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Solve quadratic equations: word problems

P(t)=2.7(0.9978)^{t-2019}

\newline

The equation models

P

, the population of Chicago, in millions, in year

t

. Which of the following statements is the best interpretation of the ordered pair

(2029,2.64)

\newline

1

\newline

(A) The model predicts that the population of Chicago will be increasing at a rate of

2

64

2029

\newline

(B) The model predicts that the population of Chicago will be decreasing at a rate of

2

64

2029

\newline

(C) The model predicts that the population of Chicago will be approximately

2

64

2029

\newline

(D) The model predicts that the population of Chicago will increase by approximately

2

64

2029

f(x)=5,000(1.012)^{t}

\newline

The function models

f

, the amount of money in Eddie's savings account

x

years after his initial deposit into the account. Which of the following statements is the best interpretation of the ordered pair

(2,5120.72)

\newline

1

\newline

(A) Eddie earned

5

120

72

dollars in interest from his savings account

2

years after his initial deposit.

\newline

5

120

72

dollars in his savings account

2

years after his initial deposit.

\newline

5

120

72

dollars in his savings account

2

months after his initial deposit.

\newline

5

120

72

dollars in his savings account, which yields

2

percent annual interest.

p(t)=31(2)^{2 t}

\newline

The equation models

p

, the amount of bacteria in colony-forming units in a bacteria culture after

t

hours of growth. Which of the following statements is the best interpretation of the ordered pair

(2,496)

\newline

1

\newline

496

colony-forming units in the bacteria culture after

2

hours of growth.

\newline

496

colony-forming units in the bacteria culture after

2

days of growth.

\newline

(C) The number of colony-forming units in the bacteria culture reaches

496

2

\newline

(D) The number of colony-forming units in the bacteria culture doubles every

496

R(q)=-0.31(q-260)^{2}+9,500

\newline

A shoe manufacturer determines that its monthly revenue,

R(q)

, in dollars, is given by the function, where

q

is the number of pairs of shoes sold each month. What is the maximum value of the company's monthly revenue in dollars?

f(x)=2,500(0.66)^{x}

\newline

The equation models

f

, the number of new social media posts associated with a trending hashtag

x

days after its peak popularity. Which of the following statements is the best interpretation of the ordered pair

(3,719)

\newline

1

\newline

(A) The trending hashtag maintains

719

new social media posts per day for

3

\newline

(B) The number of new social media posts associated with the trending hashtag increases by

719

3

\newline

3

new social media posts associated with the trending hashtag are made

719

days after the hashtag's peak popularity.

\newline

719

new social media posts associated with the trending hashtag are made

3

days after the hashtag's peak popularity.

P(t)=4.87(1.02)^{t}

\newline

The function models

P

, the population, in millions, of the city of Melbourne

t

2019

. Which of the following statements is the best interpretation of the ordered pair

(4,5.27)

\newline

1

\newline

(A) The model predicts that the population of Melbourne will increase by

4

percent per year for

5

27

2019

\newline

(B) The model predicts that the population of Melbourne will increase by

5

27

percent per year for

4

2019

\newline

(C) The model predicts that the population of Melbourne will be

4

5

27

2019

\newline

(D) The model predicts that the population of Melbourne will be

5

27

4

2019

r(x)=x(32-2 x)

\newline

The function models

r

, the revenue in thousands of dollars made by selling a particular chocolate bar at a price of

x

dollars. According to the model, as the price of the chocolate bar increases, the number of bars sold decreases. Which of the following statements is the best interpretation of the ordered pair

(4,96)

\newline

1

\newline

(A) The revenue from selling the chocolate bar at

4

dollars per bar is

96

thousand dollars.

\newline

(B) The revenue from selling the chocolate bar at

4

dollars per bar is

96

\newline

96

thousand bars of chocolate are sold at

4

dollars per bar.

\newline

96

bars of chocolate are sold at

4

dollars per bar.

A(x)=(2+2 x)^{2}

\newline

Roxane wants to add a border around a square picture. The function models

A

, the new area of the picture in square inches if it has a border

x

inches wide. Which of the following statements is the best interpretation of the ordered pair

(0.5,9)

\newline

1

\newline

0

5

-inch border is added around the picture, the new side length of the picture is

9

\newline

0

5

-inch border is added around the picture, the new perimeter of the picture is

9

\newline

0

5

-inch border is added around the picture, the new area of the picture is

9

times the original area.

\newline

0

5

-inch border is added around the picture, the new area of the picture is

9

h(t)=-16.1 t^{2}+393 t+6

\newline

The function models

h

, the height of a firework shell in feet

t

seconds after launch. Which of the following statements is the best interpretation of the ordered pair

(10,2326)

\newline

1

\newline

10

seconds after launch, the firework shell has traveled a total of

2

326

\newline

10

seconds after launch, the height of the firework shell is

2

326

\newline

(C) When the height of the firework shell is

2

326

feet, the shell is traveling at a speed of

10

feet per second.

\newline

(D) When the height of the firework shell is

2

326

feet, the shell has also traveled

10

feet horizontally.

T(n)=3.5+7 \cdot 0.9^{n}

\newline

The administrators of a factory modeled the cumulative average construction time,

T(n)

, in minutes per engine component, as a function of the total number of components their employees had previously constructed,

n

, using the function shown. After the employees had constructed a very large number of engine components, what was the cumulative average time to construct a single component according to the model?

\newline

1

\newline

3

5

minutes per engine component

\newline

7

minutes per engine component

\newline

9

8

minutes per engine component

\newline

10

5

minutes per engine component

p(t)=100\left(\frac{1}{2}\right)^{\frac{t}{2.0373}}

\newline

Radioactive isotopes are unstable atoms that decay into other atoms. Oxygen

-15

is a radioactive isotope. The function models

p

, the percentage of oxygen

-15

atoms remaining in a sample after

t

minutes. Which of the following statements is the best interpretation of the ordered pair

(4.0746,25)

\newline

1

\newline

25

percent of oxygen

-15

atoms remain in a sample after

4

0746

\newline

4

0746

percent of oxygen

-15

atoms remain in a sample after

25

\newline

25

percent of oxygen

-15

atoms in a sample decay into other atoms in

4

0746

\newline

4

0746

percent of oxygen

-15

atoms in a sample decay into other atoms in

25

f(t)=800(0.85)^{t}

\newline

The function models

f

, the value of a computer, in dollars,

t

years after its purchase. Which of the following statements is the best interpretation of the ordered pair

(5,355)

\newline

1

\newline

(A) The value of the computer increases by

355

5

years after its purchase.

\newline

(B) The value of the computer decreases by

355

5

years after its purchase.

\newline

(C) The value of the computer is

355

5

years after its purchase.

\newline

(D) The value of the computer decreases by

355

dollars per year

5

years after its purchase.

p(x)=(x-0.83)(500-100 x)

\newline

The equation models

p

, the profit in dollars made by selling tacos at a price of

x

dollars each. According to the model, as the price of a taco increases, the number of tacos sold decreases. Which of the following statements is the best interpretation of the ordered pair

(2,351)

\newline

1

\newline

351

tacos are sold at

2

dollars per taco.

\newline

(B) The profit from selling each taco is

2

351

tacos are sold.

\newline

(C) The profit from selling tacos at

2

dollars each is

351

\newline

(D) The total revenue from selling tacos at

2

dollars each is

351

p(x)=2.1 x^{2}

\newline

The function models

p

, the cost in dollars of the floor tiles needed to cover a square area with a side length of

x

feet. Which of the following statements is the best interpretation of the ordered pair

(10,210)

\newline

1

\newline

210

floor tiles are needed to cover a square area with a side length of

10

\newline

(B) The cost of the floor tiles needed to cover a square area with a side length of

10

210

\newline

(C) The cost of the floor tiles needed to cover an area of

10

210

\newline

(D) The cost of the floor tiles needed to cover an area of

210

10

k(x)=5 x^{2}

\newline

The function models

k

, the kinetic energy in joules of a silver carp with a mass of

10

kilograms as it swims at a speed of

x

meters per second. Which of the following statements is the best interpretation of the ordered pair

(2.4,28.8)

\newline

1

\newline

(A) A silver carp can swim

28

8

2

4

\newline

(B) The maximum kinetic energy of a

2

4

-kilogram silver carp is

28

8

\newline

(C) The kinetic energy of a

10

-kilogram silver carp is

2

4

joules when it is swimming at a speed of

28

8

meters per second.

\newline

(D) The kinetic energy of a

10

-kilogram silver carp is

28

8

joules when it is swimming at a speed of

2

4

meters per second.

Rui is a professional deep water free diver.

\newline

His altitude (in meters relative to sea level),

x

seconds after diving, is modeled by:

\newline

d(x)=\frac{1}{2} x^{2}-10 x

\newline

What is the lowest altitude Rui will reach?

\newline

\square

meters relative to sea level

The fish population in a certain part of the ocean (in thousands of fish) as a function of the water's temperature (in degrees Celsius) is modeled by:

\newline

p(x)=-2 x^{2}+40 x-72

\newline

Which temperatures will result in no fish (i.e.

0

\newline

Enter the lower temperature first.

\newline

Lower temperature:

\square

degrees Celsius

\newline

Higher temperature:

\square

degrees Celsius

A hovercraft takes off from a platform.

\newline

Its height (in meters),

x

seconds after takeoff, is modeled by:

\newline

h(x)=-2x^{2}+20x+48

\newline

How many seconds after takeoff will the hovercraft reach its maximum height?

\newline

\text{seconds}

Ying is a professional deep water free diver.

\newline

His altitude (in meters relative to sea level),

\newline

x

seconds after diving, is modeled by:

\newline

D(x)=\frac{1}{36}(x-60)^{2}-100

\newline

How many seconds after diving will Ying reach his lowest altitude?

\newline

\text{seconds}

Alain throws a stone off a bridge into a river below.

\newline

The stone's height (in meters above the water),

x

seconds after Alain threw it, is modeled by:

\newline

h(x)=-5x^{2}+10x+15

\newline

How many seconds after being thrown will the stone hit the water?

\newline

\text{seconds}

The power generated by an electrical circuit (in watts) as a function of its current

x

(in amperes) is modeled by:

\newline

P(x)=-12x^{2}+120x

\newline

What is the maximum power generated by the circuit?

\newline

\text{watts}

The power generated by an electrical circuit (in watts) as a function of its current

x

(in amperes) is modeled by:

\newline

P(x)=-12x^{2}+120x

\newline

What current will produce the maximum power?

\newline

A certain company's main source of income is selling cloth bracelets.

\newline

The company's annual profit (in thousands of dollars) as a function of the price of a bracelet (in dollars) is modeled by:

\newline

P(x)=-2x^{2}+16x-24

\newline

What is the maximum profit that the company can earn?

\newline

thousand dollars

A certain company's main source of income is selling cloth bracelets.

\newline

The company's annual profit (in thousands of dollars) as a function of the price of a bracelet (in dollars) is modeled by:

\newline

P(x)=-2x^{2}+16x-24

\newline

What bracelet price should the company set to earn this maximum profit?

\newline

\text{dollars}

A hovercraft takes off from a platform.

\newline

Its height (in meters),

\newline

x

seconds after takeoff, is modeled by:

\newline

h(x)=-2x^{2}+20x+48

\newline

What is the maximum height that the hovercraft will reach?

\newline

\text{meters}

Antoine stands on a balcony and throws a ball to his dog, who is at ground level.

\newline

The ball's height (in meters above the ground),

\newline

x

seconds after Antoine threw it, is modeled by:

\newline

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

\newline

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

\newline

\text{meters}

200

meters of fencing to build a rectangular garden.

\newline

The garden's area (in square meters) as a function of the garden's width

\newline

x

(in meters) is modeled by:

\newline

A(x)=-x^{2}+100x

\newline

What side width will produce the maximum garden area?

\newline

\text{meters}

200

meters of fencing to build a rectangular garden.

\newline

The garden's area (in square meters) as a function of the garden's width

\newline

x

(in meters) is modeled by:

\newline

A(x)=-x^{2}+100x

\newline

What is the maximum area possible?

\newline

The number of mosquitoes in Anchorage, Alaska (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by:

\newline

m(x)=-x^{2}+14x

\newline

What is the maximum possible number of mosquitoes?

\newline

million mosquitoes

Rui is a professional deep water free diver. His altitude (in meters relative to sea level),

x

seconds after diving, is modeled by:

\newline

d(x)=\frac{1}{2}x^{2}-10x

\newline

How many seconds after diving will Rui reach his lowest altitude?

\newline

\text{seconds}

Alain throws a stone off a bridge into a river below. The stone's height (in meters above the water),

x

seconds after Alain threw it, is modeled by:

\newline

h(x) = -5x^2 + 10x + 15

\newline

What is the height of the stone at the time it is thrown?

\newline

\text{meters}

The number of mosquitoes in Anchorage, Alaska (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by:

\newline

m(x)=-x^{2}+14x

\newline

How many centimeters of rain will produce the maximum number of mosquitoes?

\newline

\text{centimeters}

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?

\newline

An object is launched from a platform. Its height (in meters),

x

seconds after the launch, is modeled by:

\newline

h(x)=-5x^{2}+20x+60

\newline

What is the height of the object at the time of launch?

\text{meters}

An object is launched from a platform. Its height (in meters),

x

seconds after the launch, is modeled by:

\newline

h(x)=-5x^{2}+20x+60

\newline

How many seconds after launch will the object land on the ground?

\newline

\text{seconds}

A hovercraft takes off from a platform.

\newline

Its height (in meters),

\newline

x

seconds after takeoff, is modeled by:

\newline

h(x)=-3(x-3)^{2}+108

\newline

How many seconds after takeoff will the hovercraft land on the ground?

\newline

\text{seconds}

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

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

\text{seconds}

The number of mosquitoes in Minneapolis, Minnesota (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by:

\newline

m(x) = -(x - 5)^{2} + 25

\newline

How many centimeters of rainfall will produce the maximum number of mosquitoes?

\newline

\text{centimeters}

The fish population in a certain part of the ocean (in thousands of fish) as a function of the water's temperature (in degrees Celsius) is modeled by:

\newline

P(x)=-2(x-9)^{2}+200

\newline

What is the maximum number of fish?

\newline

The number of mosquitoes in Minneapolis, Minnesota (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by:

\newline

m(x) = -(x - 5)^{2} + 25

\newline

What is the maximum possible number of mosquitoes?

\newline

million mosquitoes

An object is launched from a platform. Its height (in meters),

x

seconds after the launch, is modeled by:

\newline

h(x)=-5(x-4)^{2}+180

\newline

What is the height of the object at the time of launch?

\text{meters}

Ming throws a stone off a bridge into a river below.

\newline

The stone's height (in meters above the water),

x

seconds after Ming threw it, is modeled by:

\newline

h(x)=-5(x-1)^{2}+45

\newline

How many seconds after being thrown will the stone reach its maximum height?

\newline

\text{seconds}

The power generated by an electrical circuit (in watts) as a function of its current

c

(in amperes) is modeled by:

\newline

P(c)=-20(c-3)^{2}+180

\newline

Which currents will produce no power (i.e.

0

\newline

Enter the lower current first.

\newline

\text{amperes}

\newline

Higher current:

\text{amperes}

A hovercraft takes off from a platform.

\newline

Its height (in meters),

\newline

x

seconds after takeoff, is modeled by:

\newline

h(x) = -3(x - 3)^{2} + 108

\newline

What is the height of the hovercraft at the time of takeoff?

\newline

\text{meters}

100

meters of fencing to build a rectangular garden. The garden's area (in square meters) as a function of the garden's width

x

(in meters) is modeled by:

\newline

A(x)=-(x-25)^{2}+625

\newline

What is the maximum area possible?

\newline

\text{square meters}

100

meters of fencing to build a rectangular garden. The garden's area (in square meters) as a function of the garden's width

x

(in meters) is modeled by:

A(x)=-(x-25)^{2}+625

What side width will produce the maximum garden area?

\text{meters}

Amir stands on a balcony and throws a ball to his dog, who is at ground level.

\newline

The ball's height (in meters above the ground),

x

seconds after Amir threw it, is modeled by:

\newline

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

\newline

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

\newline

meters

A certain company's main source of income is selling socks.

\newline

The company's annual profit (in millions of dollars) as a function of the price of a pair of socks (in dollars) is modeled by:

\newline

P(x)=-3(x-5)^{2}+12

\newline

What is the maximum profit that the company can earn?

\newline

million dollars

Ming throws a stone off a bridge into a river below.

\newline

The stone's height (in meters above the water),

\newline

x

seconds after Ming threw it, is modeled by:

\newline

h(x)=-5(x-1)^{2}+45

\newline

What is the maximum height that the stone will reach?

\newline

\text{meters}

A certain company's main source of income is selling socks.

\newline

The company's annual profit (in millions of dollars) as a function of the price of a pair of socks (in dollars) is modeled by:

\newline

P(x)=-3(x-5)^{2}+12

\newline

What sock price should the company set to earn a maximum profit?

\newline

\text{dollars}