Interpreting Linear Relationships in Context Alıştırma Soruları — SAT — Sayfa 3

Soru 41Soru →

A municipal recycling facility processes plastic waste. The total mass of unprocessed plastic, $M$ , in tons, remaining at the facility $h$ hours after the facility opens on a given day can be modeled by the equation $M = 142 - 8.5h$ . Which of the following is the best interpretation of the number $8.5$ in this context?

The facility has

142

tons of unprocessed plastic when it opens.

The mass of unprocessed plastic at the facility decreases by

8.5

tons each hour.

The facility takes

8.5

hours to process each ton of plastic.

The mass of unprocessed plastic at the facility decreases by

133.5

tons each hour.

Cevabı ve açıklamayı göster

Cevap: The mass of unprocessed plastic at the facility decreases by $8.5$ tons each hour.

Cevap

The mass of unprocessed plastic at the facility decreases by

8.5

tons each hour.

In the equation

M = 142 - 8.5h

, the total mass of unprocessed plastic

M

decreases by

8.5

tons for each hour

h

that passes. The number

8.5

is the magnitude of the slope of the linear relationship, which represents the rate of change. Since the coefficient of

h

is negative, the amount of unprocessed plastic decreases by

8.5

tons per hour.

Adım Adım Çözüm

1

Identify the structure of the linear equation

M = 142 - 8.5h

.

The equation is in the form

y = b + mx

, where

b

is the y-intercept (

142

) and

m

is the slope (

-8.5

).

This helps separate the initial value from the rate of change.

2

Analyze the coefficient of the independent variable

h

.

The coefficient is

-8.5

.

The coefficient of

h

represents the rate of change of the mass

M

per hour.

3

Interpret the meaning of the rate of change in context.

A rate of change of

-8.5

means that the mass of unprocessed plastic decreases by

8.5

tons for each hour that passes.

The negative sign indicates a decrease, and the unit is tons per hour.

Anahtar Kavram

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

Soru 42Soru →

A researcher uses the linear equation $T = 18.5 + 4.2d$ to estimate the temperature, $T$ , in degrees Celsius, of the Earth's crust at a depth of $d$ kilometers below the surface in a certain region. What is the estimated increase in temperature, in degrees Celsius, for each increase of 5 kilometers in depth?

Cevabı ve açıklamayı göster

Cevap: 21

Cevap

21

The coefficient of

d

in the equation is

4.2

, representing a temperature increase of

4.2

degrees Celsius for every 1 kilometer increase in depth. To find the temperature increase for an increase of 5 kilometers in depth, multiply this rate of change by 5:

4.2 \times 5 = 21

.

Adım Adım Çözüm

1

Identify the slope of the linear equation.

The slope is

4.2

.

In the equation

T = 18.5 + 4.2d

, the term

4.2d

indicates that for every 1 kilometer increase in depth (

d

), the temperature (

T

) increases by

4.2

degrees Celsius.

2

Calculate the total temperature increase for a 5-kilometer depth increase.

The increase is

21

degrees Celsius.

Since the rate of temperature increase is

4.2

degrees Celsius per kilometer, a depth increase of

5

kilometers results in a temperature increase of

4.2 \times 5 = 21

degrees Celsius.

Anahtar Kavram

Interpreting the slope of a linear equation in context as a constant rate of change.

Tahmini Süre:1m 30s

Soru 43Soru →

A software developer uses the equation $P = 1500 + 45n$ to calculate the total price $P$ , in dollars, charged to a client for a project, where $n$ represents the number of hours spent debugging the software. Which of the following is the best interpretation of the number $45$ in this context?

The increase in the total price, in dollars, for each additional hour spent debugging the software

The initial price, in dollars, charged for the project before any hours are spent debugging

The number of additional hours spent debugging for each 1-dollar increase in the total price

The total price, in dollars, charged for a project that requires exactly 1 hour of debugging

Cevabı ve açıklamayı göster

Cevap: The increase in the total price, in dollars, for each additional hour spent debugging the software

Cevap

The increase in the total price, in dollars, for each additional hour spent debugging the software

The coefficient of

n

in the equation

P = 1500 + 45n

is

45

, which represents the slope of the line. The slope measures the rate of change of the dependent variable,

P

(total price in dollars), per unit change in the independent variable,

n

(hours spent debugging). Thus,

45

represents an increase of 45 dollars in the total price for each additional hour spent debugging.

Adım Adım Çözüm

1

Identify the structure of the linear equation.

The equation

P = 1500 + 45n

is in the slope-intercept form

y = mx + b

, where

P

is the dependent variable (equivalent to

y

),

n

is the independent variable (equivalent to

x

), the coefficient

45

is the slope

m

, and the constant

1500

is the y-intercept

b

.

Understanding the components of a linear equation helps map them to their real-world meanings.

2

Interpret the meaning of the slope in context.

The slope

m = 45

represents the rate of change of the dependent variable

P

(total price in dollars) with respect to the independent variable

n

(hours spent debugging). This means that for each unit increase in

n

(1 additional hour of debugging),

P

increases by

45

units (45 dollars).

The coefficient of the independent variable in a linear model represents the unit rate of change.

Anahtar Kavram

Interpreting the slope of a linear model in context

Soru 44Soru →

A digital printing press uses paper from a large roll. The total weight $W$ , in pounds, of the paper remaining on the roll after $m$ minutes of continuous printing is given by the equation $W = 120 - 1.5m$ . According to the equation, after how many minutes of continuous printing will the weight of the remaining paper on the roll be exactly $45$ pounds?

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

The correct answer is 50.

To find the number of minutes,

m

, when the remaining weight is

45

pounds, substitute

W = 45

into the given equation

W = 120 - 1.5m

. This yields the equation

45 = 120 - 1.5m

. Subtracting

120

from both sides gives

-75 = -1.5m

. Dividing both sides by

-1.5

results in

m = 50

. Therefore, after

50

minutes, the remaining weight of the paper on the roll is

45

pounds.

Adım Adım Çözüm

1

Substitute the target weight into the linear equation.

45 = 120 - 1.5m

We are given that the weight of the remaining paper,

W

, is

45

pounds, and we need to solve for the time in minutes,

m

.

2

Isolate the variable term by subtracting

120

from both sides of the equation.

-75 = -1.5m

To solve for

m

, we first need to isolate the term containing the variable on one side of the equation.

3

Divide both sides by

-1.5

to find the value of

m

.

m = 50

Dividing by the coefficient of the variable completes the isolation and gives the final value.

Anahtar Kavram

Interpreting values and solving linear equations in context

Alternatif Yöntem

Alternatively, you can determine the total weight of paper consumed:

120 - 45 = 75

pounds. Since the paper is used at a rate of

1.5

pounds per minute, the time required is

\frac{75}{1.5} = 50

minutes.

Tahmini Süre:1m 30s

Soru 45Soru →

A company uses a fleet of refrigerated delivery trucks to transport fresh produce. The temperature inside a truck's cooling compartment, $T$ , in degrees Fahrenheit ( $^\circ\text{F}$ ), $t$ minutes after the cooling system is turned on is modeled by the equation below:

T = 68 - 1.25t

Where $0 \leq t \leq 24$ . Which of the following is the best interpretation of the number $68$ in this context?

The temperature inside the compartment, in degrees Fahrenheit, when the cooling system is first turned on

The rate, in degrees Fahrenheit per minute, at which the temperature inside the compartment decreases

The temperature inside the compartment, in degrees Fahrenheit, 1 minute after the cooling system is turned on, which is

67.25^\circ\text{F}

The number of minutes it takes for the temperature inside the compartment to reach

0^\circ\text{F}

Cevabı ve açıklamayı göster

Cevap: The temperature inside the compartment, in degrees Fahrenheit, when the cooling system is first turned on

Cevap

The temperature inside the compartment, in degrees Fahrenheit, when the cooling system is first turned on

The correct answer correctly identifies the meaning of the constant term

68

as the y-intercept of the linear equation. In this context, the y-intercept represents the value of

T

when the time

t

is

0

, which corresponds to the initial temperature inside the compartment before any cooling occurs.

Adım Adım Çözüm

1

Identify the form of the linear equation and its components.

The equation is in the slope-intercept form,

T = mt + b

, where

m = -1.25

is the slope and

b = 68

is the

T

-intercept (or vertical intercept).

Recognizing the mathematical meaning of each coefficient is the first step in interpreting them in a real-world context.

2

Determine the physical meaning of the vertical intercept (

b = 68

).

The vertical intercept occurs when the independent variable,

t

(time in minutes), is equal to

0

. Evaluating

T

at

t = 0

gives

T(0) = 68 - 1.25(0) = 68

.

Setting the independent variable to zero allows us to define the starting state of the system.

3

Interpret this starting state in the context of the problem.

At

t = 0

, the cooling system has just been turned on. Thus,

68^\circ\text{F}

represents the temperature inside the truck's cooling compartment when the system is first turned on.

Connecting the value of

T(0)

back to the scenario provides the correct interpretation of the constant

68

.

Anahtar Kavram

Interpreting the y-intercept of a linear model in context

Tahmini Süre:1m 30s

Soru 46Soru →

Due to rising global temperatures, the thickness $T$ , in meters, of a mountain glacier is decreasing at a constant rate. A researcher models the thickness of the glacier using the equation $T = 48.5 - 0.12w$ , where $w$ represents the number of weeks since the start of a monitoring study. According to the model, by how many meters does the thickness of the glacier decrease every $50$ weeks?

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The glacier's thickness decreases by 6 meters.

In the linear model

T = 48.5 - 0.12w

, the thickness

T

decreases by

0.12

meters for each increase of

1

in

w

(each week). Therefore, over a period of

50

weeks, the total decrease in thickness is

0.12 \times 50 = 6

meters.

Adım Adım Çözüm

1

Identify the weekly rate of decrease from the slope of the equation.

0.12 meters per week

In the linear model

T = 48.5 - 0.12w

, the coefficient of

w

is

-0.12

. This slope represents the rate of change, indicating a decrease of

0.12

meters in thickness per week.

2

Calculate the cumulative decrease over 50 weeks.

6

Multiply the weekly rate of decrease (

0.12

meters per week) by the duration (

50

weeks) to obtain the total decrease:

0.12 \times 50 = 6

.

Anahtar Kavram

Interpreting the slope of a linear relationship in context

Tahmini Süre:1m 30s

Soru 47Soru →

An agricultural drone is used to spray liquid fertilizer on a field. The volume of fertilizer remaining in the drone's tank, $F$ , in liters, can be modeled by the equation $F = 120 - 4.8t$ , where $t$ is the number of minutes after the drone begins spraying. What is the best interpretation of the number $4.8$ in this context?

The volume of fertilizer, in liters, in the drone's tank before the spraying operation begins.

The volume of fertilizer, in liters, that is sprayed from the drone's tank each minute.

The number of minutes it takes for the drone to spray one liter of fertilizer.

The number of minutes the drone can spray fertilizer before the tank is completely empty.

Cevabı ve açıklamayı göster

Cevap: The volume of fertilizer, in liters, that is sprayed from the drone's tank each minute.

Cevap

The volume of fertilizer, in liters, that is sprayed from the drone's tank each minute.

In the linear equation

F = 120 - 4.8t

, the variable

F

represents the remaining fertilizer in liters, and

t

represents the elapsed time in minutes. The value 4.8 is the coefficient of

t

, which represents the rate of change of the fertilizer in the tank. The negative sign in front of 4.8 indicates a decrease over time, meaning that the drone sprays 4.8 liters of fertilizer from the tank each minute.

Adım Adım Çözüm

1

Identify the role of the constant 4.8 in the given linear equation

F = 120 - 4.8t

.

The coefficient of the independent variable

t

is

-4.8

, which represents the slope of the linear relationship.

In the slope-intercept form of a linear equation, the coefficient of the variable represents the rate of change (slope).

2

Interpret the meaning of the slope in the context of the problem.

Since

F

is measured in liters and

t

is measured in minutes, the slope represents a change in liters per minute. The negative sign indicates that the volume of fertilizer is decreasing.

The rate of change is the change in the dependent variable (liters of fertilizer) divided by the change in the independent variable (minutes).

3

Select the option that correctly describes this rate of change.

The rate of decrease is 4.8 liters per minute, which means 4.8 liters of fertilizer are sprayed from the tank each minute.

This matches the definition of the rate at which fertilizer is leaving the tank.

Anahtar Kavram

Interpreting the slope of a linear equation in a real-world context.

Soru 48Soru →

A municipal water reservoir contains 250 million gallons of water. During a dry spell, water is released from the reservoir at a constant rate of 3.5 million gallons per day. Additionally, water evaporates from the reservoir at a constant rate of 0.3 million gallons per day. If no water enters the reservoir, the total amount of water $W$ , in million gallons, remaining in the reservoir after $d$ days of the dry spell is modeled by the equation $W = 250 - r d$ , where $r$ is a constant. What is the value of $r$ ?

Cevabı ve açıklamayı göster

Cevap: 3.8

Cevap

The correct value of

r

is 3.8.

The constant

r

in the linear equation

W = 250 - r d

represents the total rate, in million gallons per day, at which the water volume in the reservoir decreases. Since water is lost through both release (

3.5

million gallons per day) and evaporation (

0.3

million gallons per day), the total rate of decrease is the sum of these two rates, which is

3.5 + 0.3 = 3.8

million gallons per day. Therefore, the value of

r

is

3.8

.

Adım Adım Çözüm

1

Identify the factors causing a decrease in the reservoir's water volume.

Water is lost through release at

3.5

million gallons per day and evaporation at

0.3

million gallons per day.

Both release and evaporation contribute to the total rate of water depletion.

2

Calculate the total daily rate of water loss.

3.5 + 0.3 = 3.8

million gallons per day.

Adding the individual rates of loss yields the overall rate of decrease.

3

Compare the total daily rate of water loss to the model equation

W = 250 - r d

.

r = 3.8

.

In the linear model,

250

represents the initial amount of water, and

r

represents the constant rate at which water decreases per day. Thus,

r

is the total daily rate of water loss.

Anahtar Kavram

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

Soru 49Soru →

An electric vehicle is connected to a charging station. The energy stored in the vehicle's battery pack, $E$ , in kilowatt-hours (kWh), can be modeled by the equation $E = 0.75m + 24.0$ , where $m$ is the number of minutes the vehicle has been charging. Which of the following is the best interpretation of the number $0.75$ in this context?

The initial energy stored in the battery pack, in kilowatt-hours, before the vehicle began charging

The increase in the charging time, in minutes, for each additional kilowatt-hour of energy stored in the battery pack

The increase in the energy stored in the battery pack, in kilowatt-hours, for each additional minute the vehicle is charged

The total energy stored in the battery pack, in kilowatt-hours, after the vehicle has been charging for

1

minute

Cevabı ve açıklamayı göster

Cevap: The increase in the energy stored in the battery pack, in kilowatt-hours, for each additional minute the vehicle is charged

Cevap

The increase in the energy stored in the battery pack, in kilowatt-hours, for each additional minute the vehicle is charged

The coefficient of

m

in the linear equation

E = 0.75m + 24.0

is the slope of the line. In context, the slope represents the rate of change of the dependent variable (stored energy,

E

) per unit increase of the independent variable (time,

m

). Thus,

0.75

represents an increase of

0.75

kilowatt-hours in stored energy for each additional minute of charging.

Adım Adım Çözüm

1

Identify the form and components of the given equation.

The equation

E = 0.75m + 24.0

is in the slope-intercept form

y = mx + b

, where

m = 0.75

is the slope and

b = 24.0

is the y-intercept.

To relate the numerical values in the equation to their mathematical interpretations.

2

Interpret the meaning of the slope in context.

The slope,

0.75

, represents the unit rate of change of the dependent variable (energy

E

in kWh) per unit change of the independent variable (time

m

in minutes).

To explain what the coefficient of

m

means in real-world terms.

3

Combine the units and direction of change.

Since

0.75

is positive, the energy increases by

0.75

kWh for every additional

1

minute of charging time.

To formulate the final contextual interpretation of the slope.

Anahtar Kavram

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

Soru 50Soru →

A specialized cooling system is used to lower the temperature of a chemical solution in a laboratory. The temperature of the solution, $T$ , in degrees Celsius, can be modeled by the linear equation $T = 85.4 - 1.25m$ , where $m$ is the number of minutes since the cooling process began. According to the model, how many minutes does it take for the temperature of the solution to decrease by $15$ degrees Celsius?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

12 minutes

The correct answer is 12. In the equation

T = 85.4 - 1.25m

, the coefficient of

m

is

-1.25

, which indicates that the temperature decreases by

1.25

degrees Celsius for each minute that passes. To find how many minutes it takes for the temperature to decrease by

15

degrees Celsius, divide the total decrease by the rate of decrease:

\frac{15}{1.25} = 12

.

Adım Adım Çözüm

1

Identify the rate of change from the linear equation.

The rate of decrease is

1.25

degrees Celsius per minute.

The slope of the linear equation

T = 85.4 - 1.25m

is

-1.25

, representing the change in temperature per minute.

2

Divide the target temperature change by the rate of change to find the time.

12 minutes

To find the number of minutes for a

15

-degree decrease at a rate of

1.25

degrees per minute, calculate

\frac{15}{1.25}

.

Anahtar Kavram

Interpreting the slope of a linear relationship in context

Soru 51Soru →

A digital archiving team is scanning a large collection of historical documents. The number of unscanned documents remaining in the collection, $D$ , can be modeled by the equation $D = 14{}200 - 350t$ , where $t$ represents the number of hours the team has been scanning. Which of the following is the best interpretation of $350$ in this context?

The number of hours it takes the team to scan one document

The initial number of unscanned documents at the start of the shift

The number of documents the team scans each hour

The number of unscanned documents remaining after one hour of the shift

Cevabı ve açıklamayı göster

Cevap: The number of documents the team scans each hour

Cevap

The number of documents the team scans each hour

In the linear model

D = 14{}200 - 350t

, the coefficient

-350

is the slope of the line, which indicates the rate of change of the dependent variable. Since the remaining number of unscanned documents decreases by

350

for each hour that passes, the value

350

represents the number of documents scanned per hour by the team.

Adım Adım Çözüm

1

Identify the role of each component in the linear equation

D = 14{}200 - 350t

.

The equation is in the form

y = b + mx

, where the constant

b = 14{}200

is the y-intercept (the starting value of

D

when

t = 0

) and the coefficient

m = -350

is the slope (the constant rate of change of

D

per unit increase in

t

).

Understanding the structure of the linear model helps distinguish between initial values and rates of change.

2

Interpret the meaning of the slope in the context of the variables.

The slope of

-350

indicates that for each unit increase in the independent variable

t

(hours), the dependent variable

D

(unscanned documents remaining) decreases by

350

.

The slope represents the constant rate of change of the remaining documents over time.

3

Select the option that matches this rate of change.

A rate of

350

documents per hour means the archiving team scans

350

documents every hour.

This directly matches the definition of the slope in this context.

Anahtar Kavram

Interpreting Linear Relationships in Context

Tahmini Süre:1m 15s

Soru 52Soru →

A commercial bakery uses an automated flour silo. The mass of the flour in the silo, $F$ , in kilograms, is modeled as a linear function of the time $t$ , in hours, after the bakery opens. The table below shows the mass of the flour remaining in the silo at two different times during the day:

Time (hours), $t$	Mass of flour (kilograms), $F$
$3$	$1,850$
$7$	$1,490$

If the mass of the flour in the silo decreases at a constant rate of $r$ kilograms per hour, what is the value of $r$ ?

Cevabı ve açıklamayı göster

Cevap: 90

Cevap

90

The rate of decrease of the flour is represented by the magnitude of the slope of the linear relationship. Using the points

(3, 1850)

and

(7, 1490)

from the table, the slope is calculated as

\frac{1490 - 1850}{7 - 3} = \frac{-360}{4} = -90

. This indicates that the mass of the flour decreases by

90

kilograms per hour. Therefore, the value of

r

is

90

.

Adım Adım Çözüm

1

Identify the data points representing time and mass from the table.

(t_1, F_1) = (3, 1850)

and

(t_2, F_2) = (7, 1490)

We need two coordinates to find the slope of the linear relationship.

2

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

Slope =

\frac{1490 - 1850}{7 - 3} = -90

The slope formula

\frac{y_2 - y_1}{x_2 - x_1}

gives the rate of change of the mass of the flour per hour.

3

Determine the value of

r

based on the rate of decrease.

r = 90

The rate of decrease is the positive magnitude of the rate of change.

Anahtar Kavram

Interpreting rate of change (slope) from tabular data in a linear context

Tahmini Süre:1m 30s

Soru 53Soru →

A coffee roasting company uses a commercial roasting machine. The temperature of the roasting drum, $T$ , in degrees Fahrenheit ( $^\circ\text{F}$ ), is modeled as a linear function of the roasting time, $t$ , in minutes, where $0 \leq t \leq 10$ . After $2$ minutes of roasting, the temperature of the drum is $280^\circ\text{F}$ . After $5$ minutes of roasting, the temperature of the drum is $385^\circ\text{F}$ . What is the rate of temperature increase, in degrees Fahrenheit per minute, of the roasting drum?

Cevabı ve açıklamayı göster

Cevap: 35

Cevap

The rate of temperature increase of the roasting drum is

35

degrees Fahrenheit per minute.

The temperature,

T

, is modeled as a linear function of time,

t

. The constant rate of temperature increase corresponds to the slope of this linear function. Given two points,

(2, 280)

and

(5, 385)

, the slope is calculated as the change in temperature divided by the change in time:

\frac{385 - 280}{5 - 2} = \frac{105}{3} = 35

degrees Fahrenheit per minute.

Adım Adım Çözüm

1

Identify the coordinates representing the relationship between time and temperature.

The two points are

(2, 280)

and

(5, 385)

, where the first coordinate is the time,

t

, in minutes, and the second coordinate is the temperature,

T

, in degrees Fahrenheit.

To find the rate of change of a linear relationship, we first need to determine two points

(t_1, T_1)

and

(t_2, T_2)

from the given context.

2

Calculate the rate of temperature increase as the slope of the line passing through these two points.

The slope

m

is given by

m = \frac{385 - 280}{5 - 2} = \frac{105}{3} = 35

.

The rate of temperature increase per minute is the constant slope of the linear relationship.

Anahtar Kavram

Slope of a linear relationship in context

Soru 54Soru →

A researcher monitored the mass of a cooling block of metal over time as it underwent sublimation in a vacuum chamber. The table below shows the mass of the metal block, $M$ , in grams, for several values of time, $t$ , in hours after the sublimation process began.

Time, $t$ (hours)	Mass, $M$ (grams)
$2$	$112.4$
$5$	$104.6$
$8$	$96.8$

If the relationship between $M$ and $t$ is linear, which of the following is the best interpretation of the slope of the graph of this relationship in the $tM$ -plane?

The mass of the metal block decreases by

2.6

grams each hour.

The initial mass of the metal block before sublimation began was

117.6

grams.

The mass of the metal block decreases by

7.8

grams each hour.

The mass of the metal block after

1

hour of sublimation is

115.0

grams.

Cevabı ve açıklamayı göster

Cevap: The mass of the metal block decreases by $2.6$ grams each hour.

Cevap

The mass of the metal block decreases by

2.6

grams each hour.

The correct answer is the option stating that the mass of the metal block decreases by

2.6

grams each hour. The slope of a linear model is the rate of change, which is computed as the change in the dependent variable divided by the change in the independent variable. Here,

\frac{104.6 - 112.4}{5 - 2} = -2.6

, representing a change of

-2.6

grams per hour.

Adım Adım Çözüm

1

Calculate the slope of the linear relationship using two points from the table.

Using the points

(2, 112.4)

and

(5, 104.6)

, the slope is

\frac{104.6 - 112.4}{5 - 2} = \frac{-7.8}{3} = -2.6

.

The slope represents the constant rate of change of the dependent variable (

M

) per unit change in the independent variable (

t

).

2

Interpret the calculated slope value in the context of the problem.

A slope of

-2.6

means that the mass

M

decreases by

2.6

grams for every

1

hour increase in time

t

.

The negative sign indicates a decrease, and the rate is expressed in units of the dependent variable per unit of the independent variable (grams per hour).

Anahtar Kavram

Interpreting the slope of a linear relationship in context.

Tahmini Süre:1m 30s

Soru 55Soru →

A residential solar power system stores electricity in a battery. The total energy stored in the battery, $E$ , in kilowatt-hours (kWh), is modeled by a linear function of the time $t$ , in hours, since sunrise. After $4$ hours of sunlight, the energy stored is $18$ kWh. After $7$ hours of sunlight, the energy stored is $27$ kWh. Which of the following is the best interpretation of the slope of the graph of this function in the $tE$ -plane?

The energy stored in the battery increases by

3

kilowatt-hours per hour of sunlight.

The initial energy stored in the battery at sunrise is

6

kilowatt-hours.

It takes

3

hours of sunlight to increase the energy stored in the battery by

1

kilowatt-hour.

The energy stored in the battery increases by

9

kilowatt-hours per hour of sunlight.

Cevabı ve açıklamayı göster

Cevap: The energy stored in the battery increases by $3$ kilowatt-hours per hour of sunlight.

Cevap

The energy stored in the battery increases by

3

kilowatt-hours per hour of sunlight.

To find the slope of the linear relationship, we use the formula for the rate of change:

\text{slope} = \frac{\Delta E}{\Delta t}

. Using the two points

(4, 18)

and

(7, 27)

, the slope is

\frac{27 - 18}{7 - 4} = \frac{9}{3} = 3

kilowatt-hours per hour. In the context of the problem, the slope represents the rate at which the stored energy increases per hour of sunlight. Therefore, the energy stored in the battery increases by

3

kilowatt-hours per hour of sunlight.

Adım Adım Çözüm

1

Identify the coordinates

(t, E)

from the given values.

The two data points are

(4, 18)

and

(7, 27)

.

These points represent the time in hours and the corresponding energy stored in the battery.

2

Calculate the slope of the linear relationship using the slope formula.

The slope is

\frac{27 - 18}{7 - 4} = \frac{9}{3} = 3

.

The slope of a linear relationship represents the constant rate of change of the dependent variable (

E

) per unit change of the independent variable (

t

).

3

Interpret the meaning of the calculated slope in the context of the problem.

A slope of

3

means that the energy stored in the battery,

E

, increases by

3

kilowatt-hours (kWh) for every

1

hour of sunlight,

t

.

The units of the slope are units of

E

divided by units of

t

, which is kilowatt-hours per hour.

Anahtar Kavram

Interpreting the rate of change (slope) of a linear relationship in context.

Tahmini Süre:1m 30s

Soru 56Soru →

A forestry service uses a remote weather station powered by a solar battery system. During a period of heavy cloud cover, the remaining charge in the battery, $C$ , in watt-hours, can be modeled by the linear equation $C = 360 - 4.5h$ , where $h$ represents the number of hours since the cloud cover began. Based on the model, what is the best interpretation of the $h$ -intercept of the relationship?

The number of hours, which is 80, after the cloud cover began when the battery is completely discharged.

The initial charge of the battery, which is 360 watt-hours, when the cloud cover began.

The rate, which is 4.5 watt-hours per hour, at which the battery charge decreases.

The number of hours, which is 90, after the cloud cover began when the battery is completely discharged.

Cevabı ve açıklamayı göster

Cevap: The number of hours, which is 80, after the cloud cover began when the battery is completely discharged.

Cevap

The number of hours, which is 80, after the cloud cover began when the battery is completely discharged.

The

h

-intercept of the relationship is the value of

h

when

C = 0

. Substituting

0

for

C

in the equation

C = 360 - 4.5h

gives

0 = 360 - 4.5h

. Solving for

h

yields

4.5h = 360

, or

h = 80

. In context, this represents the number of hours after the cloud cover began when the battery is completely discharged.

Adım Adım Çözüm

1

Identify the meaning of the

h

-intercept in the context of the linear equation.

The

h

-intercept occurs when the dependent variable,

C

(remaining charge), is equal to 0.

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

2

Substitute

C = 0

into the equation and solve for

h

.

0 = 360 - 4.5h \implies 4.5h = 360 \implies h = 80

.

This algebraic isolation finds the specific value of

h

when the charge is completely depleted.

3

Interpret the resulting value of

h

in the context of the problem.

At

h = 80

hours, the remaining battery charge is 0 watt-hours, meaning the battery is fully discharged.

Connecting the mathematical coordinate

(80, 0)

back to the units of hours and watt-hours provides the physical interpretation.

Anahtar Kavram

Interpreting Linear Relationships in Context

Soru 57Soru →

A commercial printing press has a reservoir of yellow ink. The volume of yellow ink in the reservoir, $V$ , in milliliters, after printing $p$ pages of a color brochure is modeled by the equation $V = 1,200 - 0.15p$ . According to the model, what is the decrease, in milliliters, in the volume of yellow ink in the reservoir for every 100 pages printed?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The volume of yellow ink in the reservoir decreases by 15 milliliters for every 100 pages printed.

In the equation

V = 1,200 - 0.15p

, the coefficient of

p

is

-0.15

. This represents the rate of change of the volume of yellow ink in the reservoir with respect to the number of pages printed. Specifically, it means the volume decreases by

0.15

milliliters for each additional page printed. To find the decrease in volume for every 100 pages printed, multiply the rate per page by 100:

0.15 \times 100 = 15

milliliters.

Adım Adım Çözüm

1

Identify the rate of change per page from the linear equation.

The rate of change is 0.15 milliliters per page.

In the linear equation

V = 1,200 - 0.15p

, the coefficient of the independent variable

p

represents the change in the dependent variable

V

for each unit increase in

p

.

2

Calculate the decrease in ink volume for 100 pages.

15 milliliters

Since the volume decreases by 0.15 milliliters for each page printed, printing 100 pages results in a total decrease of

0.15 \times 100 = 15

milliliters.

Anahtar Kavram

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

Soru 58Soru →

A vertical farm uses an automated nutrient delivery system for a crop of lettuce. The total volume of nutrient solution, $V$ , in liters, remaining in the system's reservoir is modeled as a linear function of the time $t$ , in hours, since the system started running for the day. The table below shows several values of $t$ and the corresponding values of $V$ .

Time, $t$ (hours)	Volume, $V$ (liters)
3	415
5	385
8	340
12	280

According to the model, what was the initial volume of nutrient solution, in liters, in the reservoir when the system started running?

Cevabı ve açıklamayı göster

Cevap: 460

Cevap

460

The correct answer is 460. The remaining volume of nutrient solution,

V

, is a linear function of time,

t

, which can be written in the form

V = mt + b

, where

m

is the rate of change (slope) and

b

is the initial volume (y-intercept). The rate of change can be found using any two points from the table, for example,

(3, 415)

and

(5, 385)

:

m = \frac{385 - 415}{5 - 3} = \frac{-30}{2} = -15

liters per hour. Using the point

(3, 415)

and substituting

m = -15

,

t = 3

, and

V = 415

into the equation

V = mt + b

yields

415 = -15(3) + b

. Simplifying this expression gives

415 = -45 + b

, and adding 45 to both sides gives

b = 460

. Therefore, the initial volume of nutrient solution in the reservoir was 460 liters.

Adım Adım Çözüm

1

Calculate the constant rate of consumption (slope) of the nutrient solution.

The rate of consumption is 15 liters per hour.

The relationship between volume and time is linear. The slope

m

can be calculated from two coordinates from the table,

(3, 415)

and

(5, 385)

, as

m = \frac{385 - 415}{5 - 3} = -15

liters per hour.

2

Determine the initial volume of nutrient solution (the y-intercept) using the rate and a data point.

The initial volume is 460 liters.

Substitute the slope

m = -15

, the time

t = 3

, and the remaining volume

V = 415

into the slope-intercept equation

V = mt + b

. Solving

415 = -15(3) + b

gives

b = 460

.

Anahtar Kavram

Interpreting the y-intercept of a linear function in context as the initial value of the dependent variable when the independent variable is 0.

Tahmini Süre:1m 30s

Soru 59Soru →

A city reservoir contains water that is released at a constant rate for river conservation. The volume of water, $W$ , in millions of gallons, remaining in the reservoir $d$ days after the release begins is modeled by the equation $W = 240 - 1.8d$ . What is the best interpretation of the number $1.8$ in this context?

The reservoir loses 1 million gallons of water every 1.8 days.

The initial volume of water in the reservoir before the release begins is 1.8 million gallons.

The volume of water in the reservoir decreases by 1.8 million gallons each day.

The total duration of the controlled release is 1.8 days.

Cevabı ve açıklamayı göster

Cevap: The volume of water in the reservoir decreases by 1.8 million gallons each day.

Cevap

The volume of water in the reservoir decreases by 1.8 million gallons each day.

The equation

W = 240 - 1.8d

is in slope-intercept form, where the slope is

-1.8

and the

y

-intercept is

240

. The slope represents the rate of change of the remaining volume of water with respect to time. A slope of

-1.8

indicates that the volume of water,

W

, decreases by

1.8

million gallons for every

1

day increase in the number of days,

d

.

Adım Adım Çözüm

1

Identify the variables and constants in the given linear model.

In

W = 240 - 1.8d

,

W

is the remaining volume in millions of gallons,

d

is the time in days, 240 is the y-intercept (initial volume), and -1.8 is the slope (rate of change).

Understanding the components of a linear equation

y = mx + b

helps isolate the meaning of each constant.

2

Interpret the meaning of the slope in the context of the variables and units.

The coefficient of

d

is -1.8, which represents a rate of change of -1.8 million gallons per day.

The slope is the change in the dependent variable (

W

, in millions of gallons) per unit change in the independent variable (

d

, in days).

3

Relate the negative sign to the physical context.

A negative slope of -1.8 means that the volume decreases by 1.8 million gallons for each elapsed day.

In context, a constant release of water leads to a decrease in the remaining volume.

Anahtar Kavram

Interpreting slope in a linear relationship context

Tahmini Süre:1m 30s

Soru 60Soru →

An electric delivery van's remaining battery capacity, $B$ , in kilowatt-hours (kWh), after driving $d$ miles is modeled by a linear equation. After driving $45$ miles, the remaining battery capacity is $68$ kWh. After driving a total of $120$ miles, the remaining battery capacity is $38$ kWh. According to this model, how many miles can the van drive on a full charge before the battery capacity reaches $0$ kWh?

Cevabı ve açıklamayı göster

Cevap: 215

Cevap

215

To find the maximum distance the van can drive on a full charge, we model the relationship between the remaining battery capacity,

B

, and the distance driven,

d

, using the linear equation

B = md + B_0

. First, calculate the rate of consumption (slope,

m

) using the points

(45, 68)

and

(120, 38)

:

m = \frac{38 - 68}{120 - 45} = \frac{-30}{75} = -0.4

kWh per mile. Next, find the initial battery capacity (

B_0

) by substituting the values from one of the points:

68 = -0.4(45) + B_0

, which simplifies to

68 = -18 + B_0

, so

B_0 = 86

kWh. Finally, determine the distance

d

when the battery is fully depleted (

B = 0

):

0 = -0.4d + 86

, which yields

0.4d = 86

, and

d = 215

miles.

Adım Adım Çözüm

1

Calculate the rate of change of the battery capacity per mile driven.

The rate of change (slope) is

-0.4

kWh per mile.

The slope

m

represents the energy consumption rate and is calculated using the two data points

(45, 68)

and

(120, 38)

with the formula

m = \frac{38 - 68}{120 - 45} = \frac{-30}{75} = -0.4

.

2

Determine the initial battery capacity when the distance driven is 0 miles.

The initial battery capacity is

86

kWh.

Using the slope-intercept form

B = md + B_0

, substitute

m = -0.4

and the point

(45, 68)

to get

68 = -0.4(45) + B_0

. This simplifies to

68 = -18 + B_0

, so

B_0 = 86

.

3

Solve for the distance driven

d

when the battery capacity

B

is

0

kWh.

The distance is

215

miles.

Set

B = 0

in the linear model

B = -0.4d + 86

to get

0 = -0.4d + 86

. Solving for

d

gives

0.4d = 86

, which results in

d = \frac{86}{0.4} = 215

.

Anahtar Kavram

Interpreting and calculating slope, y-intercept, and x-intercept of a linear function in a real-world context.

Interpreting Linear Relationships in Context

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