Interpreting Linear Relationships in Context Alıştırma Soruları — SAT — Sayfa 2

Soru 21Soru →

At a municipal water desalination facility, the filtration rate of a reverse osmosis membrane decreases linearly over time due to particle accumulation. The daily volume of purified water, $V$ , in thousands of gallons per day, can be modeled as a function of the number of days, $t$ , since the membrane was last serviced. The graph of this relationship in the $tV$ -plane has a $t$ -intercept of $100$ and passes through the point $(20, 64)$ . Which of the following is the best interpretation of the slope of the graph of this relationship?

The daily volume of purified water decreases by

0.8

thousand gallons each day after the membrane is serviced.

The daily volume of purified water decreases by

1.25

thousand gallons each day after the membrane is serviced.

The daily volume of purified water decreases by

80

thousand gallons each day after the membrane is serviced.

For every decrease of

1

thousand gallons in the daily volume of purified water,

0.8

days have passed since the membrane was serviced.

Cevabı ve açıklamayı göster

Cevap: The daily volume of purified water decreases by $0.8$ thousand gallons each day after the membrane is serviced.

Cevap

The daily volume of purified water decreases by

0.8

thousand gallons each day after the membrane is serviced.

The correct answer describes that the daily volume of purified water decreases by

0.8

thousand gallons each day. The

t

-intercept of

100

indicates the point

(100, 0)

is on the graph, and the problem states the graph passes through

(20, 64)

. The slope of the relationship is

m = \frac{0 - 64}{100 - 20} = -0.8

. Since the vertical axis represents the daily volume of purified water (in thousands of gallons) and the horizontal axis represents the number of days, the slope of

-0.8

represents a decrease of

0.8

thousand gallons of water per day.

Adım Adım Çözüm

1

Identify the coordinates of two points on the line from the given context.

The

t

-intercept of

100

corresponds to the point

(100, 0)

. The second point is given directly as

(20, 64)

.

Two points on the line are needed to calculate the slope of the linear relationship.

2

Calculate the slope (

m

) using the slope formula

m = \frac{V_2 - V_1}{t_2 - t_1}

.

m = \frac{0 - 64}{100 - 20} = \frac{-64}{80} = -0.8

.

The slope of the line represents the rate of change of the daily volume of water (

V

) with respect to the elapsed days (

t

).

3

Interpret the meaning of the slope in context.

A slope of

-0.8

indicates that for every increase of

1

day in

t

, the daily volume of water

V

decreases by

0.8

units (thousands of gallons).

The slope value of

-0.8

represents a daily reduction of

0.8

thousand gallons of purified water.

Anahtar Kavram

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

Alternatif Yöntem

Instead of calculating the slope directly from the formula, one can write the equation of the line in point-slope form or slope-intercept form. Since

(100,0)

is the

t

-intercept, the line can be represented as

V = m(t - 100)

. Substituting the point

(20, 64)

gives

64 = m(20 - 100)

, which simplifies to

64 = -80m

, solving to

m = -0.8

. This slope represents the daily change in the volume

V

for each unit change in day

t

.

Tahmini Süre:2m 30s

Soru 22Soru →

For a wireless sensor node in an environmental monitoring network, the battery life $B$ , in days, when transmitting $n$ data packets per day is modeled by a linear function. Under standard operating conditions, the node's battery lasts for $120$ days when transmitting $80$ packets per day, and it lasts for $90$ days when transmitting $140$ packets per day. Under a new energy-saving firmware, the battery consumption rate per packet transmitted is reduced by $40\%$ , and the static power draw is also reduced, increasing the maximum battery life (when zero packets are transmitted) by $10\%$ . Under this new firmware, what is the battery life, in days, of a sensor node that transmits $150$ data packets per day?

Cevabı ve açıklamayı göster

Cevap: 131

Cevap

131

To solve this problem, we first determine the linear relationship

B = mn + B_0

under standard operating conditions. The rate of change (slope

m

) represents the change in battery life per packet transmitted per day:

m = \frac{90 - 120}{140 - 80} = -0.5

days per packet. Substituting this back into the linear equation gives

120 = -0.5(80) + B_0

, which simplifies to

120 = -40 + B_0

, yielding a y-intercept of

B_0 = 160

days. Under the energy-saving firmware, the battery consumption rate per packet is reduced by

40\%

. This means the rate at which battery life decreases per packet changes from

0.5

to

0.5 \times (1 - 0.40) = 0.3

, giving us a new slope of

-0.3

. The maximum battery life (y-intercept) increases by

10\%

, making the new y-intercept

160 \times 1.10 = 176

. The new linear model is

B_{\text{new}} = -0.3n + 176

. Evaluating this equation for

n = 150

packets per day yields

B_{\text{new}} = -0.3(150) + 176 = -45 + 176 = 131

days.

Adım Adım Çözüm

1

Determine the linear relationship representing standard operating conditions by using the two given points,

(80, 120)

and

(140, 90)

.

Slope

m = -0.5

days per packet and y-intercept

B_0 = 160

days.

Establishing the initial linear equation is necessary to obtain the base battery consumption rate and maximum battery capacity.

2

Calculate the new slope and y-intercept parameters under the energy-saving firmware by applying the specified percentage changes.

New slope

m_{\text{new}} = -0.3

days per packet and new y-intercept

B_{0,\text{new}} = 176

days.

To model the linear relationship under the updated power management firmware.

3

Formulate the new linear function

B_{\text{new}} = -0.3n + 176

and substitute

150

for

n

.

Battery life

B = 131

days.

To find the expected battery life at the target daily packet transmission rate.

Anahtar Kavram

Interpreting how physical rates and initial values map to the slope and y-intercept of a linear model, and applying transformations to these parameters.

Soru 23Soru →

A moving company uses the equation $C = 2.5d + 75$ to determine the total charge $C$ , in dollars, for renting a small truck and driving it $d$ miles. According to the relationship, what is the charge, in dollars, for each mile the truck is driven?

Cevabı ve açıklamayı göster

Cevap: 2.5

Cevap

The charge for each mile the truck is driven is 2.5 dollars.

In the linear model

C = 2.5d + 75

, the total cost

C

is a function of the number of miles driven

d

. The rate of change of this function represents the cost per mile driven. In the equation, this rate of change is the coefficient of

d

, which is 2.5. Therefore, the charge for each mile the truck is driven is 2.5 dollars.

Adım Adım Çözüm

1

Analyze the linear equation

C = 2.5d + 75

to determine the relationship between variables.

The total charge

C

depends on the number of miles

d

driven, with a rate of change of 2.5 dollars per mile and a base fee of 75 dollars.

Understanding the components of a linear equation helps isolate the rate of change.

2

Identify the coefficient of the independent variable

d

.

The coefficient of

d

is 2.5.

In a linear equation of the form

y = mx + b

, the coefficient of the independent variable represents the rate of change.

3

Interpret the meaning of this coefficient in the context of the problem.

The coefficient 2.5 represents the cost, in dollars, incurred per mile driven.

The question asks for the charge per mile, which corresponds to the rate of change.

Anahtar Kavram

Interpreting Linear Relationships in Context

Soru 24Soru →

A commercial cargo aircraft's total weight $W$ , in kilograms, is modeled as a linear function of the volume of fuel $F$ , in liters, in its fuel tanks. The equation modeling this relationship is:

W = 0.8F + 74,000

To comply with runway safety regulations at a destination airport, the aircraft's total weight must not exceed $85,000$ kilograms upon landing. The aircraft takes off with $18,000$ liters of fuel and consumes fuel at a constant rate of $1,700$ liters per hour of flight. What is the minimum number of hours the aircraft must fly before it can safely land?

Cevabı ve açıklamayı göster

Cevap: 2.5

Cevap

The minimum number of hours the aircraft must fly is 2.5.

To satisfy runway safety regulations, the aircraft's weight must be at most

85,000

kg. According to the weight model

W = 0.8F + 74,000

, we set

0.8F + 74,000 \leq 85,000

, which simplifies to

0.8F \leq 11,000

. Dividing by the density coefficient

0.8

yields

F \leq 13,750

liters as the maximum amount of fuel the aircraft can contain at landing. Since the aircraft takes off with

18,000

liters of fuel and consumes

1,700

liters per hour, the fuel remaining after

t

hours is

18,000 - 1,700t

. To ensure the remaining fuel is less than or equal to

13,750

liters, we solve

18,000 - 1,700t \leq 13,750

, which simplifies to

4,250 \leq 1,700t

, or

t \geq 2.5

hours. Therefore, the minimum duration of the flight is 2.5 hours.

Adım Adım Çözüm

1

Formulate the weight limit inequality using the linear relationship.

0.8F + 74,000 \leq 85,000

The aircraft's weight

W

is given by

0.8F + 74,000

, where

0.8

represents the density of the fuel in kg/L and

74,000

represents the weight of the empty aircraft and its cargo. This total weight must not exceed

85,000

kg.

2

Calculate the maximum fuel capacity allowed at landing by solving the inequality.

F \leq 13,750

Subtracting

74,000

from both sides yields

0.8F \leq 11,000

. Dividing by

0.8

gives

F \leq 13,750

liters as the fuel ceiling for landing.

3

Use the fuel consumption rate to find the minimum flight time.

t \geq 2.5

With an initial fuel volume of

18,000

liters and a burn rate of

1,700

liters/hour, the fuel remaining after

t

hours is

18,000 - 1,700t

. Setting this expression to be at most

13,750

liters yields

18,000 - 1,700t \leq 13,750

. Subtracting

18,000

gives

-1,700t \leq -4,250

, and dividing by

-1,700

results in

t \geq 2.5

hours.

Anahtar Kavram

Interpreting linear coefficients in context and setting up linear inequalities with rates to model real-world constraints.

Tahmini Süre:2m 30s

Soru 25Soru →

At a local coffee shop, the remaining weight of coffee beans $C$ , in pounds, after preparing $n$ cups of espresso is modeled by the equation $C = 80 - 0.04n$ . What is the best interpretation of the number $0.04$ in this context?

The weight of coffee beans, in pounds, used to prepare each cup of espresso

The initial weight of the coffee beans, in pounds, before any cups of espresso are prepared

The number of cups of espresso that can be prepared per pound of coffee beans

The weight of coffee beans, in pounds, remaining in the shop after preparing

80

cups of espresso

Cevabı ve açıklamayı göster

Cevap: The weight of coffee beans, in pounds, used to prepare each cup of espresso

Cevap

The weight of coffee beans, in pounds, used to prepare each cup of espresso

The correct answer is the option stating that

0.04

is the weight of coffee beans, in pounds, used to prepare each cup of espresso. In the linear relationship

C = 80 - 0.04n

, the slope is

-0.04

, which represents the rate of change of the remaining weight of coffee beans with respect to the number of cups of espresso prepared. A rate of

-0.04

pounds per cup means that for every cup prepared, the remaining coffee beans decrease by

0.04

pounds, indicating that

0.04

pounds of beans are consumed per cup.

Adım Adım Çözüm

1

Analyze the structure of the linear equation

C = 80 - 0.04n

.

The equation is in the slope-intercept form

y = mx + b

, where the dependent variable is

C

(remaining coffee beans in pounds), the independent variable is

n

(number of cups of espresso), the constant (y-intercept) is

80

, and the slope (coefficient of

n

) is

-0.04

.

Identifying the components of the linear equation helps in assigning their real-world contextual meanings.

2

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

The slope of

-0.04

represents the change in the remaining coffee beans (

C

) for every

1

-unit increase in the number of cups of espresso (

n

). This means the remaining weight decreases by

0.04

pounds per cup.

The slope of a linear model shows the constant rate of change of the dependent variable per unit of the independent variable.

3

Relate the rate of decrease to the options.

A decrease of

0.04

pounds of remaining beans per cup of espresso means that

0.04

pounds of coffee beans are used to prepare each cup of espresso.

Translating the decrease rate into consumption rate matches the physical scenario described.

Anahtar Kavram

Interpreting Linear Relationships in Context

Tahmini Süre:45s

Soru 26Soru →

A local gym charges a one-time registration fee plus a constant monthly fee. The total cost, $C$ , in dollars, for a membership of $m$ months is modeled by the equation $C = 35m + 50$ . What is the one-time registration fee, in dollars, for the gym?

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

The one-time registration fee is 50 dollars.

In the linear model

C = 35m + 50

, the constant term

50

represents the value of the function when

m = 0

. In this context,

m = 0

corresponds to

0

months of membership, meaning no monthly fees have been incurred yet. Therefore, the value of

50

represents the initial, one-time registration fee.

Adım Adım Çözüm

1

Analyze the linear equation

C = 35m + 50

to identify the slope and the y-intercept.

The slope is

35

(the coefficient of

m

) and the y-intercept is

50

(the constant term).

In a linear equation of the form

y = mx + b

, the constant

b

represents the value of

y

when

x = 0

(the y-intercept), and the coefficient

m

represents the rate of change (the slope).

2

Interpret the meaning of the y-intercept in the given context.

At

m = 0

months, the total cost

C

is

C = 35(0) + 50 = 50

dollars.

The cost at

0

months represents the upfront, one-time fee before any monthly fees are added, which is the registration fee.

Anahtar Kavram

Interpreting the y-intercept of a linear function in a real-world context.

Soru 27Soru →

A deep-sea research submersible's internal cabin pressure, $P$ , in atmospheres ( $\text{atm}$ ), is modeled by a linear function of its depth below the ocean surface, $d$ , in meters. At the surface ( $d = 0$ ), the internal pressure is $1.0\text{ atm}$ . For every increase in depth of $100\text{ meters}$ , the internal pressure increases by $0.05\text{ atm}$ . The submersible descends from the surface at a constant rate of $2.5\text{ meters per second}$ . What is the rate of increase of the internal cabin pressure, in $\text{atm}$ per hour, as the submersible descends?

Cevabı ve açıklamayı göster

Cevap: 4.5

Cevap

The rate of increase of the internal cabin pressure is

4.5

atmospheres per hour.

The rate of change of internal pressure with depth is

\frac{0.05\text{ atm}}{100\text{ m}} = 0.0005\text{ atm/m}

. Since the submersible descends at a rate of

2.5\text{ m/s}

, we convert this rate of depth change to hours:

2.5\text{ m/s} \times 3600\text{ s/hr} = 9000\text{ m/hr}

. Multiplying the pressure change per meter by the depth change per hour gives the rate of change of the internal pressure per hour:

0.0005\text{ atm/m} \times 9000\text{ m/hr} = 4.5\text{ atm/hr}

.

Adım Adım Çözüm

1

Calculate the rate of pressure increase per meter of depth.

0.0005\text{ atm/m}

The pressure increases by

0.05\text{ atm}

for every

100\text{ meters}

of depth, giving a rate of change of

\frac{0.05\text{ atm}}{100\text{ m}} = 0.0005\text{ atm/m}

.

2

Determine the distance descended by the submersible in one hour.

9000\text{ meters}

With

3600\text{ seconds}

in one hour and a descent speed of

2.5\text{ m/s}

, the submersible descends a total of

2.5\text{ m/s} \times 3600\text{ s} = 9000\text{ meters}

in one hour.

3

Calculate the rate of internal pressure increase per hour.

4.5\text{ atm/hr}

Multiply the rate of change of pressure per meter (

0.0005\text{ atm/m}

) by the hourly descent distance (

9000\text{ m}

) to obtain

0.0005 \times 9000 = 4.5\text{ atm/hr}

.

Anahtar Kavram

Interpreting Linear Relationships in Context

Alternatif Yöntem

Write the cabin pressure as a function of time

t

, in seconds:

P(t) = 1.0 + 0.0005(2.5t) = 1.0 + 0.00125t

. The slope of this line,

0.00125\text{ atm/s}

, represents the rate of increase per second. To convert this rate to hours, multiply by

3600\text{ seconds/hour}

:

0.00125 \times 3600 = 4.5\text{ atm/hr}

.

Tahmini Süre:2m 0s

Soru 28Soru →

A geophysicist models the temperature, $T$ , in degrees Celsius ( $^\circ\text{C}$ ), of a rock layer during a deep-crust drilling project using a linear function of the depth, $d$ , in kilometers ( $\text{km}$ ), below the surface. According to the model, for every increase in depth of $0.8\text{ km}$ , the temperature of the rock increases by $22^\circ\text{C}$ . At a depth of $2.4\text{ km}$ , the temperature of the rock is $81^\circ\text{C}$ . According to the model, at what depth, in kilometers, will the temperature of the rock be $114^\circ\text{C}$ ?

Cevabı ve açıklamayı göster

Cevap: 3.6

Cevap

The temperature of the rock will reach

114^\circ\text{C}

at a depth of

3.6

kilometers.

The correct answer is

3.6

. The rate of change of temperature with depth is

\frac{22^\circ\text{C}}{0.8\text{ km}} = 27.5^\circ\text{C/km}

. The linear relationship between temperature

T

and depth

d

can be modeled by

T = 27.5d + T_0

, where

T_0

is the temperature at the surface. Substituting the known values

d = 2.4

and

T = 81

into the model gives

81 = 27.5(2.4) + T_0

, which simplifies to

81 = 66 + T_0

. Solving for

T_0

yields

T_0 = 15

. Thus, the model is

T = 27.5d + 15

. To find the depth when the temperature is

114^\circ\text{C}

, substitute

T = 114

into the model:

114 = 27.5d + 15

. Subtracting

15

from both sides gives

99 = 27.5d

, and dividing by

27.5

yields

d = 3.6

.

Adım Adım Çözüm

1

Find the rate of change (slope) of the temperature with respect to depth.

The slope is

27.5^\circ\text{C/km}

.

The temperature increases by

22^\circ\text{C}

for every

0.8\text{ km}

of depth, so the rate of change is

\frac{22^\circ\text{C}}{0.8\text{ km}} = 27.5^\circ\text{C/km}

.

2

Set up a linear model and find the surface temperature (y-intercept).

The linear model is

T = 27.5d + 15

.

Using the slope-intercept form

T = md + T_0

and substituting the given values

d = 2.4

and

T = 81

gives

81 = 27.5(2.4) + T_0

, which simplifies to

81 = 66 + T_0

, so

T_0 = 15

.

3

Substitute the target temperature into the linear model and solve for the target depth.

d = 3.6

Substitute

T = 114

into the equation

T = 27.5d + 15

to get

114 = 27.5d + 15

. Subtracting

15

from both sides gives

99 = 27.5d

. Dividing both sides by

27.5

yields

d = 3.6

.

Anahtar Kavram

Interpreting slope and solving for values in a linear relationship context.

Alternatif Yöntem

Find the required temperature increase:

114^\circ\text{C} - 81^\circ\text{C} = 33^\circ\text{C}

. Set up a proportion using the rate of

22^\circ\text{C}

increase per

0.8\text{ km}

to find the change in depth

\Delta d

:

\frac{33^\circ\text{C}}{\Delta d} = \frac{22^\circ\text{C}}{0.8\text{ km}} \implies \Delta d = \frac{33 \times 0.8}{22} = 1.2\text{ km}

. Add this change in depth to the initial depth to find the final depth:

2.4 + 1.2 = 3.6\text{ km}

.

Tahmini Süre:2m 0s

Soru 29Soru →

An IT administrator uses the equation $T = 0.06r + 40$ to model the total time $T$ , in milliseconds, required to retrieve $r$ records from a cloud database. The constant term in the model represents the total setup time, which consists of network latency and database connection overhead. If the database connection overhead is 20% of the total setup time, which of the following is the best interpretation of the value 8 in this model?

The increase in database connection overhead, in milliseconds, for each additional record retrieved.

The network latency, in milliseconds, for retrieving any number of records.

The database connection overhead, in milliseconds, for retrieving any number of records.

The rate at which the total time increases, in milliseconds per record, for the first 8 records retrieved.

Cevabı ve açıklamayı göster

Cevap: The database connection overhead, in milliseconds, for retrieving any number of records.

Cevap

The database connection overhead, in milliseconds, for retrieving any number of records.

In the linear model

T = 0.06r + 40

, the constant term of

40

represents the total setup time in milliseconds, which is independent of the number of records

r

retrieved. The database connection overhead is

20\%

of this total setup time, which equals

0.20 \times 40 = 8

milliseconds. Since it is part of the constant term, it represents a fixed overhead of

8

milliseconds for retrieving any number of records.

Adım Adım Çözüm

1

Identify the constant term in the linear model

T = 0.06r + 40

.

The constant term (y-intercept) is

40

, which represents the total setup time of

40

milliseconds when

r = 0

records are retrieved.

In a linear relationship of the form

y = mx + b

, the constant term

b

represents the initial value or y-intercept.

2

Calculate the database connection overhead.

The database connection overhead is

20\%

of the

40

milliseconds total setup time, which is

0.20 \times 40 = 8

milliseconds.

The problem states that the database connection overhead is 20% of the total setup time.

3

Interpret the calculated value of 8 in the context of the model.

Since the

40

milliseconds is a constant term that does not depend on the number of records

r

, the

8

milliseconds represents a fixed database connection overhead for any number of records retrieved.

The constant term in a linear model represents a fixed value that does not scale with the independent variable.

Anahtar Kavram

Interpreting the constant term (y-intercept) and its components in a linear relationship context.

Soru 30Soru →

During a dry season, the volume of water $V$ , in millions of gallons, in a municipal reservoir $d$ days after the start of a monitoring period is modeled by the equation $V = 248 - 1.8d$ , where $0 \leq d \leq 60$ . According to the model, at what rate, in gallons per minute, is water leaving the reservoir?

Cevabı ve açıklamayı göster

Cevap: 1250

Cevap

The reservoir is losing water at a rate of

1,250

gallons per minute.

The slope of the linear equation

V = 248 - 1.8d

is

-1.8

, indicating a decrease of

1.8

million gallons of water per day. To express this rate in gallons per minute, we perform the unit conversion. First, convert

1.8

million gallons to gallons:

1.8 \times 1,000,000 = 1,800,000

gallons. Second, convert

1

day to minutes:

24 \text{ hours/day} \times 60 \text{ minutes/hour} = 1,440

minutes. Finally, divide the total gallons by the total minutes:

\frac{1,800,000 \text{ gallons}}{1,440 \text{ minutes}} = 1,250

gallons per minute.

Adım Adım Çözüm

1

Identify the daily rate of water leaving the reservoir from the given linear equation.

The daily rate is

1.8

million gallons per day.

The slope of the linear equation

V = 248 - 1.8d

is

-1.8

, which represents the rate of change of the volume with respect to time in days. The negative sign indicates a decrease, meaning water is leaving at a rate of

1.8

million gallons per day.

2

Convert the volume from million gallons to gallons.

1,800,000

gallons.

To convert from millions of gallons to gallons, multiply the value by

1,000,000

, resulting in

1.8 \times 1,000,000 = 1,800,000

.

3

Convert the time unit from days to minutes.

1,440

minutes.

There are

24

hours in a day and

60

minutes in an hour, so

1 \text{ day} = 24 \times 60 = 1,440 \text{ minutes}

.

4

Calculate the rate in gallons per minute.

1,250

gallons per minute.

Divide the volume in gallons by the time in minutes:

\frac{1,800,000 \text{ gallons}}{1,440 \text{ minutes}} = 1,250

.

Anahtar Kavram

Interpreting the slope of a linear relationship in context and performing unit conversions on rates.

Soru 31Soru →

A commercial 3D printer uses a polymer filament to print prototype parts. The remaining mass of the filament spool, $M$ , in grams, is modeled by a linear function of the total printing time, $t$ , in minutes. When the printer has been running for $20$ minutes, the spool has $850$ grams of filament remaining. When the printer has been running for $50$ minutes, the spool has $685$ grams of filament remaining. According to the model, how many grams of filament does the printer use per minute?

Cevabı ve açıklamayı göster

Cevap: 5.5

Cevap

The printer uses 5.5 grams of filament per minute.

The relationship between the printing time

t

and the remaining mass of the filament

M

is linear. The slope of this line represents the change in filament mass per unit of time. Based on the given coordinates

(20, 850)

and

(50, 685)

, the slope is calculated as

\frac{685 - 850}{50 - 20} = \frac{-165}{30} = -5.5

grams per minute. This means that the remaining mass decreases by

5.5

grams for each additional minute the printer runs, which corresponds to a usage rate of

5.5

grams of filament per minute.

Adım Adım Çözüm

1

Identify the data points representing time and remaining filament mass from the context.

The two points are

(20, 850)

and

(50, 685)

.

These points allow us to calculate the rate at which the mass of the filament is changing over time.

2

Calculate the change in remaining mass and the change in printing time.

Change in mass is

685 - 850 = -165

grams. Change in time is

50 - 20 = 30

minutes.

Finding the differences in the dependent variable (mass) and independent variable (time) is the standard method to calculate a rate of change.

3

Calculate the rate of filament usage per minute.

\frac{-165\text{ grams}}{30\text{ minutes}} = -5.5

grams per minute. The rate of usage is the positive magnitude, which is

5.5

grams per minute.

The slope of the linear relationship is negative because the mass is decreasing, but the rate of consumption/usage is represented as a positive quantity.

Anahtar Kavram

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

Soru 32Soru →

An agricultural drone is used to spray fertilizer on a crop field. The volume of fertilizer remaining in the drone's tank, $F$ , in liters, after the drone has been spraying for $t$ minutes is modeled by the equation $F = 240 - 3.2t$ , where $0 \leq t \leq 75$ . What is the best interpretation of the number $3.2$ in this context?

The initial volume of fertilizer, in liters, in the tank before spraying begins.

The number of minutes required for the drone to spray

1

liter of fertilizer.

The volume of fertilizer, in liters, sprayed from the tank each minute.

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

Cevabı ve açıklamayı göster

Cevap: The volume of fertilizer, in liters, sprayed from the tank each minute.

Cevap

The volume of fertilizer, in liters, sprayed from the tank each minute.

In the linear relationship

F = 240 - 3.2t

, the variable

t

represents time in minutes, and

F

represents the remaining volume in liters. The coefficient of

t

, which is

-3.2

, represents the rate of change of the volume of fertilizer in the tank. A slope of

-3.2

means that the volume decreases by

3.2

liters for every additional minute of spraying. Therefore, the number

3.2

represents the volume of fertilizer, in liters, that is sprayed from the tank each minute.

Adım Adım Çözüm

1

Identify the structure of the linear equation.

The equation is of the form

y = mx + b

, or in this case,

F = -3.2t + 240

, where the slope is

-3.2

and the

F

-intercept (initial value) is

240

.

Analyzing the components of a linear function helps determine what each constant represents in context.

2

Determine the meaning of the slope in context.

The slope is the coefficient of the independent variable

t

(time in minutes). The value

-3.2

indicates that for each increase of

1

minute in spraying time, the remaining volume of fertilizer,

F

, decreases by

3.2

liters.

Slope represents the rate of change of the dependent variable relative to the independent variable.

3

Interpret the absolute value of the slope.

The value

3.2

represents the magnitude of this rate, which is the amount of fertilizer sprayed out of the tank per minute (

3.2

liters per minute).

Matching the mathematical rate of change to the corresponding verbal description provides the correct interpretation.

Anahtar Kavram

Slope in a linear model represents the constant rate of change of the dependent variable per unit increase of the independent variable.

Tahmini Süre:1m 15s

Soru 33Soru →

A commercial drone delivery service models the total power remaining in a drone's battery, $P(w)$ , as a percentage, after carrying a payload of weight $w$ kilograms for a fixed delivery distance of 10 kilometers. The relationship is modeled by the linear function $P(w) = -8.5w + 92$ . Which of the following is the best interpretation of the value 92 in this context?

The decrease in the remaining battery power, as a percentage, for each increase of 1 kilogram in payload weight.

The remaining battery power, as a percentage, after the drone carries a payload of 1 kilogram for the delivery distance.

The remaining battery power, as a percentage, after the drone carries a payload of 0 kilograms for the delivery distance.

The payload weight, in kilograms, for which the remaining battery power after the delivery is 0 percent.

Cevabı ve açıklamayı göster

Cevap: The remaining battery power, as a percentage, after the drone carries a payload of 0 kilograms for the delivery distance.

Cevap

The remaining battery power, as a percentage, after the drone carries a payload of 0 kilograms for the delivery distance.

The linear equation is written in the slope-intercept form

P(w) = mw + b

, where

b = 92

is the vertical intercept (y-intercept). This intercept represents the value of the dependent variable,

P(w)

, when the independent variable,

w

, is equal to

0

. In this context,

w

is the payload weight in kilograms and

P(w)

is the remaining battery power as a percentage. Thus, the value

92

represents the remaining battery power, as a percentage, when the drone carries a payload of

0

kilograms.

Adım Adım Çözüm

1

Identify the structure of the linear function and the component being interpreted.

The linear equation is given in slope-intercept form,

P(w) = mw + b

, where

m = -8.5

is the slope and

b = 92

is the

P(w)

-intercept (y-intercept).

Understanding the components of a linear function allows us to relate them to their contextual definitions.

2

Evaluate the value of the function at the

P(w)

-intercept where the independent variable is zero.

Substitute

w = 0

into the equation:

P(0) = -8.5(0) + 92 = 92

.

The y-intercept represents the value of the dependent variable when the independent variable is equal to 0.

3

Interpret the meaning of

w = 0

and

P(0) = 92

in the given context.

Since

w

represents the payload weight in kilograms and

P(w)

represents the remaining battery power as a percentage,

w = 0

corresponds to a payload of 0 kilograms, and

P(0) = 92

corresponds to a remaining battery power of 92%.

Applying the units and definitions of the variables completes the contextual interpretation of the constant.

Anahtar Kavram

Interpreting the y-intercept of a linear function in a real-world context

Soru 34Soru →

A researcher monitors the temperature of a chemical compound during an experiment. The temperature, $T$ , in degrees Celsius, of the compound $m$ minutes after heating begins can be modeled by a linear relationship. The temperature increases by $4.5$ degrees Celsius every $2$ minutes. If the temperature of the compound was $-12$ degrees Celsius when heating began, after how many minutes will the temperature of the compound reach $15$ degrees Celsius?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The temperature of the compound will reach

15

degrees Celsius after

12

minutes.

The correct answer is

12

because the temperature increases at a constant rate of

2.25

degrees Celsius per minute. Starting from an initial temperature of

-12

degrees Celsius, the equation is

T = 2.25m - 12

. Setting

T = 15

gives

15 = 2.25m - 12

, which simplifies to

27 = 2.25m

, resulting in

m = 12

.

Adım Adım Çözüm

1

Calculate the constant rate of temperature increase per minute.

2.25

degrees Celsius per minute

This rate represents the slope of the linear relationship between time and temperature.

2

Formulate the linear equation representing the temperature

T

as a function of the elapsed minutes

m

.

T = 2.25m - 12

The slope is

2.25

and the vertical intercept is the initial temperature of

-12

degrees Celsius.

3

Set

T = 15

and solve the linear equation for

m

.

m = 12

This determines the exact number of minutes needed for the temperature to reach

15

degrees Celsius.

Anahtar Kavram

Interpreting and solving linear equations modeled from context

Soru 35Soru →

To restore a depleted wetland, conservationists pump water into a basin. The volume of water in the basin, $W$ , in thousands of gallons, $t$ hours after the pumping begins is modeled by the equation $W = 12.5t + 85$ . According to the model, how many hours of pumping are required for the volume of water in the basin to increase by $150$ thousand gallons?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

12

The linear equation is given in slope-intercept form,

W = mt + b

, where

m = 12.5

is the slope and

b = 85

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

m = 12.5

represents the rate at which water is pumped into the basin, which is

12.5

thousand gallons per hour. To find the number of hours required for the volume of water to increase by

150

thousand gallons, divide the total increase by the rate:

\frac{150}{12.5} = 12

hours.

Adım Adım Çözüm

1

Identify the rate of water volume increase per hour from the equation.

The rate is

12.5

thousand gallons per hour.

In the linear equation

W = 12.5t + 85

, the coefficient of

t

(the slope) represents the rate of change of the water volume with respect to time.

2

Set up a relation to find the time

t

for a volume increase of

150

thousand gallons.

12.5t = 150

The change in volume is equal to the rate of change multiplied by the time elapsed.

3

Solve the equation for

t

.

t = 12

Dividing both sides of the equation

12.5t = 150

by

12.5

isolates

t

.

Anahtar Kavram

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

Tahmini Süre:1m 30s

Soru 36Soru →

A municipal water treatment facility utilizes 3 identical backup filtration units to process stormwater runoff. The average volume of water, $V(t)$ , in thousands of gallons, remaining to be filtered per unit $t$ hours after the units are activated is modeled by the equation:

3V(t) + 14.4t = 162

where $0 \le t \le 10$ . Which of the following is the best interpretation of the number $14.4$ in this context?

The initial total volume of water, in thousands of gallons, remaining to be filtered by all 3 units combined.

The rate, in thousands of gallons per hour, at which water is filtered by each individual unit, which is 4.8.

The rate, in thousands of gallons per hour, at which water is filtered by all 3 units combined.

The rate, in thousands of gallons per hour, at which water is filtered by each individual unit, which is 4.6.

Cevabı ve açıklamayı göster

Cevap: The rate, in thousands of gallons per hour, at which water is filtered by all 3 units combined.

Cevap

The rate, in thousands of gallons per hour, at which water is filtered by all 3 units combined.

The correct answer is the option stating that 14.4 represents the rate, in thousands of gallons per hour, at which water is filtered by all 3 units combined. In the given equation

3V(t) + 14.4t = 162

, we can isolate the total volume remaining,

3V(t)

, as

3V(t) = 162 - 14.4t

. In this form, the rate of change of the total volume is

-14.4

, indicating that the combined remaining volume decreases by 14.4 thousand gallons each hour.

Adım Adım Çözüm

1

Express the total volume of water remaining to be filtered in terms of the variable

V(t)

.

Since

V(t)

is the average volume of water remaining per unit and there are 3 identical units, the total volume of water remaining to be filtered by all 3 units combined is

3V(t)

.

This allows us to analyze the total filtration system rather than the average per unit.

2

Isolate the term representing the total volume remaining,

3V(t)

, in the given equation.

3V(t) = 162 - 14.4t

This puts the equation into a standard linear form

y = mx + b

where

y = 3V(t)

is the dependent variable (total volume remaining) and

x = t

is the independent variable (time in hours).

3

Interpret the slope and the y-intercept of the isolated linear equation.

The constant term 162 is the initial total volume of water (in thousands of gallons) remaining at

t = 0

. The coefficient of

t

, which is

-14.4

, represents the change in the total volume remaining per hour.

In a linear model

y = mx + b

, the slope

m

represents the rate of change of the dependent variable per unit increase of the independent variable.

4

Determine the contextual meaning of the absolute value of the rate of change, 14.4.

A rate of change of

-14.4

thousand gallons per hour means that the total volume of water remaining decreases by 14.4 thousand gallons each hour. Therefore, 14.4 is the rate, in thousands of gallons per hour, at which water is filtered by all 3 units combined.

The rate at which the remaining volume decreases is equal to the rate at which the system filters the water.

Anahtar Kavram

Interpreting coefficients and constants in a linear relationship within a real-world context, particularly when variables represent averages or totals.

Soru 37Soru →

A ride-share driver uses a mobile application to track their daily net earnings. The driver's net earnings, $E$ , in dollars, after completing $n$ rides in a day can be modeled by the equation $E = 18n - 25$ . Which of the following is the best interpretation of the number $25$ in this context?

The driver's net earnings, in dollars, for each completed ride.

The driver's daily starting cost, in dollars, before any rides are completed.

The driver's net earnings, in dollars, after completing exactly one ride.

The number of completed rides for which the driver's net earnings are equal to 0 dollars.

Cevabı ve açıklamayı göster

Cevap: The driver's daily starting cost, in dollars, before any rides are completed.

Cevap

The driver's daily starting cost, in dollars, before any rides are completed.

In the linear equation

E = 18n - 25

, the variable

E

represents net earnings and

n

represents the number of rides. The constant term

-25

represents the value of

E

when

n = 0

. In this context, starting with

-25

dollars in net earnings means the driver has a daily starting cost or fee of

25

dollars before completing any rides.

Adım Adım Çözüm

1

Identify the component of the linear equation

E = 18n - 25

associated with the number

25

.

The number

25

is part of the constant term

-25

in the equation.

Linear equations in context are typically written in the form

y = mx + b

, where

m

is the slope and

b

is the y-intercept.

2

Determine the contextual meaning of the constant term (y-intercept) in this model.

The constant term

-25

represents the value of

E

when

n = 0

.

Setting the independent variable

n

(number of rides) to

0

gives the starting value of the dependent variable

E

(net earnings).

3

Interpret the negative value in terms of daily operations.

An initial net earning of

-25

dollars means the driver starts with a cost or loss of

25

dollars.

A negative starting value indicates a cost or fee that must be paid before any revenue is generated.

Anahtar Kavram

Interpreting the y-intercept of a linear model in context.

Soru 38Soru →

A botanist models the rate of water transpiration of a plant species under various temperature conditions. The table below shows the estimated transpiration rate, $R$ , in milligrams of water per square decimeter of leaf area per hour ( $\text{mg}/(\text{dm}^2\cdot\text{h})$ ), at various ambient temperatures, $t$ , in degrees Celsius ( $^\circ\text{C}$ ).

Temperature ( $t$ )	Transpiration rate ( $R$ )
$15$	$84.5$
$20$	$108.5$
$25$	$132.5$
$30$	$156.5$

The relationship between the ambient temperature and the transpiration rate is linear. Based on the model, what is the estimated increase in the transpiration rate, in milligrams of water per square meter of leaf area per minute, for each increase of $1$ degree Celsius in the ambient temperature? (Note: $1\text{ square meter} = 100\text{ square decimeters}$ )

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

To find the estimated increase in transpiration rate per degree Celsius in the new units, we first determine the rate of change in the original units from the table. The change in temperature is

20 - 15 = 5^\circ\text{C}

, and the corresponding change in transpiration rate is

108.5 - 84.5 = 24\text{ mg}/(\text{dm}^2\cdot\text{h})

. The rate of change is

\frac{24}{5} = 4.8\text{ mg}/(\text{dm}^2\cdot\text{h})

per

1^\circ\text{C}

temperature increase. Converting this rate to square meters, we multiply by

100

because

1\text{ square meter} = 100\text{ square decimeters}

, giving

480\text{ mg}/(\text{m}^2\cdot\text{h})

. Finally, to convert to minutes, we divide by

60

because

1\text{ hour} = 60\text{ minutes}

, which yields

\frac{480}{60} = 8\text{ mg}/(\text{m}^2\cdot\text{min})

per

1^\circ\text{C}

temperature increase.

Adım Adım Çözüm

1

Find the rate of change of the transpiration rate with respect to temperature from the given data table.

Slope =

4.8\text{ mg}/(\text{dm}^2\cdot\text{h})

per

1^\circ\text{C}

The rate of change represents the increase in transpiration rate for each

1^\circ\text{C}

increase in temperature in the original units.

2

Convert the unit of area in the rate of change from square decimeters to square meters.

Rate =

480\text{ mg}/(\text{m}^2\cdot\text{h})

per

1^\circ\text{C}

Since

1\text{ square meter} = 100\text{ square decimeters}

, the rate per square meter is

100

times the rate per square decimeter.

3

Convert the unit of time in the rate of change from hours to minutes.

Rate =

8\text{ mg}/(\text{m}^2\cdot\text{min})

per

1^\circ\text{C}

Since there are

60

minutes in an hour, dividing the hourly rate by

60

gives the rate per minute.

Anahtar Kavram

Interpreting the slope of a linear relationship in context and performing unit conversions.

Tahmini Süre:2m 30s

Soru 39Soru →

A commercial hydroponic farm monitors the volume of nutrient solution in a reservoir. The table below shows the volume of solution remaining, $V$ , in liters, after $t$ hours of operation.

Time ( $t$ , hours)	Volume ( $V$ , liters)
0	450
10	432
20	414
30	396

The relationship between the time and the volume of remaining solution can be modeled by a linear equation. Which of the following is the best interpretation of the slope of the graph of this equation in the $tV$ -plane?

The nutrient solution is being depleted at a rate of 1.8 liters per hour.

The initial volume of the nutrient solution in the reservoir is 450 liters.

The reservoir will be completely empty after 250 hours.

The volume of the nutrient solution decreases by 1 liter every 1.8 hours.

Cevabı ve açıklamayı göster

Cevap: The nutrient solution is being depleted at a rate of 1.8 liters per hour.

Cevap

The nutrient solution is being depleted at a rate of 1.8 liters per hour.

The slope of a linear function represents the constant rate of change of the dependent variable with respect to the independent variable. By selecting two data points from the table, such as

(0, 450)

and

(10, 432)

, the slope is calculated as

\frac{432 - 450}{10 - 0} = -1.8

. Since the volume

V

is measured in liters and the time

t

is in hours, this slope represents a decrease of

1.8

liters of nutrient solution per hour of operation. Therefore, the nutrient solution is being depleted at a rate of

1.8

liters per hour.

Adım Adım Çözüm

1

Calculate the slope using two points from the table.

Using the points

(0, 450)

and

(10, 432)

, the slope is

\frac{432 - 450}{10 - 0} = \frac{-18}{10} = -1.8

.

The slope of a linear relationship is given by the change in the dependent variable (Volume) divided by the change in the independent variable (Time).

2

Interpret the unit of the slope in context.

The unit of the slope is liters per hour (

\text{L/h}

), representing a change of

-1.8

liters for every hour of operation.

Determining the units of the rate helps verify what physical quantity the slope represents.

3

Relate the negative sign of the slope to the context of depletion.

A negative slope of

-1.8

means the volume of nutrient solution decreases by

1.8

liters each hour, which translates to a depletion rate of

1.8

liters per hour.

Connecting the mathematical sign to the real-world action clarifies whether the quantity is increasing or decreasing.

Anahtar Kavram

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

Tahmini Süre:1m 30s

Soru 40Soru →

A commercial bakery uses a mixing bowl that initially contains some flour. A machine adds flour to the bowl at a constant rate. After the machine has been running for $8$ minutes, the total mass of the flour in the bowl is $14$ kilograms. After the machine has been running for $20$ minutes, the total mass of the flour in the bowl is $29$ kilograms. The relationship between the total mass of the flour in the bowl, $M$ , in kilograms, and the time the machine has been running, $t$ , in minutes, is linear. What is the mass of the flour, in kilograms, in the bowl before the machine starts running?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4

The relationship between the total mass of the flour,

M

, and the time,

t

, is linear and can be represented by the equation

M = mt + b

, where

m

is the rate at which flour is added and

b

is the initial mass of the flour in the bowl. Using the two given points,

(8, 14)

and

(20, 29)

, the slope

m

is calculated as

m = \frac{29 - 14}{20 - 8} = \frac{15}{12} = 1.25

kilograms per minute. Substituting

m = 1.25

and the point

(8, 14)

into the linear equation gives

14 = 1.25(8) + b

, which simplifies to

14 = 10 + b

. Solving for

b

yields

b = 4

. Therefore, the mass of the flour in the bowl before the machine starts running is

4

kilograms.

Adım Adım Çözüm

1

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

The rate is

1.25

kilograms per minute.

The slope of a linear relationship represents the constant rate at which flour is added to the bowl.

2

Set up the linear equation and solve for the y-intercept.

The initial mass is

4

kilograms.

The y-intercept represents the initial mass of the flour in the bowl at

t = 0

minutes.

Anahtar Kavram

Interpreting the y-intercept of a linear relationship in context

Interpreting Linear Relationships in Context

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