Interpreting Linear Relationships in Context Practice Questions — SAT — Page 4

Question 61Question →

A team of glaciologists monitors the thickness of an alpine glacier during the summer season. The thickness of the glacier, $T$ , in meters, can be modeled by the equation $T = 54.5 - 0.12d$ , where $d$ represents the number of days since the start of the summer season, for $0 \leq d \leq 150$ . Which of the following is the best interpretation of the number $0.12$ in this context?

The initial thickness of the glacier, in meters, at the start of the summer season

The number of days it takes for the thickness of the glacier to decrease by

1

meter

The decrease in the thickness of the glacier, in meters, each day during the summer season

The total number of days from the start of the summer season until the glacier melts completely

Show answer & explanation

Answer: The decrease in the thickness of the glacier, in meters, each day during the summer season

Answer

The correct answer is the option stating that 0.12 is the decrease in the thickness of the glacier, in meters, each day during the summer season.

The linear model is given by

T = 54.5 - 0.12d

, which is in the slope-intercept form

y = mx + b

, where

m

is the slope and

b

is the y-intercept. In this equation, the slope

m

is

-0.12

, which represents the rate of change of the glacier's thickness per day. Since the slope is negative, the thickness decreases by

0.12

meters each day. Therefore, the number

0.12

represents the decrease in the thickness of the glacier, in meters, each day during the summer season.

Step-by-Step Solution

1

Identify the components of the linear equation

T = 54.5 - 0.12d

.

The constant term is

54.5

, and the coefficient of the variable

d

is

-0.12

.

To interpret a linear relationship in context, we must distinguish between the initial value (y-intercept) and the rate of change (slope).

2

Determine the meaning of the slope in this context.

The slope is

-0.12

meters per day, meaning the glacier's thickness

T

decreases by

0.12

meters for each day

d

that passes.

The coefficient of the independent variable in a linear equation represents the rate of change of the dependent variable per unit of the independent variable.

3

Relate the number

0.12

to the rate of change.

The positive value

0.12

represents the magnitude of this rate of change, which is the amount of decrease in thickness per day.

Since the slope is negative, the change is a decrease, and the rate of this decrease is

0.12

meters per day.

Key Concept

Interpreting the slope of a linear relationship in a real-world context

Question 62Question →

A commercial cargo ship is unloading shipping containers at a port. The total mass of the ship and its remaining cargo, $M$ , in kilotonnes (kt), is a linear function of the number of hours, $h$ , since the unloading process began. After $3$ hours of unloading, the total mass of the ship and its cargo is $116$ kt. After $8$ hours of unloading, the total mass is $98$ kt. According to this model, what is the mass, in kilotonnes, of the cargo that is unloaded each hour?

Show answer & explanation

Answer: 3.6

Answer

The mass of the cargo unloaded each hour is

3.6

kilotonnes (or the equivalent fraction

\frac{18}{5}

).

The correct answer is

3.6

. Since the relationship between the total mass of the ship and cargo,

M

, and the elapsed time,

h

, is linear, the rate at which cargo is unloaded corresponds to the magnitude of the slope of the linear function. Using the two data points

(3, 116)

and

(8, 98)

, the slope

m

can be calculated as

\frac{98 - 116}{8 - 3} = \frac{-18}{5} = -3.6

. The negative sign indicates that the mass is decreasing, meaning that

3.6

kilotonnes of cargo are unloaded each hour.

Step-by-Step Solution

1

Set up the linear relationship model.

M = mh + b

Since the relationship between total mass

M

and time

h

is linear, it can be modeled by a linear equation where

m

is the slope (unloading rate) and

b

is the y-intercept (initial mass).

2

Determine the two data points from the given information.

(3, 116)

and

(8, 98)

After

3

hours of unloading, the mass is

116

kt, and after

8

hours, the mass is

98

kt.

3

Calculate the slope (rate of change) of the linear relationship.

m = -3.6

The slope is calculated as

m = \frac{98 - 116}{8 - 3} = \frac{-18}{5} = -3.6

.

4

Interpret the slope's value in context to find the amount of cargo unloaded per hour.

3.6

The slope is

-3.6

kilotonnes per hour, which means the total mass decreases by

3.6

kilotonnes each hour. Therefore, the mass of the cargo unloaded each hour is

3.6

kilotonnes.

Key Concept

Interpreting the slope of a linear relationship in context

Estimated Time:1m 30s

Question 63Question →

A commercial coffee roaster heats a large batch of coffee beans. The temperature $T$ , in degrees Celsius ( $^\circ\text{C}$ ), of the beans $m$ minutes after heating begins is modeled by the equation $T = 18m + 31$ , where $m \le 10$ . Which of the following is the best interpretation of $18$ in this context?

The increase in the temperature of the beans, in degrees Celsius, for each minute of heating.

The initial temperature of the beans, in degrees Celsius, when heating begins.

The number of minutes it takes for the temperature of the beans to increase by

1^\circ\text{C}

.

The temperature of the beans, in degrees Celsius, after

18

minutes of heating.

Show answer & explanation

Answer: The increase in the temperature of the beans, in degrees Celsius, for each minute of heating.

Answer

The increase in the temperature of the beans, in degrees Celsius, for each minute of heating.

The equation is in the slope-intercept form

T = 18m + 31

, where

18

is the slope and

31

is the y-intercept. In this context,

T

represents the temperature of the beans and

m

represents the time in minutes. The slope,

18

, represents the rate of change of the temperature with respect to time, which means the temperature increases by

18^\circ\text{C}

for each minute of heating.

Step-by-Step Solution

1

Identify the component of the linear equation being questioned.

The number

18

is the coefficient of the variable

m

in the linear equation

T = 18m + 31

.

We need to determine what role the coefficient of the independent variable plays in a linear model.

2

Determine the mathematical meaning of the coefficient in a linear equation of the form

y = mx + b

.

The coefficient of

m

, which is

18

, represents the slope or the rate of change of

T

with respect to

m

.

In a linear function, the slope represents the change in the dependent variable per unit change in the independent variable.

3

Interpret the rate of change in the context of the problem.

Since

T

is the temperature in degrees Celsius (

^\circ\text{C}

) and

m

is the time in minutes, a rate of change of

18

means that the temperature increases by

18^\circ\text{C}

for every

1

minute of heating.

This matches the definition of the slope in the given context.

Key Concept

Interpreting the slope of a linear relationship in context

Estimated Time:1m 0s

Question 64Question →

A scientist measures the density of a core sample of ice as a function of depth. The density $d$ , in grams per cubic centimeter ( $\text{g/cm}^3$ ), of the ice at a depth of $x$ meters below the glacier surface is modeled by the equation $d = 0.0004x + 0.917$ . According to the model, what is the depth, in meters, at which the ice density is $0.935\text{ g/cm}^3$ ?

Show answer & explanation

Answer: 45

Answer

The depth is 45 meters.

To find the depth at which the density is

0.935\text{ g/cm}^3

, substitute

0.935

for the density

d

in the model equation, yielding

0.935 = 0.0004x + 0.917

. Subtracting

0.917

from both sides gives

0.018 = 0.0004x

. Dividing both sides by

0.0004

results in

x = 45

. Therefore, the depth is 45 meters.

Step-by-Step Solution

1

Substitute the target density of

0.935\text{ g/cm}^3

for

d

in the given model equation.

0.935 = 0.0004x + 0.917

This sets up the linear equation to solve for the corresponding depth

x

.

2

Subtract

0.917

from both sides of the equation.

0.018 = 0.0004x

This isolates the term containing the variable

x

on one side of the equation.

3

Divide both sides of the equation by

0.0004

to find the value of

x

.

x = 45

Dividing

0.018

by

0.0004

solves for the depth

x

in meters.

Key Concept

Solving linear relationships in context for the independent variable given a value of the dependent variable.

Question 65Question →

A community library is moving its collection of books to a new building. The number of books, $B$ , remaining to be packed $h$ hours after the packing begins is modeled by the linear equation $B = 12,500 - 450h$ , where $0 \leq h \leq 25$ . Which of the following is the best interpretation of $450$ in this context?

The number of books packed each hour

The initial number of books that need to be packed

The number of hours it takes to pack each book, which is

\frac{1}{450}

hour

The total number of hours it takes to pack all the books

Show answer & explanation

Answer: The number of books packed each hour

Answer

The number of books packed each hour

In the linear equation

B = 12,500 - 450h

, the coefficient of

h

is

-450

. This value represents the rate of change of the dependent variable,

B

(remaining books), with respect to the independent variable,

h

(hours). A slope of

-450

means that for every hour that passes, the number of remaining books decreases by

450

, which means

450

books are packed each hour.

Step-by-Step Solution

1

Identify the variable and constants in the given linear equation

B = 12,500 - 450h

.

B

represents the number of remaining books,

h

represents the number of hours spent packing,

12,500

is the constant term (

y

-intercept), and

-450

is the coefficient of

h

(slope).

Understanding the components of the linear model is essential to interpreting their real-world meanings.

2

Analyze the rate of change (slope) of the equation.

The slope is

-450

, which indicates that the number of remaining books decreases by

450

for every

1

hour that passes.

The coefficient of the independent variable in a linear equation in context represents the rate of change of the dependent variable with respect to the independent variable.

3

Translate the rate of change of the remaining books into the rate of packing.

A decrease of

450

remaining books per hour is equivalent to packing

450

books per hour.

Since books are being packed to be moved, a reduction in the remaining unpacked books corresponds directly to the number of books packed.

Key Concept

Interpreting the slope (rate of change) of a linear equation in a real-world context.

Estimated Time:1m 30s

Question 66Question →

An environmentalist studies the sound level of a waterfall. The sound level, $S$ , in decibels ( $\text{dB}$ ), at a distance of $d$ meters from the base of the waterfall is modeled by the equation $S = 92 - 0.15d$ , where $0 \leq d \leq 200$ . What is the best interpretation of the $S$ -intercept of the graph of this equation in the $dS$ -plane?

The sound level of the waterfall decreases by

0.15\text{ decibels}

for each meter of distance from its base.

The sound level of the waterfall is

91.85\text{ decibels}

at its base.

The sound level of the waterfall is

92\text{ decibels}

at its base.

The distance from the base where the sound level is

0\text{ decibels}

is

92\text{ meters}

.

Show answer & explanation

Answer: The sound level of the waterfall is $92\text{ decibels}$ at its base.

Answer

The sound level of the waterfall is

92\text{ decibels}

at its base.

The

S

-intercept of the graph in the

dS

-plane occurs where the horizontal variable,

d

, is equal to

0

. Substituting

d = 0

into the equation yields

S = 92 - 0.15(0) = 92

. Since

d

represents the distance from the base of the waterfall in meters and

S

represents the sound level in decibels, this means that at a distance of

0

meters (the base of the waterfall), the sound level is

92\text{ decibels}

.

Step-by-Step Solution

1

Identify the definition of the

S

-intercept in the context of the

dS

-plane.

The

S

-intercept is the point on the graph where the independent variable,

d

(distance), is equal to

0

.

By definition, the vertical intercept of a graph occurs where the horizontal coordinate is zero.

2

Substitute

d = 0

into the linear equation.

S = 92 - 0.15(0) = 92

.

Evaluating the equation at

d = 0

determines the value of the sound level at zero meters from the base.

3

Interpret the mathematical result in the real-world context.

At a distance of

0

meters (which is the base of the waterfall), the sound level is

92\text{ decibels}

.

This connects the coordinate point

(0, 92)

to the physical properties of the scenario.

Key Concept

Interpreting the vertical intercept of a linear relationship in context

Estimated Time:1m 30s

Question 67Question →

A botanist is studying the transpiration rate of a certain plant species under controlled conditions. The transpiration rate $T$ , in grams of water vapor per square meter of leaf area per hour ( $\text{g/m}^2/\text{h}$ ), is modeled by a linear function of the relative humidity $H$ , expressed as a decimal where $0.20 \leq H \leq 0.80$ . The model is represented by the equation $T = -4.8H + C$ , where $C$ is a constant. Based on this model, if the relative humidity increases by $0.15$ , what is the corresponding decrease in the transpiration rate, in $\text{g/m}^2/\text{h}$ ?

Show answer & explanation

Answer: 0.72

Answer

The correct answer is 0.72.

In the linear equation

T = -4.8H + C

, the coefficient of

H

(which is

-4.8

) represents the slope of the line. The slope indicates that for every increase of

1.0

in the relative humidity

H

, the transpiration rate

T

decreases by

4.8

grams of water vapor per square meter of leaf area per hour. To find the decrease in transpiration rate corresponding to an increase of

0.15

in relative humidity, multiply the rate of change by the change in relative humidity:

4.8 \times 0.15 = 0.72

. Thus, the transpiration rate decreases by

0.72

grams of water vapor per square meter of leaf area per hour.

Step-by-Step Solution

1

Identify the slope of the linear model.

The slope is

-4.8

.

The coefficient of

H

in the linear equation

T = -4.8H + C

represents the rate of change of the transpiration rate with respect to the relative humidity.

2

Calculate the change in the transpiration rate (

\Delta T

) for an increase of

0.15

in the relative humidity (

\Delta H = 0.15

).

\Delta T = -4.8 \times 0.15 = -0.72

The change in the dependent variable is equal to the slope multiplied by the change in the independent variable.

3

Determine the magnitude of the decrease.

The decrease in the transpiration rate is

0.72

.

The negative sign in the change

\Delta T = -0.72

represents a decrease, so the amount of decrease is

0.72

.

Key Concept

Interpreting the slope of a linear relationship in context as the rate of change of the dependent variable with respect to the independent variable.

Question 68Question →

A manufacturing plant uses a heating chamber for curing composite materials. The temperature $T$ , in degrees Celsius ( $^\circ\text{C}$ ), of the chamber $m$ minutes after the heating element is turned on is modeled by the equation $T = 3.5m + 22$ , where $0 \leq m \leq 60$ . The chamber must reach a target temperature of $180^\circ\text{C}$ for the curing process. Which of the following is the best interpretation of the value $\frac{180 - 22}{3.5}$ in this context?

The rate, in degrees Celsius per minute, at which the temperature of the chamber increases

The temperature, in degrees Celsius, of the heating chamber after

180

minutes

The number of minutes it takes for the heating chamber to reach the target temperature of

180^\circ\text{C}

The number of minutes it takes for the temperature of the chamber to increase by

180^\circ\text{C}

Show answer & explanation

Answer: The number of minutes it takes for the heating chamber to reach the target temperature of $180^\circ\text{C}$

Answer

The number of minutes it takes for the heating chamber to reach the target temperature of

180^\circ\text{C}

The correct answer represents the time needed to reach the target temperature. To find when the temperature

T

reaches

180^\circ\text{C}

, we substitute

180

for

T

in the equation

T = 3.5m + 22

, giving

180 = 3.5m + 22

. Solving for time

m

requires subtracting the initial temperature of

22^\circ\text{C}

from the target temperature of

180^\circ\text{C}

, and then dividing by the heating rate of

3.5^\circ\text{C}

per minute. This algebraic isolation results in

m = \frac{180 - 22}{3.5}

, representing the duration in minutes to reach that target.

Step-by-Step Solution

1

Identify the meaning of each term in the linear model.

In the equation

T = 3.5m + 22

,

T

represents the chamber's temperature,

m

represents time in minutes,

3.5

represents the rate of temperature increase per minute, and

22

represents the initial temperature of the chamber.

Understanding the components of the linear equation helps contextualize the mathematical operations performed in the expression.

2

Set up the equation using the target temperature.

The target temperature is

180^\circ\text{C}

. Setting

T = 180

gives the linear equation

180 = 3.5m + 22

.

This establishes the relationship between the target temperature and the time

m

required to reach it.

3

Solve for the variable

m

representing time.

Subtract

22

from both sides to get

180 - 22 = 3.5m

, then divide both sides by

3.5

to isolate

m

, yielding

m = \frac{180 - 22}{3.5}

.

Isolating

m

shows that the expression is mathematically equivalent to the time, in minutes, at which the temperature reaches

180^\circ\text{C}

.

Key Concept

Interpreting expressions derived from linear relationships in real-world contexts

Alternative Method

Use dimensional analysis to verify the units of the expression. The numerator

(180 - 22)

represents the difference between two temperatures in degrees Celsius (

^\circ\text{C}

), which yields a temperature change in

^\circ\text{C}

. The denominator,

3.5

, represents the rate of change in degrees Celsius per minute (

^\circ\text{C}/\text{min}

). Dividing

^\circ\text{C}

by

^\circ\text{C}/\text{min}

results in minutes:

\frac{^\circ\text{C}}{^\circ\text{C}/\text{min}} = \text{min}

. This confirms the expression represents a time duration.

Estimated Time:1m 30s

Question 69Question →

An agricultural irrigation system draws water from a storage tank at a constant rate. The volume of water, $V$ , in gallons, remaining in the tank $m$ minutes after the irrigation begins is given by the equation $V = 12,000 - 45m$ . According to the model, after how many minutes of irrigation will there be exactly $8,400$ gallons of water remaining in the tank?

Show answer & explanation

Answer: 80

Answer

The irrigation system will have exactly

8,400

gallons of water remaining after

80

minutes.

To find the number of minutes,

m

, when the volume of water remaining is exactly

8,400

gallons, substitute

8,400

for

V

in the given linear model:

8,400 = 12,000 - 45m

. Subtracting

12,000

from both sides of the equation yields

-3,600 = -45m

. Dividing both sides of the equation by

-45

yields

m = 80

. Therefore, after

80

minutes, there will be exactly

8,400

gallons of water remaining in the tank.

Step-by-Step Solution

1

Substitute the target volume of

8,400

gallons for

V

in the linear model equation.

8,400 = 12,000 - 45m

We want to find the number of minutes,

m

, when the remaining volume,

V

, is exactly

8,400

gallons.

2

Subtract

12,000

from both sides of the equation.

-3,600 = -45m

This isolates the variable term

-45m

on one side of the equation.

3

Divide both sides by

-45

to solve for

m

.

m = 80

This isolates

m

to find the number of minutes elapsed.

Key Concept

Solving linear equations in context

Question 70Question →

A commercial drone descends from a mapping mission at a constant rate. The altitude of the drone, $A$ , in meters, $t$ seconds after it begins its descent is modeled by the equation $A = -4.5t + 360$ . The graph of this equation in the $tA$ -plane is shown below.

Which of the following is the best interpretation of the $t$ -intercept of the graph in this context?

The number of seconds it takes for the drone to reach sea level.

The altitude, in meters, of the drone when it begins its descent.

The rate, in meters per second, at which the drone descends.

The number of seconds it takes for the drone to descend 1 meter.

Show answer & explanation

Answer: The number of seconds it takes for the drone to reach sea level.

Answer

The

t

-intercept represents the number of seconds it takes for the drone to reach sea level.

The correct answer is the option stating that the intercept represents the number of seconds it takes for the drone to reach sea level. The

t

-intercept is the point on the graph where

A = 0

. In this context,

A = 0

means the drone's altitude is

0

meters (sea level), and

t

represents the time elapsed in seconds. Setting

A = 0

in the equation

A = -4.5t + 360

gives

t = 80

, meaning it takes

80

seconds for the drone to reach sea level.

Step-by-Step Solution

1

Identify the meaning of the

t

-intercept on the graph.

The

t

-intercept is the point on the graph where the vertical coordinate,

A

, is equal to

0

.

By definition, the horizontal intercept of a function occurs when the dependent variable (altitude,

A

) is zero.

2

Relate the condition

A = 0

to the context.

An altitude of

A = 0

meters corresponds to sea level.

The variable

A

represents altitude in meters, so

A = 0

represents the level where altitude is zero, which is sea level.

3

Interpret the corresponding value of

t

.

The

t

-coordinate represents the time, in seconds, when the altitude is 0.

Since

t

represents the number of seconds after the descent began, the

t

-intercept

(80, 0)

indicates that it takes

80

seconds for the drone to reach sea level.

Key Concept

Interpreting the intercepts of a linear relationship in context.

Interpreting Linear Relationships in Context

Answer

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