Algebra Alıştırma Soruları — SAT — Sayfa 15

Soru 281Soru →

During a chemical reaction, the temperature $T$ , in degrees Celsius, of a solution $s$ seconds after the reaction begins is modeled by the equation $T = -0.04(s - 150) + 92$ , where $150 \le s \le 900$ . According to the model, how many minutes does it take for the temperature of the solution to decrease by $12$ degrees Celsius?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

To find the number of minutes it takes for the temperature to decrease by

12^\circ\text{C}

, we first determine the rate of temperature change from the linear model. The equation is given in the form

T = m(s - s_0) + T_0

, where the slope

m = -0.04

represents the rate of change of temperature in degrees Celsius per second. Thus, the temperature decreases at a rate of

0.04^\circ\text{C}

per second. To achieve a total decrease of

12^\circ\text{C}

, the time in seconds required is

\frac{12}{0.04} = 300

seconds. Converting

300

seconds to minutes gives

\frac{300}{60} = 5

minutes.

Adım Adım Çözüm

1

Identify the rate of change from the linear equation.

The rate of temperature decrease is

0.04^\circ\text{C}

per second.

The slope of the linear equation

T = -0.04(s - 150) + 92

is

-0.04

, which represents a change of

-0.04^\circ\text{C}

for every

1

second increase in time.

2

Calculate the time in seconds for a decrease of

12^\circ\text{C}

.

300

seconds

Divide the target temperature change of

-12^\circ\text{C}

by the rate of change of

-0.04^\circ\text{C}

per second:

\frac{-12}{-0.04} = 300

seconds.

3

Convert the time from seconds to minutes.

5

minutes

Since there are

60

seconds in

1

minute, divide

300

seconds by

60

:

\frac{300}{60} = 5

minutes.

Anahtar Kavram

Interpreting Linear Relationships in Context

Soru 282Soru →

A nutritionist is designing a meal plan containing $x$ grams of protein and $y$ grams of carbohydrates. The meal plan must satisfy the following system of inequalities:

\begin{aligned} y &\ge 1.5x + 15 \\ 3x + 2y &\le 120 \end{aligned}

What is the maximum possible number of grams of protein, $x$ , that can be included in the meal plan?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The maximum possible number of grams of protein that can be included is 15.

To find the maximum possible value of

x

, we determine the region defined by the system of inequalities. The system restricts the values to the region above the line

y = 1.5x + 15

and below the line

3x + 2y = 120

. Since the first inequality limits

y

from below and the second limits

y

from above, the feasible region narrows as

x

increases, terminating at the intersection of the two boundary lines. Substituting

y = 1.5x + 15

into

3x + 2y = 120

gives

3x + 2(1.5x + 15) = 120

. Simplifying this yields

3x + 3x + 30 = 120

, which simplifies further to

6x = 90

, giving

x = 15

. Thus, the maximum value of

x

is 15.

Adım Adım Çözüm

1

Identify the boundary lines of the system of inequalities.

The boundary lines are

y = 1.5x + 15

and

3x + 2y = 120

.

The maximum value of

x

under these linear constraints occurs at the intersection of the boundary lines of the feasible region.

2

Substitute the expression for

y

from the first boundary equation into the second equation.

3x + 2(1.5x + 15) = 120

This allows us to solve for

x

by eliminating

y

.

3

Simplify the equation and solve for

x

.

3x + 3x + 30 = 120 \implies 6x + 30 = 120 \implies 6x = 90 \implies x = 15

.

Solving the linear equation gives the

x

-coordinate of the intersection point.

4

Verify that this point lies in the feasible region and represents the maximum possible value of

x

.

At

x=15

,

y=37.5

. Since

y \ge 1.5x + 15

restricts the region above the line and

3x + 2y \le 120

restricts it below the line, the region lies to the left of the intersection point

(15, 37.5)

. Thus, the maximum value of

x

is 15.

Confirming the geometry of the feasible region ensures the intersection point is indeed the maximum value.

Anahtar Kavram

Solving systems of linear inequalities to find the boundaries and extreme values of a feasible region.

Soru 283Soru →

Consider the system of inequalities below:

\begin{aligned} 3x + y &> 5 \\ x - 2y &> 4 \end{aligned}

Which of the following coordinate pairs $(x, y)$ is a solution to the system?

(3, 1)

(2, -1)

(4, -2)

(3, 2)

Cevabı ve açıklamayı göster

Cevap: $(4, -2)$

Cevap

(4, -2)

The coordinate pair

(4, -2)

is the correct answer because substituting these values into both inequalities of the system produces true statements:

3(4) + (-2) = 10 > 5

and

4 - 2(-2) = 8 > 4

.

Adım Adım Çözüm

1

Substitute the coordinates of the candidate point into the first inequality,

3x + y > 5

.

For

(4, -2)

, we get

3(4) + (-2) = 12 - 2 = 10

. Since

10 > 5

, the first inequality is satisfied.

A coordinate pair must satisfy both inequalities in the system to be a solution.

2

Substitute the coordinates of the candidate point into the second inequality,

x - 2y > 4

.

For

(4, -2)

, we get

4 - 2(-2) = 4 + 4 = 8

. Since

8 > 4

, the second inequality is also satisfied.

Since both inequalities are true for

(4, -2)

, it is a valid solution to the system.

Anahtar Kavram

A coordinate pair

(x, y)

is a solution to a system of linear inequalities if and only if it makes all inequalities in the system true when substituted.

Tahmini Süre:1m 30s

Soru 284Soru →

A chemist needs to mix a $10\%$ acid solution with a $30\%$ acid solution to create a $100$ -milliliter mixture. If the final mixture must be $18\%$ acid, how many milliliters of the $10\%$ acid solution should the chemist use?

40

60

70

80

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

60 milliliters

The correct answer is 60 milliliters. Defining

x

as the volume of the

10\%

solution and

y

as the volume of the

30\%

solution gives the system of equations

x + y = 100

and

0.10x + 0.30y = 18

. Substituting

y = 100 - x

into the second equation yields

0.10x + 30 - 0.30x = 18

. Simplifying this equation gives

-0.20x = -12

, which simplifies to

x = 60

.

Adım Adım Çözüm

1

Define variables for the volume of each solution and set up the system of equations representing the total volume and the total amount of pure acid.

Let

x

be the number of milliliters of the

10\%

acid solution, and let

y

be the number of milliliters of the

30\%

acid solution. The system is:

\begin{aligned} x + y &= 100 \\ 0.10x + 0.30y &= 18 \end{aligned}

To represent the physical relationships between the two solutions mathematically.

2

Solve the first equation for

y

in terms of

x

and substitute this expression into the second equation.

y = 100 - x

0.10x + 0.30(100 - x) = 18

To reduce the system to a single linear equation in terms of

x

.

3

Distribute the coefficients, combine like terms, and solve for

x

.

0.10x + 30 - 0.30x = 18

-0.20x + 30 = 18

-0.20x = -12

x = 60

To isolate

x

and determine the volume of the

10\%

acid solution.

Anahtar Kavram

Solving systems of linear equations in two variables using substitution or elimination.

Soru 285Soru →

A water utility company charges a flat monthly connection fee plus a constant rate per gallon of water consumed. In a certain month, a household that consumed $2,800$ gallons of water was billed a total of $62.00$ dollars, and a household that consumed $4,500$ gallons of water was billed a total of $87.50$ dollars. What is the flat monthly connection fee, in dollars, charged by the water utility company?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The flat monthly connection fee charged by the water utility company is 20 dollars.

The flat monthly connection fee is 20 dollars. A linear relationship can be modeled by the equation

y = mx + b

, where

x

represents the water consumed in gallons,

y

represents the total bill in dollars,

m

represents the rate per gallon, and

b

represents the flat monthly connection fee. Using the data for the two households, we get the points

(2800, 62)

and

(4500, 87.5)

. The rate per gallon is the slope of the line passing through these points:

m = \frac{87.5 - 62}{4500 - 2800} = \frac{25.5}{1700} = 0.015

dollars per gallon. Substituting this slope and the point

(2800, 62)

into the equation gives

62 = 0.015(2800) + b

, which simplifies to

62 = 42 + b

. Solving for

b

yields

20

dollars.

Adım Adım Çözüm

1

Define the linear relationship using the variables

x

for water consumed in gallons and

y

for the total bill in dollars.

The relationship can be written as

y = mx + b

, where

m

represents the cost per gallon of water and

b

represents the flat monthly connection fee.

This establishes the linear framework needed to solve for the unknown constants using the provided data points.

2

Calculate the constant rate of change (slope

m

) using the coordinates of the two households:

(2800, 62)

and

(4500, 87.5)

.

m = \frac{87.5 - 62}{4500 - 2800} = \frac{25.5}{1700} = 0.015

dollars per gallon.

The slope of a linear function represents the constant rate of change between the two variables.

3

Substitute the calculated slope

m = 0.015

and the point

(2800, 62)

into the equation

y = mx + b

to find the flat fee

b

.

62 = 0.015(2800) + b \implies 62 = 42 + b \implies b = 20

.

The flat monthly connection fee corresponds to the vertical intercept (

y

-intercept) of the linear equation.

Anahtar Kavram

Linear Equations in Two Variables

Soru 286Soru →

A custom t-shirt printing company uses the function $C(n) = 12n + 45$ to determine the total cost, in dollars, for an order of $n$ t-shirts. What is the best interpretation of the value $45$ in this function?

The price, in dollars, charged for each printed t-shirt.

The setup fee, in dollars, charged for the entire order regardless of the number of t-shirts printed.

The total cost, in dollars, for an order of exactly

12

t-shirts.

The minimum number of t-shirts that must be purchased in a single order.

Cevabı ve açıklamayı göster

Cevap: The setup fee, in dollars, charged for the entire order regardless of the number of t-shirts printed.

Cevap

The setup fee, in dollars, charged for the entire order regardless of the number of t-shirts printed.

In the linear function

C(n) = 12n + 45

, the term

45

is a constant that does not depend on the variable

n

(the number of t-shirts). Therefore, it represents the initial or flat setup cost in dollars for placing an order, which is charged regardless of the number of t-shirts printed.

Adım Adım Çözüm

1

Identify the structure of the linear function

C(n) = 12n + 45

.

The function is in the slope-intercept form

y = mx + b

, where

m = 12

is the slope and

b = 45

is the y-intercept.

Understanding the components of a linear function helps determine which part of the equation corresponds to which real-world quantity.

2

Determine the meaning of the constant term (y-intercept) in the context of the problem.

The constant term

45

represents the value of

C(n)

when

n = 0

, which is the cost before any t-shirts are added, representing a flat setup fee.

Evaluating the function at

n = 0

isolates the base cost or initial charge.

Anahtar Kavram

Interpreting Linear Relationships in Context

Soru 287Soru →

A landscape designer purchased a total of

35

plants, consisting of boxwood shrubs and fern plants, for a total of \

510$. Each boxwood shrub cost \

18

, and each fern plant cost \$

12

. If the designer purchased

b

boxwood shrubs and

f

fern plants, what is the value of

f - b$?

15

5

27

29

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The correct answer is 5, representing the difference between the 20 fern plants and 15 boxwood shrubs.

To find the value of

f - b

, we can set up a system of linear equations based on the given information. The total number of plants purchased is

35

, which gives the equation

b + f = 35

. The total cost of the plants is \

510$, with boxwood shrubs costing \

18

each and fern plants costing \$

12

each, giving the equation

18b + 12f = 510

. Solving the first equation for

f

gives

f = 35 - b

. Substituting this into the second equation yields

18b + 12(35 - b) = 510

. Distributing and simplifying gives

18b + 420 - 12b = 510

, which simplifies to

6b + 420 = 510

. Subtracting

420

from both sides gives

6b = 90

, so

b = 15

. Substituting

b = 15

back into

b + f = 35

gives

15 + f = 35

, so

f = 20

. The value of

f - b

is

20 - 15 = 5$.

Adım Adım Çözüm

1

Define variables and write the system of linear equations representing the problem.

Let

b

be the number of boxwood shrubs and

f

be the number of fern plants. The system of equations is:

\begin{aligned} b + f &= 35 \\ 18b + 12f &= 510 \end{aligned}

We need to translate the word problem into a mathematical system of two linear equations with two variables.

2

Express one variable in terms of the other from the first equation and substitute it into the second equation.

From

b + f = 35

, we get

f = 35 - b

.
Substituting this into the second equation gives:

18b + 12(35 - b) = 510

Substitution is a standard method to reduce a system of two equations to a single equation in one variable.

3

Solve the resulting single-variable equation for

b

.

18b + 420 - 12b = 510

6b + 420 = 510

6b = 90

b = 15

By distributing the coefficient and combining like terms, we isolate and solve for the variable b.

4

Substitute the value of

b

back into the first equation to solve for

f

, and then calculate

f - b

.

15 + f = 35 \Rightarrow f = 20

f - b = 20 - 15 = 5

Finding the individual values of both variables allows us to calculate the required difference.

Anahtar Kavram

Solving systems of linear equations in two variables using substitution or elimination, and evaluating a linear combination of the solution.

Alternatif Yöntem

Instead of substitution, the system can be solved using elimination. Multiply the first equation,

b + f = 35

, by

12

to get

12b + 12f = 420

. Subtract this from the second equation,

18b + 12f = 510

, to eliminate

f

:

(18b - 12b) + (12f - 12f) = 510 - 420

, which simplifies to

6b = 90

, giving

b = 15

. Then, find

f = 20

and compute

f - b = 5

.

Tahmini Süre:1m 30s

Soru 288Soru →

In the inequality $-2(3x - k) - 5(x + 1) > 17$ , $k$ is an integer constant. If the maximum integer value of $x$ that satisfies the inequality is $2$ , what is the least possible value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 23

Cevap

The correct answer is 23.

By simplifying the inequality to

x < \frac{2k - 22}{11}

, we establish that the upper bound of the solution interval must be strictly greater than

2

but less than or equal to

3

for

2

to be the maximum integer solution. Solving the resulting compound inequality

2 < \frac{2k - 22}{11} \le 3

yields

22 < k \le 27.5

. The smallest integer within this range is

23

.

Adım Adım Çözüm

1

Distribute the constants on the left side of the inequality.

-6x + 2k - 5x - 5 > 17

Applying the distributive property simplifies the expression and removes the parentheses.

2

Combine like terms on the left side and isolate the

x

term.

-11x + 2k - 5 > 17 \implies -11x > 22 - 2k

Grouping

x

terms together and moving the constant terms to the other side prepares the inequality for division.

3

Divide both sides by

-11

and flip the inequality sign.

x < \frac{22 - 2k}{-11} \implies x < \frac{2k - 22}{11}

Dividing by a negative number reverses the direction of the inequality sign from greater-than (

>

) to less-than (

<

).

4

Set up the inequality for the maximum integer solution to be

2

.

2 < \frac{2k - 22}{11} \le 3

For

2

to be the largest integer satisfying

x < L

(where

L

is the boundary),

2

must be strictly less than

L

, and

L

must be less than or equal to the next integer,

3

.

5

Solve the compound inequality for the parameter

k

.

22 < 2k - 22 \le 33 \implies 44 < 2k \le 55 \implies 22 < k \le 27.5

Multiplying all parts by 11, adding 22, and dividing by 2 isolates

k

.

6

Find the least integer value of

k

in the interval

(22, 27.5]

.

23

The integers that satisfy

22 < k \le 27.5

are

23, 24, 25, 26,

and

27

. The least of these values is

23

.

Anahtar Kavram

Solving linear inequalities in one variable involving parameter bounds, negative coefficients, and integer solution constraints.

Soru 289Soru →

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 290Soru →

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 291Soru →

In the $xy$ -plane, a line has a $y$ -intercept of $(0, d)$ and passes through the point $(4, d - 3)$ , where $d$ is a constant. If the line also passes through the point $(12, k)$ , which of the following expressions represents the value of $k$ ?

d + 9

d - 16

d - 9

d - 36

Cevabı ve açıklamayı göster

Cevap: $d - 9$

Cevap

d - 9

The correct answer represents the value of

k

as

d - 9

. The slope

m

of the line passing through

(0, d)

and

(4, d - 3)

is

\frac{(d - 3) - d}{4 - 0} = -\frac{3}{4}

. Using the

y

-intercept

(0, d)

, the equation of the line is

y = -\frac{3}{4}x + d

. Substituting the point

(12, k)

gives

k = -\frac{3}{4}(12) + d = d - 9

.

Adım Adım Çözüm

1

Calculate the slope of the line using the points

(0, d)

and

(4, d - 3)

.

m = -\frac{3}{4}

The slope formula is

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

. Substituting the given coordinates yields

m = \frac{(d - 3) - d}{4 - 0} = -\frac{3}{4}

.

2

Write the equation of the line using the slope-intercept form.

y = -\frac{3}{4}x + d

Since the line's

y

-intercept is

(0, d)

, the constant

b

in the slope-intercept equation

y = mx + b

is equal to

d

.

3

Substitute the point

(12, k)

into the equation of the line to solve for

k

.

k = d - 9

Substituting

x = 12

and

y = k

into

y = -\frac{3}{4}x + d

gives

k = -\frac{3}{4}(12) + d = -9 + d

, which simplifies to

d - 9

.

Anahtar Kavram

Determining the equation and coordinates of a line in the coordinate plane given its slope and intercepts.

Soru 292Soru →

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 293Soru →

A software programmer is writing test cases for a new application. The programmer must write at least 15 test cases in total, consisting of $x$ unit tests and $y$ integration tests. Each unit test takes 10 minutes to write, and each integration test takes 30 minutes to write. If the programmer has at most 300 minutes to write all the test cases, what is the maximum number of integration tests the programmer can write?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

7

Adım Adım Çözüm

1

Set up the system of inequalities representing the given constraints.

x + y \ge 15

and

10x + 30y \le 300

The total number of tests must be at least 15, and the total time taken by writing

x

unit tests (10 minutes each) and

y

integration tests (30 minutes each) cannot exceed 300 minutes.

2

Simplify the time inequality and combine it with the total test count constraint to isolate

y

.

x + 3y \le 30

. Substituting

x \ge 15 - y

into this inequality yields

(15 - y) + 3y \le 30 \implies 15 + 2y \le 30

.

Simplification and substitution help find the upper bound for the number of integration tests.

3

Solve for

y

and determine the maximum integer value.

2y \le 15 \implies y \le 7.5

. The largest integer satisfying this inequality is 7.

The number of integration tests must be a whole number, so we round down to the nearest integer.

4

Verify that a valid integer number of unit tests (

x

) exists when

y = 7

.

When

y = 7

, we get

x \ge 15 - 7 \implies x \ge 8

and

x + 3(7) \le 30 \implies x \le 9

. The integers

x = 8

and

x = 9

both satisfy the conditions.

We must confirm that the maximum value of

y

is achievable with an integer number of unit tests.

Anahtar Kavram

Systems of Linear Inequalities in Two Variables

Soru 294Soru →

In the $xy$ -plane, the graph of the linear equation $3x + 5y = c$ , where $c$ is a constant, passes through the point $(4, 3)$ . What is the $x$ -coordinate of the point on this graph where the $y$ -coordinate is $-3$ ?

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

The correct answer is 14.

To find the correct answer, first substitute the given point

(4, 3)

into the equation

3x + 5y = c

to determine the value of the constant

c

:

3(4) + 5(3) = 12 + 15 = 27

. Thus, the equation is

3x + 5y = 27

. Next, substitute

-3

for

y

in this equation to find the corresponding

x

-coordinate:

3x + 5(-3) = 27 \implies 3x - 15 = 27

. Adding

15

to both sides gives

3x = 42

, and dividing by

3

yields

x = 14

.

Adım Adım Çözüm

1

Substitute the point

(4, 3)

into the equation

3x + 5y = c

to solve for

c

.

c = 27

Since the point lies on the graph of the equation, its coordinates must satisfy the equation.

2

Substitute

y = -3

and

c = 27

into the equation

3x + 5y = c

.

3x - 15 = 27

We want to find the

x

-coordinate of the point on the line when the

y

-coordinate is

-3

.

3

Solve the equation

3x - 15 = 27

for

x

.

x = 14

Isolating

x

gives the

x

-coordinate of the point.

Anahtar Kavram

Using a known point on a line to find a constant coefficient or constant term, and using the resulting equation to find other coordinates.

Alternatif Yöntem

Alternatively, you can write the equation in slope-intercept form. Solving

3x + 5y = c

for

y

gives

y = -\frac{3}{5}x + \frac{c}{5}

. The slope of the line is

-\frac{3}{5}

. Since the slope is constant, the change in

y

divided by the change in

x

between

(4, 3)

and

(x, -3)

is equal to the slope:

\frac{-3 - 3}{x - 4} = -\frac{3}{5} \implies \frac{-6}{x - 4} = -\frac{3}{5}

. Cross-multiplying gives

-30 = -3(x - 4) \implies 10 = x - 4 \implies x = 14

.

Tahmini Süre:1m 30s

Soru 295Soru →

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 296Soru →

A logistics company uses two types of boxes, small and large, to ship items. A shipment of $8$ small boxes and $5$ large boxes has a total weight of $180$ pounds. A second shipment of $6$ small boxes and $10$ large boxes has a total weight of $260$ pounds. What is the weight, in pounds, of one large box?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The weight of one large box is 20 pounds.

The correct weight of one large box is 20 pounds. By formulating the system of linear equations representing the total weight of each shipment (

8s + 5L = 180

and

6s + 10L = 260

), we can eliminate

L

by multiplying the first equation by 2, resulting in

16s + 10L = 360

. Subtracting the second equation from this gives

10s = 100

, which solves to

s = 10

. Substituting

s = 10

back into

8s + 5L = 180

yields

80 + 5L = 180

, which simplifies to

5L = 100

, meaning

L = 20

.

Adım Adım Çözüm

1

Set up a system of two linear equations based on the shipments.

Let

s

be the weight of a small box and

L

be the weight of a large box. The system of equations is:

\begin{aligned} 8s + 5L &= 180 \\ 6s + 10L &= 260 \end{aligned}

Translating the verbal statements into mathematical symbols represents the problem systemically.

2

Multiply the first equation by 2 to align the coefficients of

L

.

16s + 10L = 360

This sets up the variable

L

to have the same coefficient in both equations, allowing for elimination.

3

Subtract the second equation from the modified first equation to solve for

s

.

(16s + 10L) - (6s + 10L) = 360 - 260

10s = 100

s = 10

Subtracting eliminates the variable

L

, leaving a single-variable linear equation to solve.

4

Substitute

s = 10

back into the first equation to solve for

L

.

8(10) + 5L = 180

80 + 5L = 180

5L = 100

L = 20

Substituting the value of the solved variable determines the value of the remaining variable.

Anahtar Kavram

Solving systems of linear equations in two variables using elimination or substitution.

Soru 297Soru →

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 298Soru →

In the $xy$ -plane, a system of inequalities consists of the following:

y > -\frac{1}{2}x + 2

y \leq 3x - 1

Which of the following coordinate pairs $(x, y)$ is a solution to this system?

(1, 1)

(2, 2)

(1, 4)

(2, 0)

Cevabı ve açıklamayı göster

Cevap: $(2, 2)$

Cevap

The coordinate pair

(2, 2)

The coordinate pair

(2, 2)

is the correct answer because substituting

x = 2

and

y = 2

into both inequalities yields true statements. For the first inequality,

2 > -\frac{1}{2}(2) + 2

simplifies to

2 > 1

, which is true. For the second inequality,

2 \leq 3(2) - 1

simplifies to

2 \leq 5

, which is also true. Since the point satisfies both inequalities, it lies in the solution region.

Adım Adım Çözüm

1

Understand the definition of a solution to a system of inequalities.

A coordinate pair

(x, y)

is a solution to a system of inequalities if and only if it satisfies both inequalities simultaneously when substituted.

This establishes the verification method for checking the options.

2

Substitute the coordinate pair

(2, 2)

into the first inequality:

y > -\frac{1}{2}x + 2

.

2 > -\frac{1}{2}(2) + 2 \implies 2 > -1 + 2 \implies 2 > 1

.

This determines if the coordinate pair satisfies the first boundary condition.

3

Substitute the coordinate pair

(2, 2)

into the second inequality:

y \leq 3x - 1

.

2 \leq 3(2) - 1 \implies 2 \leq 6 - 1 \implies 2 \leq 5

.

This determines if the coordinate pair satisfies the second boundary condition.

4

Conclude whether both statements are true.

Since

2 > 1

is true and

2 \leq 5

is true, the coordinate pair

(2, 2)

is a solution to the system.

Both conditions must be met for the coordinate pair to belong to the solution set.

Anahtar Kavram

Verifying coordinate solutions for systems of linear inequalities

Tahmini Süre:1m 30s

Soru 299Soru →

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 300Soru →

An online service provider offers two monthly subscription plans. Under Plan A, the customer pays a flat monthly fee of $C$ dollars. Under Plan B, the monthly cost, in dollars, is determined by the expression $1.5(20 - x) - 0.8(3x - 5)$ , where $x$ is the number of premium features the customer uses. The provider wants Plan B to be strictly cheaper than Plan A for any customer who uses more than 4 premium features. If $C$ is an integer, what is the minimum possible value of $C$ ?

Cevabı ve açıklamayı göster

Cevap: 19

Cevap

19

To find the minimum integer value of

C

, we simplify Plan B's cost expression to

34 - 3.9x

and set up the inequality

34 - 3.9x < C

. Solving for

x

by dividing by

-3.9

and reversing the inequality sign gives

x > \frac{34 - C}{3.9}

. For Plan B to be cheaper than Plan A for all customers using more than 4 features, the solution set

x > \frac{34 - C}{3.9}

must contain the interval

x > 4

. This requires the boundary point to be at most 4, so

\frac{34 - C}{3.9} \le 4

. Solving this inequality yields

C \ge 18.4

. The smallest integer value greater than or equal to

18.4

is 19.

Adım Adım Çözüm

1

Simplify the cost expression for Plan B

34 - 3.9x

To combine like terms and express Plan B's cost in standard linear form.

2

Set up the inequality stating Plan B is strictly cheaper than Plan A

34 - 3.9x < C

Plan B is cheaper than Plan A when its cost is less than

C

dollars.

3

Solve the inequality for

x

in terms of

C

x > \frac{34 - C}{3.9}

Isolating

x

allows us to find the threshold number of premium features, remembering to reverse the inequality direction when dividing by the negative coefficient

-3.9

.

4

Relate the threshold condition to the given minimum number of premium features

\frac{34 - C}{3.9} \le 4

For Plan B to be cheaper for any

x > 4

, the solution interval

x > \frac{34 - C}{3.9}

must cover the entire interval

x > 4

. Thus, the boundary point must be at most 4.

5

Solve the boundary inequality for

C

C \ge 18.4

Multiplying by

3.9

and isolating

C

gives the lower bound for the cost of Plan A.

6

Find the minimum integer value for

C

19

Since

C

must be an integer and at least

18.4

, the smallest integer that satisfies this inequality is 19.

Anahtar Kavram

Solving linear inequalities in one variable with parameter constraints and real-world conditions.

Algebra

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