes this situation:

\newline

-120 \cdot 3=-360

\newline

-360

\newline

1

\newline

360

residents left Planet

X

3

\newline

360

residents left Planet

X

\newline

360

residents left on Planet X.

An arctic cold front caused the temperature to drop a total of

20^{\circ} \mathrm{C}

. The temperature was dropping at a rate of

10^{\circ} \mathrm{C}

per hour. The temperature dropped an equal amount each hour.

\newline

The following equation describes this situation.

\newline

-20 \div-10=2

\newline

2

\newline

1

\newline

(A) The temperature dropped

20^{\circ} \mathrm{C}

2

\newline

(B) The temperature dropped

10^{\circ} \mathrm{C}

2

\newline

(C) The temperature is dropping at a rate of

2^{\circ} \mathrm{C}

Emanuel surveyed a random sample of

50

subscribers to Auto Wheel magazine about the number of cars that they own. Of the subscribers surveyed,

15

2

vehicles. There are

340

subscribers to Auto Wheel magazine.

\newline

Based on the data, what is the most reasonable estimate for the number of Auto Wheel magazine subscribers who own fewer than

2

\newline

1

\newline

15

\newline

84

\newline

93

\newline

102

Bounce Club surveyed a random sample of

55

of their members about their favorite type of music. Of the members surveyed,

22

said that jazz is their favorite type of music. There are

422

Bounce Club members.

\newline

Based on the data, what is the most reasonable estimate for the number of Bounce Club members whose favorite type of music is jazz music?

\newline

1

\newline

22

\newline

141

\newline

\mathbf{1 6 9}

\newline

253

Mr. Randolph is putting the finishing touches on his backyard flower bed. The flower bed is

25

6

feet wide, and it is bordered on two sides by the backyard fence. Mr. Randolph plans to put a pathway around the other two sides of the flower bed that is the same width on both sides.

\newline

The total area of the flower bed and pathway in square feet can be modeled by the expression

(x + 25)(x + 6)

x

is the width of the pathway in feet. This expression can also be written in the form

x^2 + 31x + 150

. What does the quantity

x^2 + 31x

represent in the expression?

\newline

\newline

(A)the area of the flower bed in square feet

\newline

(B)the area of the pathway in square feet

\newline

(C)the perimeter around the outside of the flower bed and pathway in feet

\newline

(D)the length of the flower bed and pathway together in feet

The Rock Rollers released their first album, Metamorphic, when they were still mostly unknown, and the value of the album initially declined. As the band became more famous and the album became more popular, its value started to increase. The value of the album in dollars can be modeled by the expression

0.2t^2 - 4t + 25

t

is the time in years after it was first released.

\newline

What does the number

25

represent in the expression?

\newline

\newline

(A)the time in years from when the album was released until its value was the same as when it was first released

\newline

(B)the time in years from when the album was released until its value was the least

\newline

(C)the album's value in dollars when it was worth the least

\newline

(D)the album's value in dollars when it was first released

Jaylen launches his new high-powered model rocket upward during the Rocket Week demonstration. The rocket flies far up into the sky before falling back down to the ground. The rocket has an initial velocity of

147\,\text{meters per second}

. Therefore, the rocket's height above the ground in meters,

t

seconds after being launched, can be modeled by the expression

-4.9t^2 + 147t

. This expression can be written in factored form as

-4.9t(t - 30)

\newline

What does the number

30

represent in the expression?

\newline

(A)the time in seconds from when the rocket is launched until it hits the ground

\newline

(B)the height in meters of the rocket when it is launched

\newline

(C)the time in seconds from when the rocket is launched until it reaches the top of its flight

\newline

(D)the height in meters of the rocket when it reaches the top of its flight

McClure Sisters Inc., an architectural firm, is in the early phases of planning the building for the Sports and Leisure Museum. The museum will be twice as long as it is wide to accommodate the exhibit hall for sports and the exhibit hall for leisure. The museum will also have a

15

-foot-wide parking lot along two sides.

\newline

The total area of the museum and parking lot together in square feet can be modeled by the expression

(2x + 15)(x + 15)

x

is the width of the museum in feet. This expression can also be written in the form

2x^2 + 45x + 225

. What does the quantity

45x + 225

represent in the expression?

\newline

\newline

(A)the area of the museum in square feet

\newline

(B)the length of the museum and parking lot together in feet

\newline

(C)the perimeter around the outside of the museum and parking lot in feet

\newline

(D)the area of the parking lot in square feet

Rockin' Jon, a music club owner, is planning for the next annual Rockin' Jon's New Year's Eve Jamfest. He has noticed that when he charges more for tickets for the show, fewer people buy tickets. Rockin' Jon's total revenue from selling tickets for the Jamfest, in dollars, can be modeled by the expression

p(576 - 16p)

p

is the price per ticket in dollars. This expression can be written in factored form as

-16p(p - 36)

\newline

What does the number

36

represent in the expression?

\newline

(A) Rockin' Jon's minimum total revenue in dollars

\newline

(B) the price per Jamfest ticket in dollars so that Rockin' Jon's total revenue is zero

\newline

(C) Rockin' Jon's maximum total revenue in dollars

\newline

(D) the price per Jamfest ticket in dollars that maximizes Rockin' Jon's total revenue

The value of a limited-edition porcelain doll decreased at first, but then the doll became harder to find, and the value started to increase. The value of the doll can be modeled by the expression

0.25t^2 - 3t + 24

t

is the time in years after it was first released.

\newline

What does the number

24

represent in the expression?

\newline

\newline

(A)the time in years from when the doll was released until its value was the same as when it was first released

\newline

(B)the doll's value in dollars when it was worth the least

\newline

(C)the time in years from when the doll was released until its value was the least

\newline

(D)the doll's value in dollars when it was first released

Seth, a carpenter, is helping a client remodel her backyard patio entryway. The entryway will have a set of double doors that are trimmed with moulding. The double doors will be square, and the moulding will be

2

inches wide along the top and sides of the door. The total area of the double doors and moulding in square inches can be modeled by the expression

(x + 4)(x + 2)

x

is the width of the double doors in inches. This expression can also be written in the form

x^2 + 6x + 8

. What does the quantity

6x + 8

represent in the expression?

\newline

(A)the area of the moulding in square inches

\newline

(B)the height of the double doors and moulding together in inches

\newline

(C)the perimeter around the outside of the double doors and moulding in inches

\newline

(D)the area of the double doors in square inches

During a match of badminton, Josh swings his racquet and hits the birdie high up into the air. The birdie's height above the ground in meters can be modeled by the expression

1.4 + 6t - 4.9t^2

t

is the time in seconds after Josh hits the birdie.

\newline

What does the number

1.4

represent in the expression?

\newline

(A)the height in meters of the birdie when Josh hits it

\newline

(B)the time in seconds from when Josh hits the birdie until it hits the ground

\newline

(C)the time in seconds from when Josh hits the birdie until it reaches its highest point

\newline

(D)the height in meters of the birdie when it reaches its highest point

Mr. Larry, of Larry's Landscaping, is designing a backyard garden for a client. For visual appeal, the client has requested that the length of the garden be equal to twice the width. The client has also asked for the garden to have a rock edging

2

feet wide around three sides, since the backyard fence will be along the fourth side.

\newline

The total area of the garden and edging in square feet can be modeled by the expression

(2x + 4)(x + 2)

x

is the width of the garden in feet. This expression can also be written in the form

2x^2 + 8x + 8

\newline

What does the quantity

8x + 8

represent in the expression?

\newline

\newline

(A)the length of the garden and edging together in feet

\newline

(B)the area of the edging in square feet

\newline

(C)the area of the garden in square feet

\newline

(D)the perimeter around the outside of the edging in feet

When Bazillium released its signature trading-card game, Wandering Wizards, the value of a first-edition deck decreased at first. As the game got more popular and the deck became more rare, its value started to increase. The value of a first-edition deck in dollars can be modeled by the expression

0.25t^2 - 4t + 28

t

is the time in years after it was first released.

\newline

What does the number

28

represent in the expression?

\newline

(A)the time in years from when the deck was released until its value was the least

\newline

(B)the deck's value in dollars when it was first released

\newline

(C)the time in years from when the deck was released until its value was the same as when it was first released

\newline

(D)the deck's value in dollars when it was worth the least

Ann wants to print and frame a photo she recently took at the beach. She plans for the print to have the dimensions

10

8

inches, and she also plans to put a border around the photo that has an equal width all around.

\newline

The total area of the photo and border in square inches can be modeled by the expression

(10 + 2x)(8 + 2x)

x

is the width of the border in inches. This expression can also be written in the form

4x^2 + 36x + 80

\newline

What does the quantity

4x^2 + 36x

represent in the expression?

\newline

\newline

(A)the perimeter around the outside of the border in inches

\newline

(B)the area of the photo in square inches

\newline

(C)the length of the photo and border together in inches

\newline

(D)the area of the border in square inches

\newline

Venture Properties is looking to design a new food-truck park. The dimensions of the rectangular park will be

110

80

feet. The parking lot will be along two sides of the park, and it will be the same width on both sides.

\newline

The total area of the park and parking lot together in square feet can be modeled by the expression

(x + 80)(x + 110)

x

is the width of the parking lot in feet.

\newline

What does the quantity

x + 110

represent in the expression?

\newline

(A)the length of the park and parking lot together in feet

\newline

(B)the perimeter around the outside of the park and parking lot in feet

\newline

(C)the area of the parking lot in square feet

\newline

(D)the total area of the park and parking lot together in square feet

Liz just returned from a trip to visit some of the national parks with her family, and she wants to print her favorite picture from the trip to hang on her wall. She is not quite sure what size print she wants to get, although she does know that she wants the print to be square. She also wants to add a border that is

3

inches wide around the sides of the photo.

\newline

The total area of the photo and border in square inches can be modeled by the expression

(x + 6)(x + 6)

x

is the width of the photo in inches. This expression can also be written in the form

x^2 + 12x + 36

. What does the quantity

12x + 36

represent in the expression?

\newline

\newline

(A)the length of the photo and border together in inches

\newline

(B)the area of the photo in square inches

\newline

(C)the area of the border in square inches

\newline

(D)the perimeter around the outside of the border in inches

The Pool Company installs custom-built swimming pools. One of its customers would like a rectangular pool with a sidewalk of equal width around the entire perimeter. To fit in the backyard, the pool and sidewalk together will have to be

18

36

\newline

The area of the pool in square feet can be modeled by the expression

(18 - 2x)(36 - 2x)

x

is the width of the sidewalk in feet.

\newline

What does the quantity

36 - 2x

represent in the expression?

\newline

\newline

(A)the perimeter around the outside of the pool in feet

\newline

(B)the area of the pool in square feet

\newline

(C)the area of the sidewalk in square feet

\newline

(D)the length of the pool in feet

Levi kicks a soccer ball up into the air with an initial upward velocity of

14.7

meters per second. Therefore, the ball's height above the ground in meters,

t

seconds after it is kicked, can be modeled by the expression

-4.9t^2 + 14.7t

. This expression can be written in factored form as

-4.9t(t - 3)

\newline

What does the number

3

represent in the expression?

\newline

(A)the height in meters of the ball when it is kicked

\newline

(B)the time in seconds from when the ball is kicked until it hits the ground

\newline

(C)the time in seconds from when the ball is kicked until it reaches its highest point

\newline

(D)the height in meters of the ball when it reaches its highest point

y Association (I

952

\newline

enator Josepb R. McCariby of Wisconsin, bitberto unknown to fame, rocheted in headlines in

1950

wben be declaredt in a political speecb that there were scon f known communists in the State Department.

A pot of piping hot stew has been removed from the stove and left to cool.

\newline

The relationship between the elapsed time,

m

, in minutes, since the stew was removed from the stove, and the temperature of the stew,

T(m)

{ }^{\circ} C

, is modeled by the following function.

\newline

T(m)=20+50 \cdot 10^{-0.04 m}

\newline

How many minutes will it take for the stew to cool to a temperature of

30^{\circ} \mathrm{C}

\newline

Round your answer, if necessary, to the nearest hundredth.

\newline

\square

Amir stands on a balcony and throws a ball to his dog, who is at ground level. The ball's height (in meters above the ground),

x

seconds after Amir threw it, is modeled by

h(x)=- (x+1)(x-7)

. What is the maximum height that the ball will reach?

1.13(0.55 p)

represents the amount Chee paid for dinner at a restaurant after she applied a discount coupon and added tip. Which changes to the original price could have resulted in the expression?

\newline

0.13 \%

tip and then a discount of

0.45 \%

\newline

0.45 \%

13 \%

\newline

13 \%

tip and then a discount of

45 \%

\newline

55 \%

13 \%

Water Wizards and Pat's Pipes are two companies that offer plumbing and heating services. Both charge a base fee for all service appointments and an hourly labor rate for repairs. The two companies charge the same base fee, but Water Wizards charges a higher hourly rate than Pat's Pipes. If the total amount Water Wizards charges for

x

hours of repairs is modeled by the expression

45 x+60

, which expression could represent the total amount Pat's Pipes would charge for the same amount of time spent on repairs?

\newline

55 x+60

\newline

45 x+50

\newline

35 x+60

\newline

45 x+70

Amir stands on a balcony and throws a ball to his dog who is at ground level. The ball's height (in meters above the ground),

x

seconds after Amir threw it, is modeled by:

h(x) = -(x-2)^2 + 16

. What is the height of the ball at the time it is thrown.

The equation for line

j

can be written as

y=-\frac{5}{7} x-10

k

, which is parallel to line

j

, includes the point

(-9,5)

. What is the equation of line

k

\newline

Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

r

has an equation of

y-9=3(x-10)

s

includes the point

(-2,-7)

and is parallel to line

r

. What is the equation of line

s

\newline

Write the equation in slope-intercept form. Write the numbers in the equation as simplified proper fractions, improper fractions, or integers.

The Rock Rollers released their first album, Metamorphic, when they were still mostly unknown, and the value of the album initially declined. As the band became more famous and the album became more popular, its value started to increase. The value of the album in dollars can be modeled by the expression

0.2t^2-4t+25

t

is the time in years after it was first released. What does the number

25

represent in the expression?

An accountant for a certain state models the sales tax revenue that the state has earned over several years. The model shows that sales tax revenue grew by

15\%

2

years ago, when it started to grow by

\$179.5

million per year. If the sales tax revenue

2

\$2.1

billion, approximately how much lower is the sales tax revenue this year than it would be if it had continued growing by

15\%

\newline

1

=1,000

\newline

1

\newline

\$136

\newline

\$318

\newline

\$2.05

\newline

\$5.24

Maya shaved her head and then began letting her hair grow. She represents the length

\ell

of her hair, in centimeters,

m

months after she shaved her head using the equation

\ell=1.25 \mathrm{~m}

\newline

1

25

mean in this situation?

\newline

1

\newline

1

25

months since Maya shaved her head.

\newline

(B) Maya's hair grows

1

25

centimeters per month.

\newline

(C) Maya's hair takes

1

25

months to grow a centimeter longer.

Which of the following is a correct interpretation of the expression

2-(-7) ?

\newline

1

\newline

2

on the number line and move

7

\newline

2

on the number line and move

7

\newline

-7

on the number line and move

2

\newline

-7

on the number line and move

2

Which of the following is a correct interpretation of the expression

3-(-4) ?

\newline

1

\newline

3

on the number line and move

4

\newline

3

on the number line and move

4

\newline

-4

on the number line and move

3

\newline

-4

on the number line and move

3

Lucy is a dress maker. She sews

\frac{4}{7}

\frac{3}{4}

hour. Lucy sews at a constant rate.

\newline

At this rate, how many dresses does Lucy sew in one hour? Include fractions of dresses if applicable.

\newline

A ferry traveled

\frac{1}{6}

of the distance between two ports in

\frac{3}{7}

hour. The ferry travels at a constant rate.

\newline

At this rate, what fraction of the distance between the two ports can the ferry travel in one hour?

\newline

of the distance

Which of the following are true statements about the expression

-8+(-8) ?

\newline

2

\newline

(A) The expression describes the number that is

8

-8

on the number line.

\newline

(B) The expression describes the number that is

8

to the right of

-8

on the number line.

\newline

(C) The expression equals

-16

\newline

(D) The expression equals

0

Which of the following is a correct interpretation of the expression

4-(-3)

\newline

1

\newline

4

on the number line and move

3

\newline

4

on the number line and move

3

\newline

-3

on the number line and move

4

\newline

-3

on the number line and move

4

Which of the following are true statements about the expression

-5+(-5) ?

\newline

2

\newline

(A) The expression describes the number that is

5

-5

on the number line.

\newline

(B) The expression describes the number that is

5

to the right of

-5

on the number line.

\newline

(C) The expression equals

-10

\newline

(D) The expression equals

0

Which of the following are true statements about the expression

4-(-4)

\newline

2

\newline

(A) The expression describes the number that is

4

4

on the number line.

\newline

(B) The expression describes the number that is

4

to the right of

4

on the number line.

\newline

(C) The expression equals

0

\newline

(D) The expression equals

8

A school charges

\$ 6

per ticket to a dance. They spend

\$ 300

on music and decorations.

\newline

6 n-300

describes the amount of profit the school makes from the dance if

n

students buy tickets.

\newline

6 n

represent in this context?

\newline

1

\newline

(A) The amount of money the school charges for all

n

\newline

(B) The amount of money the school spends on the dance

\newline

(C) The number of students who buy tickets

\newline

(D) The profit the school earns per ticket

Ray is buying some ginger roots to brew some fresh ginger ale. The price of the ginger roots is

G

, and Ray has a coupon for

10 \%

\newline

0.9 G

represents the price Ray pays for the ginger roots after using the coupon.

\newline

0

9

represent in this context?

\newline

1

\newline

(A) The amount of the discount from the coupon

\newline

(B) The percent, as a decimal, of the discount from the coupon

\newline

(C) The percent, as a decimal, of the original price that Ray pays

\newline

(D) The amount Ray pays after using the coupon

Camilla borrows a book from the library for

d

days. The library charges a late fee of

0

10

dollars per day that the book is late.

\newline

If Camilla returns the book more than

21

days after she borrowed it, the expression

0.10(d-21)

\newline

represents the total late fee Camilla owes.

\newline

(d-21)

represent in this context?

\newline

1

\newline

(A) The number of days Camilla borrows the book

\newline

(B) The late fee Camilla owes per day the book is late

\newline

(C) The late fee Camilla owes per day she borrowed the book

\newline

(D) The number of days the book is late

1.08 s+1.02 b

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

s

is the value of stocks and

b

is the value of bonds in the portfolio at the beginning of the year.

\newline

What is the predicted end-of-year value of a portfolio that begins the year with

\$ 200

\$ 100

\newline

\$

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.

\newline

How much does an ice cream cone with

5

\newline

\$

100+20 m

gives the volume of water in Eduardo's pool (in liters) after Eduardo spends

m

minutes filling his pool.

\newline

What is the volume of water in Eduardo's pool after he fills it for

5 \frac{1}{4}

\newline

4 x

gives the perimeter of a square with a side length of

x

\newline

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

\frac{5}{7}

\newline

...