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Resources

Evaluate two-variable equations: word problems

A company orders boxed lunches from a deli, which all cost the same price. The relationship between the number of boxed lunches ordered,

x

, and the total cost in dollars of the lunches,

y

, is represented by a graph drawn in the xy-plane.

\newline

(9,63)

lies on the graph, what does the ordered pair

(9,63)

\newline

9

lunches that cost

\$ 63.00

\newline

9

lunches that cost a total of

\$ 63.00

\newline

63

lunches that cost

\$ 9.00

\newline

63

lunches that cost a total of

\$ 9.00

A professional baseball team sells regular tickets and premium tickets to all of its home games. At a recent game, the average price for a regular ticket was

\$52.46

, and the average price for a premium ticket was

\$175.85

. Which of the following equations shows a relationship where the number of regular tickets,

r

, and the number of premium tickets,

p

, that the team sells generates

\$5,000,000

in ticket sales?

\newline

1

\newline

\frac{p}{52.46} + \frac{r}{175.85} = 5,000,000

\newline

\frac{r}{52.46} + \frac{p}{175.85} = 5,000,000

\newline

52.46p + 175.85r = 5,000,000

\newline

52.46r + 175.85p = 5,000,000

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=1.25m

1.25

mean in this situation?

\newline

1

\newline

(A) It has been

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.

Wyoli plans to enclose a rectangular area for his sheep to graze, with approximately

200

feet of fencing. One side will be bordered by a straight section of river, the other three by the fence, and the enclosure has a width of

x

ft, as seen in the figure. Which of the following functions best models the enclosed area, in square feet?

Kaori is taking a free-throw.

\newline

H(d)

models the basketball's height (in meters) at a horizontal distance of

d

meters from Kaori.

\newline

What does the statement

H(R) = 4

\newline

1

\newline

(A) The ball was at the same height at the horizontal distances of

4

R

\newline

(B) At a horizontal distance of

R

meters from Kaori, the ball's height was equal to

4

\newline

(C) At a horizontal distance of

4

meters from Kaori, the ball's height was equal to

R

To preserve the ecological balance of a reserve, a number of lions were released in the reserve in

2000

. Initially, there were only

5

lions in the reserve. After the observation of a few years, a function

L(n)=(n+1)^2+(n+1)+15

was developed to represent the number of lions in the reserve, where

L(n)

represents the number of lions and

n

represents the number of years after which they were released. Assume that no lion died during this period. Based on the above function, which of the following statements are correct? (without calculator)

\newline

10

lions were released in the reserve in

2000

\newline

(II) The increase in the number of lions in the third year was

8

\newline

(III) The total number of lions in the beginning of the third year was

27

\newline

\newline

\newline

\newline

A lake near the Arctic Circle is covered by a

h

-meter-thick sheet of ice during the cold winter months. When spring arrives, the warm air gradually melts the ice, causing its thickness to decrease at a constant rate. After

t

weeks, the sheet is only

d

meters thick. Let

T

represent the ice sheet's thickness (in meters) after

t

weeks. Complete the equation for the relationship between the thickness and number of weeks.

Apply the distributive property to factor out the greatest common factor.

\newline

22c+33d=_______

The rate of change

\frac{d P}{d t}

of the number of fox at a national park is modeled by a logistic differential equation. The maximum capacity of the park is

888

8

PM, the number of fox at the national park is

210

and is increasing at a rate of

36

fox per day. Write a differential equation to describe the situation.

\newline

\frac{d P}{d t}=\square

The rate of change

\frac{d P}{d t}

of the number of wolves at a national park is modeled by a logistic differential equation. The maximum capacity of the park is

955

8

PM, the number of wolves at the national park is

218

and is increasing at a rate of

26

wolves per day. Write a differential equation to describe the situation.

\newline

\frac{d P}{d t}=\square \text { Submit Answer }

The rate of change

\frac{d P}{d t}

of the number of algae in a test tube is modeled by a logistic differential equation. The maximum capacity of the tube is

580

8 \mathrm{AM}

, the number of algae in the test tube is

155

and is increasing at a rate of

31

algae per minute. Write a differential equation to describe the situation.

\newline

\frac{d P}{d t}=\square

Kaori is taking a free-throw.

\newline

H(d)

models the basketball's height (in meters) at a horizontal distance of

d

meters from Kaori.

\newline

What does the statement

H(R) = 4

\newline

1

\newline

(A) The ball was at the same height at the horizontal distances of

4

R

\newline

(B) At a horizontal distance of

R

meters from Kaori, the ball's height was equal to

4

\newline

(C) At a horizontal distance of

4

meters from Kaori, the ball's height was equal to

R

C=16,500+130m

\newline

The total cost,

C

, in dollars, to purchase a planchet machine and use it for

m

months is given by the equation. What is the best interpretation of

130

as shown in the given equation?

\newline

1

\newline

(A) The cost increases by

130

dollars for each additional month of use.

\newline

(B) The cost to purchase the machine and not use it is

130

\newline

(C) After purchase, the machine can be used for up to

130

\newline

(D) The average cost per month to purchase and use the machine is

130

Four players are playing a game together. They each gain or lose points based on their actions.

\newline

50

points for finding a green mushroom.

\newline

100

points for completing a level.

\newline

100

points for getting hit by a fireball.

\newline

75

points for stepping on a spike.

\newline

Which two friends had the same amount of change to their scores?

\newline

1

\newline

(A) Amani and Ajeet

\newline

(B) Damian and Julius

\newline

(C) Amani and Julius

\newline

(D) Ajeet and Damian

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

C=290+155h

. 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.

s=-11.8(t-1.2)^{2}+17

\newline

The equation models the horizontal distance,

s

, in centimeters, of a particular image on a computer screen from the left-hand edge of the screen,

t

seconds after appearing. If the image moves in from the left-hand edge of the screen during an on-screen animation, how far to the right does the image travel?

\newline

1

\newline

1.2

\newline

5.2

\newline

17

\newline

34

p(m)=100\left(\frac{1}{2}\right)^{\frac{m}{27}}

\newline

-199

is a radioactive isotope that decays over time. The half-life of a radioactive isotope is the amount of time it takes for

\frac{1}{2}

of the isotopes in a sample to decay. The function models the percent of bismuth

-199

remaining in a sample after

m

minutes. What is the meaning of the number

27

in the function?

\newline

1

\newline

(A) The amount of bismuth-

199

in the sample decreases by

27 \%

\frac{1}{2}

\newline

(B) The sample is initially composed of

27 \%

199

\newline

(C) The half-life of bismuth

-199

13

5

\newline

(D) The half-life of bismuth

-199

27

Carbon dating involves the measurement of the amount of carbon

-14

in a sample. Archaeologists sometimes use carbon dating to estimate the number of years since an organism died. The function shows the ratio,

R(t)

-14

remaining in a sample to the original amount of carbon

-14

t

\newline

R(t)=2^{-0.00175 t}

\newline

If one sample was

3

times as old as another sample of the same material, how would the carbon

-14

ratios of the samples relate?

\newline

1

\newline

(A) The ratio of the older sample would be

3

times the ratio of the newer sample.

\newline

(B) The ratio of the newer sample would be

3

times the ratio of the older sample.

\newline

(C) The ratio of the older sample would be the cube of the ratio of the newer sample.

\newline

(D) The ratio of the newer sample would be the cube of the ratio of the older sample.

Josiah is cooking a piece of lamb. Before he puts it in the oven, the internal temperature of the lamb is

41^\circ C

below the final temperature it needs to reach. The outside of the lamb heats up faster than the inside. Josiah takes the lamb out of the oven when the internal temperature is still

6^\circ C

below the final temperature it needs to reach, so the inside and outside can balance out to the right final temperature.

\newline

Josiah wonders how much the internal temperature had changed when he removed the lamb from the oven.

\newline

Which of the following equations matches the situation above?

\newline

1

\newline

41+6=?

\newline

-41+?=-6

\newline

41+?=6

A forest ranger travels up a river by boat to collect soil samples. The boat travels at a speed of

v

miles per hour for a distance of

2

miles. After collecting the samples, the ranger travels the same distance back, but the boat travels

5

miles per hour slower than it did on the way up the river. If the ranger spent

3

hours travelling up and back, which of the following equations could be used to determine the speed of the boat on the way up the river?

\newline

1

\newline

2v^{2}-13v+17=0

\newline

3v^{2}-15v-4=0

\newline

3v^{2}-11v+10=0

\newline

3v^{2}-19v+10=0

The equation models

h

, the height, in meters, of a soccer ball

t

seconds after the goalkeeper kicks it. Which of the following statements is the best interpretation of the ordered pair

(3,10.4)

\newline

1

\newline

3

seconds after the kick, the distance between the soccer ball and the goalkeeper is

10.4

\newline

3

seconds after the kick, the soccer ball is traveling at a speed of

10.4

meters per second.

\newline

3

seconds after the kick, the height of the soccer ball is

10.4

\newline

10.4

seconds after the kick, the height of the soccer ball is

3

11(1.022)^{t}

models the per capita gross domestic product (GDP) of the US, in thousands of dollars, as a function of the number of years since

1950

\newline

1

022

represent in this expression?

\newline

1

\newline

(A) The per capita GDP in the US was

\$ 1022

1950

\newline

(B) The maximum per capita GDP occurred

1

022

1950

\newline

(C) The per capita GDP in the US is multiplied by approximately

1

022

The value of a computer is decreasing at a rate that is proportional at any time to the value of the computer at that time.

\newline

The computer is worth

\$ 850

initially, and it is worth

\$ 306

2

\newline

How much will the computer be worth after

5

\newline

1

\newline

\$ 0

\newline

\$ 66

\newline

\$ 79

b(t)

gives the temperature (in degrees Celsius) of a pan of brownies by time

t

\newline

\int_{0}^{6} b^{\prime}(t) d t=93

\newline

1

\newline

t=6

minutes, the temperature of the brownies is

93

degrees Celsius.

\newline

t=6

minutes, the brownies have an instantaneous rate of temperature change of

93

degrees Celsius per minute.

\newline

(C) During the first six minutes, the temperature of the brownies increases an average of

93

degrees Celsius per minute.

\newline

(D) The temperature of the brownies increased by

93

degrees Celsius in the first six minutes.

The number of subscribers to a magazine is changing at a rate of

r(t)

subscribers per month (where

t

is time in months).

\newline

\int_{8}^{10} r^{\prime}(t) d t=7

\newline

1

\newline

(A) As of month

10

, the magazine had

7

\newline

(B) The average rate of change in subscribers between month

8

10

7

subscribers per month.

\newline

(C) The number of subscribers increased by

7

t=8

t=10

\newline

(D) The rate of change of number of subscribers increased by

7

subscribers per month between

t=8

t=10

The number of people who have adopted a new fashion trend is increasing at a rate of

r(t)

people per month (where

t

is the time in months).

\newline

\int_{5}^{6} r(t) d t

\newline

1

\newline

(A) The growth in the number of people to adopt the fashion trend during the sixth month

\newline

(B) The time it took to change from

5

6

people who had adopted the fashion trend

\newline

(C) The instantaneous rate of change of the number of people to adopt the fashion trend when

t=6

\newline

(D) The total number of people who have adopted the fashion trend by

t=6

The amount of money in Jim's bank account changes at a rate of

r(t)

dollars per month (where

t

is time in months).

\newline

\int_{0}^{1} r(t) d t=-100

\newline

1

\newline

(A) The amount of money in Jim's account increases by

\$ 100

\newline

(B) Over the first month, the amount of money in Jim's account decreased by

\$ 100

\newline

(C) The amount of money in Jim's account decreases by

\$ 100

\newline

(D) After one month, the amount of money in Jim's account was

-\$ 100

A grocery store receives a shipment of bananas. The percent of the shipment that is still suitable for sale is decreasing at a rate of

r(t)

percent of the original shipment per day (where

t

is the time in days). At

t=2

, the grocery store found

85 \%

of the shipment to be suitable for sale.

\newline

0.85+\int_{2}^{4} r(t) d t=0.6 \text { mean? }

\newline

1

\newline

(A) As of the fourth day,

60 \%

of the bananas suitable on the second day are still suitable.

\newline

(B) Between the second and fourth days,

60 \%

of the remaining bananas became unsuitable for sale per day.

\newline

(C) Between the second and fourth day, another

60 \%

of the bananas became unsuitable for sale.

\newline

(D) As of the fourth day,

60 \%

of the shipment was still suitable for sale.

The number of people exposed to a certain illness is increasing at a rate of

r(t)

people per day (where

t

is the time in days). At

t=12

115

people exposed to the illness.

\newline

115+\int_{12}^{20} r(t) d t

\newline

1

\newline

(A) The change in the number of people exposed to the illness between days

12

20

\newline

(B) The change in the rate at which people are exposed to the illness between days

12

20

\newline

(C) The time it takes for

20

more people to be exposed to the illness

\newline

(D) The number of people exposed to the illness at

t=20

The population of a town grows at a rate of

r(t)

people per year (where

t

is time in years).

\newline

\int_{2}^{4} r(t) d t

\newline

1

\newline

(A) The average rate at which the population grew between the second and the fourth year.

\newline

(B) The change in number of people between the second and the fourth year.

\newline

(C) The number of people in the town on the fourth year.

\newline

(D) The time it took for the town to grow from a population of

2

people to a population of

4

The population of a town grows at a rate of

r(t)

people per year (where

t

is time in years). At

t=3

, the town's population was

1000

\newline

1000+\int_{3}^{8} r(t) d t=1500

\newline

1

\newline

(A) The town's population grew by

1500

t=3

t=8

\newline

(B) The town's population grew at a rate of

1500

people per year at

t=8

\newline

(C) The average rate at which the population grew between

t=3

t=8

1500

people per year.

\newline

t=8

, the town's population was

1500

The approval rating of a certain politician is changing at a rate of

r(t)

percent of the population per month (where

t

is the time in months).

\newline

\int_{0}^{3} r(t) d t

\newline

1

\newline

(A) The rate of change of the approval rating of the politician when

t=3

\newline

(B) The average rate of change of the approval rating during the first three months

\newline

(C) The change in the percent of the population who approves of the politician during the first three months

\newline

(D) The percent of the population who approves of the politician when

t=3

b(t)

gives the temperature (in degrees Celsius) of a pan of brownies by time

t

\newline

\int_{0}^{6} b^{\prime}(t) d t=93

\newline

1

\newline

t=6

minutes, the temperature of the brownies is

93

degrees Celsius.

\newline

(B) The temperature of the brownies increased by

93

degrees Celsius in the first six minutes.

\newline

(C) During the first six minutes, the temperature of the brownies increases an average of

93

degrees Celsius per minute.

\newline

t=6

minutes, the brownies have an instantaneous rate of temperature change of

93

degrees Celsius per minute.

The approval rating of a certain politician is changing at a rate of

r(t)

percent of the population per month (where

t

is the time in months).

\newline

\int_{0}^{3} r(t) d t

\newline

1

\newline

(A) The rate of change of the approval rating of the politician when

t=3

\newline

(B) The average rate of change of the approval rating during the first three months

\newline

(C) The percent of the population who approves of the politician when

t=3

\newline

(D) The change in the percent of the population who approves of the politician during the first three months

...