Algebra Alıştırma Soruları — SAT — Sayfa 16

Soru 301Soru →

For a constant $a$ , the inequality $5x - a(3 - 2x) \ge 4$ has a solution set of the form $x \le d$ , where $d$ is a constant. Which of the following must be true about the value of $a$ ?

a < -\frac{5}{2}

a > -\frac{5}{2}

a > \frac{5}{2}

a < \frac{5}{2}

Cevabı ve açıklamayı göster

Cevap: $a < -\frac{5}{2}$

Cevap

The value of

a

must satisfy

a < -\frac{5}{2}

.

To find the correct range for

a

, we first expand the inequality

5x - a(3 - 2x) \ge 4

using the distributive property, which yields

5x - 3a + 2ax \ge 4

. Grouping the

x

terms gives

(5 + 2a)x \ge 3a + 4

. The problem states that the solution set is of the form

x \le d

. Because the inequality sign flipped from greater-than-or-equal-to (

\ge

) to less-than-or-equal-to (

\le

), the coefficient of

x

must be negative. Setting the coefficient

5 + 2a < 0

and solving for

a

gives

a < -\frac{5}{2}

.

Adım Adım Çözüm

1

Expand the inequality to separate the terms.

5x - 3a + 2ax \ge 4

Apply the distributive property to the term

-a(3 - 2x)

, paying close attention to the signs:

-a \times 3 = -3a

and

-a \times (-2x) = 2ax

.

2

Group and factor the terms containing

x

on the left side, and move the constant terms to the right side.

(5 + 2a)x \ge 3a + 4

Factor out

x

from the terms

5x

and

2ax

to isolate the variable, and add

3a

to both sides of the inequality.

3

Analyze the relationship between the coefficient of

x

and the inequality sign of the solution set.

5 + 2a < 0

The original inequality has a greater-than-or-equal-to sign (

\ge

), but the given solution set is of the form

x \le d

(less-than-or-equal-to). For the inequality sign to reverse when dividing both sides by the coefficient of

x

, the coefficient

(5 + 2a)

must be negative.

4

Solve the inequality for

a

.

a < -\frac{5}{2}

Subtract 5 from both sides to get

2a < -5

, then divide both sides by 2.

Anahtar Kavram

Solving linear inequalities in one variable with symbolic coefficients and applying the inequality sign reversal rule when multiplying or dividing by a negative value.

Soru 302Soru →

An athlete participates in a charity event where they earn money by swimming laps in a pool and running laps on a track. The athlete earns $s$ dollars for each lap swum and $r$ dollars for each lap run. Completing 10 laps in the pool and 15 laps on the track raises a total of $135. Completing 12 laps in the pool and 20 laps on the track raises a total of$ 176. What is the value of $r$ ?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

r

is 7.

The correct answer is 7. To find this, we translate the word problem into a system of two linear equations:

10s + 15r = 135

and

12s + 20r = 176

. Simplifying these equations by dividing by their common factors yields

2s + 3r = 27

and

3s + 5r = 44

. Multiplying the first equation by 3 gives

6s + 9r = 81

, and multiplying the second equation by 2 gives

6s + 10r = 88

. Subtracting the two equations eliminates

s

and gives

r = 7

.

Adım Adım Çözüm

1

Write the system of linear equations representing the athlete's earnings.

The system is:

10s + 15r = 135

12s + 20r = 176

To represent the relationships between the number of laps completed and the total amount raised.

2

Simplify both equations to work with smaller coefficients.

Divide the first equation by 5:

2s + 3r = 27

Divide the second equation by 4:

3s + 5r = 44

Simplifying equations reduces the likelihood of arithmetic errors.

3

Eliminate the variable

s

to solve for

r

.

Multiply the first simplified equation by 3 and the second by 2:

3(2s + 3r) = 3(27) \implies 6s + 9r = 81

2(3s + 5r) = 2(44) \implies 6s + 10r = 88

Subtract the first equation from the second:

(6s + 10r) - (6s + 9r) = 88 - 81 \implies r = 7

Using the elimination method allows us to solve directly for the target variable,

r

.

Anahtar Kavram

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

Soru 303Soru →

In the $xy$ -plane, the graph of the linear equation $y - 4 = m(x - 6)$ , where $m$ is a constant, has a $y$ -intercept of $(0, 7)$ . What is the $x$ -coordinate of the $x$ -intercept of the line?

-4

-2

14

8

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

The correct answer is 14. This is the x-coordinate of the point where the line crosses the x-axis, which is (14, 0).

To find the value of

m

, substitute the y-intercept

(0, 7)

into the equation:

7 - 4 = m(0 - 6)

, which simplifies to

3 = -6m

, so

m = -\frac{1}{2}

. The equation of the line is

y - 4 = -\frac{1}{2}(x - 6)

. The x-intercept occurs when

y = 0

. Substituting

y = 0

gives

-4 = -\frac{1}{2}(x - 6)

, which simplifies to

8 = x - 6

, so

x = 14

.

Adım Adım Çözüm

1

Substitute the y-intercept

(0, 7)

into the given equation to solve for the constant

m

.

7 - 4 = m(0 - 6) \implies 3 = -6m \implies m = -\frac{1}{2}

The y-intercept point

(0, 7)

must satisfy the equation of the line, allowing us to find the slope

m

.

2

Write the linear equation using the calculated value of

m

.

y - 4 = -\frac{1}{2}(x - 6)

Knowing the slope allows us to write the complete linear equation in point-slope form.

3

Find the x-intercept of the line by setting

y = 0

and solving for

x

.

0 - 4 = -\frac{1}{2}(x - 6) \implies -4 = -\frac{1}{2}(x - 6) \implies 8 = x - 6 \implies x = 14

The x-intercept is the point on the line where the y-coordinate is equal to 0.

Anahtar Kavram

Finding the intercepts and slope of a linear equation in two variables.

Alternatif Yöntem

Alternatively, write the equation in slope-intercept form

y = mx + b

. Since the y-intercept is

(0, 7)

, we have

b = 7

, so the equation is

y = mx + 7

. Since the line passes through the point

(6, 4)

(determined by the point-slope form

y - 4 = m(x - 6)

), substitute this point into the equation:

4 = m(6) + 7 \implies -3 = 6m \implies m = -\frac{1}{2}

. Thus, the equation is

y = -\frac{1}{2}x + 7

. Setting

y = 0

for the x-intercept gives

0 = -\frac{1}{2}x + 7 \implies \frac{1}{2}x = 7 \implies x = 14

.

Tahmini Süre:1m 30s

Soru 304Soru →

A manufacturer plans to produce two models of smartphones: Model X and Model Y. The assembly of each Model X smartphone requires $2.5$ hours, and the assembly of each Model Y smartphone requires $4$ hours. A total of $800$ hours of assembly time is allocated for producing these two models. If the manufacturer produces $120$ Model X smartphones, what is the maximum number of Model Y smartphones that can be produced using the remaining allocated assembly time?

125

128

150

275

Cevabı ve açıklamayı göster

Cevap: 125

Cevap

The maximum number of Model Y smartphones that can be produced is 125.

The correct answer is the value that satisfies the linear relation under the given constraints. Setting up the equation

2.5x + 4y = 800

and substituting

120

for

x

results in

300 + 4y = 800

. Subtracting

300

from both sides yields

4y = 500

, and dividing by

4

gives

125

Model Y smartphones.

Adım Adım Çözüm

1

Set up a linear equation representing the total assembly time.

2.5x + 4y = 800

, where

x

is the number of Model X smartphones and

y

is the number of Model Y smartphones.

The total assembly time is the sum of the time spent on Model X (

2.5

hours per phone) and Model Y (

4

hours per phone), which must equal the allocated

800

hours.

2

Substitute the given number of Model X smartphones into the equation.

2.5(120) + 4y = 800

, which simplifies to

300 + 4y = 800

.

The problem states that the manufacturer produces

120

Model X smartphones.

3

Isolate the variable representing Model Y smartphones to find its value.

4y = 500

, which simplifies to

y = 125

.

Subtracting

300

from both sides of the equation and then dividing by

4

isolates the variable

y

.

Anahtar Kavram

Linear Equations in Two Variables

Tahmini Süre:1m 30s

Soru 305Soru →

An online retailer uses a linear relationship to determine the shipping cost of an order based on the total weight of the items in the order. The table below shows the shipping cost, $y$ , in dollars, for an order with a total weight of $x$ pounds.

Weight ( $x$ pounds)	Shipping Cost ( $y$ dollars)
$2$	$9.50$
$6$	$17.50$

What is the shipping cost, in dollars, for an order with a total weight of $11$ pounds?

Cevabı ve açıklamayı göster

Cevap: 27.5

Cevap

The correct shipping cost for an order weighing 11 pounds is 27.5 dollars.

The linear relationship has a constant rate of change (slope). Using the coordinates from the table,

(2, 9.50)

and

(6, 17.50)

, the slope

m

is calculated as

\frac{17.50 - 9.50}{6 - 2} = \frac{8}{4} = 2

. Using the point-slope form, the equation of the line is

y - 9.50 = 2(x - 2)

, which simplifies to

y = 2x + 5.50

. Substituting

x = 11

gives

y = 2(11) + 5.50 = 27.5

.

Adım Adım Çözüm

1

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

m = 2

Determines the rate at which the shipping cost increases per additional pound of weight.

2

Formulate the linear equation representing the relationship.

y = 2x + 5.50

Establishes a functional relationship to calculate costs for any given weight.

3

Evaluate the function at the target weight of 11 pounds.

y = 27.5

Determines the final cost of shipping for the specified weight of 11 pounds.

Anahtar Kavram

Determining and evaluating linear functions represented in tabular form

Tahmini Süre:1m 30s

Soru 306Soru →

Company A and Company B both rent moving trucks. The total cost $A(d)$ , in dollars, for renting a truck from Company A for $d$ days is given by $A(d) = 35d + 75$ . The table below shows some values of the total cost $B(d)$ , in dollars, for renting a truck from Company B for $d$ days.

Days ( $d$ )	Total Cost ( $B(d)$ )
2	150
5	300

If the relationship between $d$ and $B(d)$ is linear, for how many days of rental will the total cost at Company B be exactly $95$ dollars more than the total cost at Company A?

2

3

5

8

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The total cost at Company B will be exactly 95 dollars more than the total cost at Company A for a rental of 8 days.

To find the number of days for which Company B's cost is exactly 95 dollars more than Company A's cost, we first find the linear function for Company B. The rate of change is

\frac{300 - 150}{5 - 2} = 50

dollars per day. The initial cost is

150 - 50(2) = 50

dollars, giving

B(d) = 50d + 50

. We then set up the equation

B(d) - A(d) = 95

, which is

(50d + 50) - (35d + 75) = 95

. Simplifying this equation gives

15d - 25 = 95

. Adding 25 to both sides gives

15d = 120

, and dividing by 15 gives

d = 8

.

Adım Adım Çözüm

1

Determine the linear cost function for Company B,

B(d) = md + b

, using the points

(2, 150)

and

(5, 300)

from the table.

The slope is

m = \frac{300 - 150}{5 - 2} = 50

. Using

(2, 150)

to find the y-intercept:

150 = 50(2) + b \Rightarrow b = 50

. Thus,

B(d) = 50d + 50

.

We need to find the equation representing the total cost of Company B to set up the comparison.

2

Set up an equation where the total cost at Company B is equal to the total cost at Company A plus 95 dollars.

B(d) = A(d) + 95 \Rightarrow 50d + 50 = (35d + 75) + 95

.

This represents the condition that Company B's cost is 95 dollars more than Company A's cost.

3

Simplify the equation and solve for the number of days,

d

.

50d + 50 = 35d + 170 \Rightarrow 15d = 120 \Rightarrow d = 8

.

Isolating the variable

d

gives the exact number of days required.

Anahtar Kavram

Comparing linear functions and solving linear equations representing real-world contexts.

Soru 307Soru →

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 308Soru →

During a chemistry experiment, the temperature of a liquid increases at a constant rate. The temperature of the liquid is $24^\circ\text{C}$ at $5$ minutes after the experiment starts, and $48^\circ\text{C}$ at $15$ minutes after the experiment starts. The temperature $T$ , in degrees Celsius, of the liquid $t$ minutes after the experiment starts can be modeled by the equation $T = mt + b$ , where $m$ and $b$ are constants. What is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

12

The correct answer is 12. By translating the given information into two points on the line,

(5, 24)

and

(15, 48)

, we can find the slope

m = \frac{48 - 24}{15 - 5} = 2.4

. Substituting this slope and the point

(5, 24)

into

T = mt + b

yields

24 = 2.4(5) + b

, which simplifies to

24 = 12 + b

. Solving for

b

gives

b = 12

.

Adım Adım Çözüm

1

Identify the data points representing the relationship between time and temperature.

The two data points are

(t_1, T_1) = (5, 24)

and

(t_2, T_2) = (15, 48)

.

These points represent coordinates

(t, T)

on the line representing the temperature over time.

2

Calculate the slope

m

of the linear equation.

m = \frac{48 - 24}{15 - 5} = \frac{24}{10} = 2.4

The slope represents the constant rate of temperature increase per minute.

3

Substitute the slope

m

and the coordinates of one point into the equation

T = mt + b

to solve for the y-intercept

b

.

24 = 2.4(5) + b \implies 24 = 12 + b \implies b = 12

This determines the value of the constant

b

, which corresponds to the initial temperature of the liquid.

Anahtar Kavram

Determining the y-intercept of a linear model in two variables given two points.

Soru 309Soru →

An online store sells two types of monthly subscription plans: a basic plan for $12 per month and a premium plan for$ 20 per month. In April, the store had a total of 150 active subscribers for these two plans and collected a total of $2,280 in subscription fees. How many of the active subscribers in April were on the premium plan?

15

50

60

90

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

60

By representing the number of basic subscribers as

x

and premium subscribers as

y

, we set up the system of equations

x + y = 150

and

12x + 20y = 2280

. Solving for

x

in the first equation gives

x = 150 - y

. Substituting this into the second equation yields

12(150 - y) + 20y = 2280

. Distributing the 12 and combining like terms gives

1800 + 8y = 2280

. Subtracting 1800 from both sides gives

8y = 480

, and dividing by 8 results in

y = 60

. Therefore, there were 60 premium subscribers.

Adım Adım Çözüm

1

Define variables for the unknown quantities: let

x

represent the number of basic subscribers and let

y

represent the number of premium subscribers.

The system of equations is:

x + y = 150

and

12x + 20y = 2280

.

Translate the given information about total subscribers and total revenue into algebraic equations.

2

Express

x

in terms of

y

using the first equation (

x = 150 - y

) and substitute it into the second equation.

The substituted equation is:

12(150 - y) + 20y = 2280

, which expands to

1800 - 12y + 20y = 2280

.

Eliminate one variable to create a single equation with only one variable,

y

.

3

Simplify the equation and isolate

y

.

1800 + 8y = 2280 \implies 8y = 480 \implies y = 60

.

Combine like terms and use basic operations to solve for the number of premium subscribers.

Anahtar Kavram

Solving a system of two linear equations using substitution or elimination.

Tahmini Süre:1m 30s

Soru 310Soru →

A point $(x, y)$ in the coordinate plane satisfies the system of inequalities below.

y \geq 2x - 4

y \leq -x + 5

x \geq 0

y \geq 0

What is the maximum possible value of $x$ for this point?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The maximum possible value of

x

that satisfies the system of inequalities is 3.

To find the maximum possible value of

x

for a point satisfying the system of inequalities, we identify the vertices of the region. The upper-right boundary of the region is formed by the intersection of the lines

y = 2x - 4

and

y = -x + 5

. Solving

2x - 4 = -x + 5

gives

3x = 9 \implies x = 3

. At this point,

y = -3 + 5 = 2

, which satisfies the constraints

x \geq 0

and

y \geq 0

. The other boundary vertices of the region are

(0, 0)

,

(2, 0)

, and

(0, 5)

. Comparing the x-coordinates of these vertices (

0

,

2

, and

3

), we see that the maximum possible value of

x

is

3

.

Adım Adım Çözüm

1

Find the intersection point of the boundary lines

y = 2x - 4

and

y = -x + 5

.

x = 3

Setting the two boundary line equations equal to each other (

2x - 4 = -x + 5

) allows us to find the x-coordinate where the boundaries cross.

2

Substitute

x = 3

back into one of the boundary equations to find the y-coordinate.

y = 2

This yields the intersection point

(3, 2)

which lies on both boundary lines.

3

Verify that the intersection point

(3, 2)

satisfies the other constraints:

x \geq 0

and

y \geq 0

.

3 \geq 0

and

2 \geq 0

(both true)

The point must lie within the first quadrant to be a valid solution.

4

Determine the remaining boundary vertices of the solution region in the first quadrant.

Vertices are

(0, 0)

,

(0, 5)

,

(2, 0)

, and

(3, 2)

.

Comparing all vertices will confirm if

(3, 2)

indeed provides the maximum value of

x

.

Anahtar Kavram

Systems of Linear Inequalities in Two Variables

Soru 311Soru →

A logistics company uses a delivery drone to transport two types of packages. Let $x$ represent the number of Type A packages and $y$ represent the number of Type B packages in a single flight. The drone can carry at most 12 packages in total. Additionally, to balance the drone, the total weight of the cargo must be at least 15 pounds. Each Type A package weighs 2 pounds, and each Type B package weighs 1 pound. Which of the following combinations of Type A and Type B packages is a viable shipment for the drone?

3 Type A packages and 8 Type B packages

4 Type A packages and 5 Type B packages

5 Type A packages and 6 Type B packages

6 Type A packages and 7 Type B packages

Cevabı ve açıklamayı göster

Cevap: 5 Type A packages and 6 Type B packages

Cevap

5 Type A packages and 6 Type B packages

The correct combination of 5 Type A packages and 6 Type B packages satisfies all system requirements. The total package count of 11 is less than or equal to the drone's limit of 12 packages (

5 + 6 \leq 12

). Furthermore, the total weight of 16 pounds meets the minimum balance requirement of 15 pounds (

2(5) + 6 \geq 15

).

Adım Adım Çözüm

1

Write the system of linear inequalities that represents the given constraints.

The capacity constraint is

x + y \leq 12

, and the weight constraint is

2x + y \geq 15

, where

x

and

y

are non-negative integers.

To define the mathematical boundaries for a valid shipment.

2

Substitute the package counts from the correct combination into both inequalities to verify they are satisfied.

For 5 Type A packages (

x = 5

) and 6 Type B packages (

y = 6

):
- Package count:

5 + 6 = 11 \leq 12

(True)
- Cargo weight:

2(5) + 6 = 16 \geq 15

(True)

To confirm that the chosen combination satisfies all system requirements.

Anahtar Kavram

Systems of Linear Inequalities in Two Variables

Tahmini Süre:1m 30s

Soru 312Soru →

A coffee shop sells two sizes of bagged coffee: a standard bag and a deluxe bag. On Monday, the shop sold $8$ standard bags and $5$ deluxe bags for a total of $\$108$ . On Tuesday, the shop sold $6$ standard bags and $10$ deluxe bags for a total of $\$156$ . What is the price, in dollars, of one deluxe bag of coffee?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The price of one deluxe bag of coffee is

12

dollars.

By setting up a system of equations, we represent the total cost on Monday as

8s + 5d = 108

and on Tuesday as

6s + 10d = 156

. Multiplying the first equation by

2

gives

16s + 10d = 216

. Subtracting the second equation yields

10s = 60

, so

s = 6

. Substituting

s = 6

into the first equation yields

8(6) + 5d = 108

, which simplifies to

48 + 5d = 108

, then

5d = 60

, and finally

d = 12

.

Adım Adım Çözüm

1

Define the variables and write the system of equations.

Let

s

be the price of a standard bag and

d

be the price of a deluxe bag. The system is:

8s + 5d = 108

6s + 10d = 156

To translate the word problem into mathematical equations.

2

Multiply the first equation by

2

to align the coefficients of

d

.

16s + 10d = 216

To prepare for elimination of the variable

d

.

3

Subtract the second equation from the new equation to solve for

s

.

10s = 60

, which simplifies to

s = 6

.

To eliminate

d

and find the price of a standard bag.

4

Substitute the value of

s

back into one of the original equations to solve for

d

.

8(6) + 5d = 108 \implies 48 + 5d = 108 \implies 5d = 60 \implies d = 12

.

To find the price of a deluxe bag.

Anahtar Kavram

Solving systems of linear equations using the elimination method.

Tahmini Süre:1m 30s

Soru 313Soru →

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 314Soru →

In the inequality $3(4 - 2x) - 5(x - c) \geq -7$ , where $c$ is a constant, the solution set consists of all values of $x$ such that $x \leq 3$ . What is the value of $c$ ?

Cevabı ve açıklamayı göster

Cevap: 2.8

Cevap

2.8

To find the value of

c

, first simplify the inequality

3(4 - 2x) - 5(x - c) \geq -7

by expanding the terms using the distributive property, which yields

12 - 6x - 5x + 5c \geq -7

. Combining like terms gives

-11x + 12 + 5c \geq -7

. Next, isolate the variable term by subtracting

12

and

5c

from both sides to obtain

-11x \geq -19 - 5c

. Dividing both sides of the inequality by

-11

and reversing the inequality sign results in

x \leq \frac{19 + 5c}{11}

. Given that the solution set consists of all values of

x

such that

x \leq 3

, the boundary value

\frac{19 + 5c}{11}

must equal

3

. Solving the equation

\frac{19 + 5c}{11} = 3

gives

19 + 5c = 33

, which simplifies to

5c = 14

and results in

c = 2.8

.

Adım Adım Çözüm

1

Distribute the constants on the left side of the inequality.

12 - 6x - 5x + 5c \geq -7

Applying the distributive property simplifies the parentheses.

2

Combine the variable terms.

-11x + 12 + 5c \geq -7

Grouping like terms simplifies the inequality.

3

Subtract

12

and

5c

from both sides of the inequality.

-11x \geq -19 - 5c

This isolates the term containing

x

.

4

Divide both sides by

-11

and reverse the direction of the inequality.

x \leq \frac{19 + 5c}{11}

Dividing by a negative number requires flipping the inequality sign to maintain equivalence.

5

Set the algebraic boundary

\frac{19 + 5c}{11}

equal to the given boundary value of

3

and solve for

c

.

c = 2.8

Since the solution set is

x \leq 3

, the boundary values must be equivalent.

Anahtar Kavram

Solving multi-step linear inequalities in one variable containing parameters.

Soru 315Soru →

The graph of the equation $ax + by = 24$ , where $a$ and $b$ are constants, is a line in the $xy$ -plane. If this line passes through the points $(2, 9)$ and $(6, 3)$ , what is the value of $a + b$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

a + b

is

5

.

Substituting the coordinates of the two points

(2, 9)

and

(6, 3)

into the given linear equation

ax + by = 24

yields a system of two equations:

2a + 9b = 24

and

6a + 3b = 24

. Simplifying the second equation gives

2a + b = 8

, which means

b = 8 - 2a

. Substituting this expression for

b

into the first equation yields

2a + 9(8 - 2a) = 24

, which simplifies to

-16a = -48

, or

a = 3

. Plugging

a = 3

back into

b = 8 - 2a

gives

b = 2

. Thus, the value of

a + b

is

3 + 2 = 5

.

Adım Adım Çözüm

1

Substitute the point

(2, 9)

into the equation

ax + by = 24

.

2a + 9b = 24

Since the line passes through the point

(2, 9)

, the coordinates must satisfy the equation of the line.

2

Substitute the point

(6, 3)

into the equation

ax + by = 24

.

6a + 3b = 24

Since the line passes through the point

(6, 3)

, the coordinates must satisfy the equation of the line.

3

Solve the system of equations for

a

and

b

.

a = 3

and

b = 2

To find the values of the constants

a

and

b

, we solve the linear system: (1)

2a + 9b = 24

and (2)

6a + 3b = 24

. Dividing the second equation by 3 gives

2a + b = 8

, or

b = 8 - 2a

. Substituting this into the first equation gives

2a + 9(8 - 2a) = 24 \implies 2a + 72 - 18a = 24 \implies -16a = -48 \implies a = 3

. Then,

b = 8 - 2(3) = 2

.

4

Add the values of

a

and

b

.

a + b = 5

The question asks for the value of

a + b

.

Anahtar Kavram

Solving systems of linear equations derived from coordinate substitution in a two-variable linear equation.

Tahmini Süre:1m 30s

Soru 316Soru →

For a constant $k$ , the inequality $2(3 - x) - k(x + 5) > 4$ has the solution set $x < -3$ . What is the value of $k$ ?

-1

2

4

-0.5

Cevabı ve açıklamayı göster

Cevap: $4$

Cevap

4

The correct answer is

4

. Expanding the inequality

2(3 - x) - k(x + 5) > 4

gives

6 - 2x - kx - 5k > 4

. Grouping the

x

terms yields

(-2 - k)x > 5k - 2

. Because the given solution set is

x < -3

, the inequality sign must flip, indicating that the coefficient

-2 - k

is negative. Dividing by this negative coefficient yields

x < \frac{5k - 2}{-2 - k}

. Setting the boundary expression equal to the boundary of the solution set gives

\frac{5k - 2}{-2 - k} = -3

. Multiplying both sides by

-2 - k

results in

5k - 2 = 6 + 3k

. Subtracting

3k

and adding

2

to both sides gives

2k = 8

, which simplifies to

k = 4

.

Adım Adım Çözüm

1

Distribute the constants in the inequality

6 - 2x - kx - 5k > 4

To clear parentheses and prepare to group terms.

2

Group the

x

terms and constant terms

(-2 - k)x > 5k - 2

To isolate the variable

x

on one side of the inequality.

3

Divide by the coefficient of

x

and flip the inequality direction

x < \frac{5k - 2}{-2 - k}

Since the solution set is

x < -3

, the direction of the inequality must flip from

>

to

<

, meaning the coefficient

-2 - k

must be negative.

4

Set the boundary value equal to

-3

and solve for

k

k = 4

The boundary of the solution set must be equal to

-3

. Solving

\frac{5k - 2}{-2 - k} = -3

yields

5k - 2 = 6 + 3k

, which simplifies to

2k = 8

, so

k = 4

.

Anahtar Kavram

Solving linear inequalities in one variable with symbolic coefficients, accounting for direction flips when dividing by negative quantities.

Soru 317Soru →

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 318Soru →

A community theater sold adult tickets and student tickets for its opening weekend performances. On Friday, the theater sold $20$ adult tickets and $40$ student tickets for a total of $\$760$ . On Saturday, the theater sold $35$ adult tickets and $30$ student tickets for a total of $\$930$ . What is the price, in dollars, of one adult ticket?

10

16

18

30

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

The price of one adult ticket is 18 dollars.

The price of one adult ticket is

18

. By setting up the system of linear equations representing the total sales on Friday (

20x + 40y = 760

) and Saturday (

35x + 30y = 930

), where

x

is the adult ticket price and

y

is the student ticket price, we can simplify them to

x + 2y = 38

and

7x + 6y = 186

. Multiplying the first simplified equation by

3

gives

3x + 6y = 114

. Subtracting this equation from

7x + 6y = 186

eliminates

y

and yields

4x = 72

, which simplifies to

x = 18

.

Adım Adım Çözüm

1

Define variables and write the system of linear equations based on the given information.

Let

x

represent the price of an adult ticket and

y

represent the price of a student ticket. The system of equations is:

20x + 40y = 760

35x + 30y = 930

Variables represent the unknown quantities, and the equations relate these quantities to the total sales on Friday and Saturday.

2

Simplify both equations by dividing by their greatest common divisors.

Divide the first equation by

20

:

x + 2y = 38

Divide the second equation by

5

:

7x + 6y = 186

Simplifying equations reduces the coefficient values and minimizes the chance of arithmetic errors in later steps.

3

Use the elimination method to solve for

x

. Multiply the first simplified equation by

3

to align the

y

-coefficients.

3(x + 2y) = 3(38) \Rightarrow 3x + 6y = 114

Matching the coefficients of

y

allows us to eliminate the variable

y

by subtracting the equations.

4

Subtract the equation obtained in Step 3 from the second simplified equation.

(7x + 6y) - (3x + 6y) = 186 - 114 \Rightarrow 4x = 72

Subtracting the equations eliminates

y

, leaving a single linear equation in terms of

x

.

5

Solve for

x

by dividing both sides of the equation by

4

.

x = 18

Isolating

x

provides the price of one adult ticket.

Anahtar Kavram

Solving a system of linear equations using substitution or elimination.

Alternatif Yöntem

Instead of using elimination on the simplified equations, you can use substitution. Express

x

in terms of

y

using the first simplified equation:

x = 38 - 2y

. Substitute this expression into the second equation:

7(38 - 2y) + 6y = 186

. Expanding this gives

266 - 14y + 6y = 186

, which simplifies to

266 - 8y = 186

. Subtracting

266

from both sides gives

-8y = -80

, which solves to

y = 10

. Finally, substitute

y = 10

back into

x = 38 - 2y

to find

x = 38 - 2(10) = 18

.

Tahmini Süre:1m 30s

Soru 319Soru →

In the $xy$ -plane, a point with coordinates $(x, y)$ lies in the region defined by the system of inequalities below.

y + 2x \leq 12

x - 2y \leq 6

x \geq 2

What is the maximum possible value of $y$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

The system of inequalities bounds the solution set. The upper boundary is given by

y \leq -2x + 12

. Because the slope is negative, the maximum value of

y

on this boundary occurs at the smallest possible value of

x

. The constraint

x \geq 2

dictates that the minimum value of

x

is 2. Substituting

x = 2

into the boundary equation yields

y = -2(2) + 12 = 8

. Checking this coordinate against the third inequality,

2 - 2(8) = -14 \leq 6

, verifies that

(2, 8)

is a valid solution.

Adım Adım Çözüm

1

Express the first two inequalities in terms of y.

y \leq -2x + 12

and

y \geq \frac{1}{2}x - 3

This helps identify the upper and lower boundaries of the solution region.

2

Determine the boundary line that limits the maximum values of y.

The upper boundary is the line

y = -2x + 12

.

Since the inequality is

y \leq -2x + 12

, any solution must lie on or below this line.

3

Find the maximum value of y on this boundary line given the constraint

x \geq 2

.

y \leq -2(2) + 12 = 8

Since the slope of the boundary line is negative, y is maximized when x is at its minimum value, which is 2.

4

Verify that the point

(2, 8)

satisfies the inequality

x - 2y \leq 6

.

2 - 2(8) = -14 \leq 6

, which is true.

This confirms that the point

(2, 8)

is indeed in the solution set of the system.

Anahtar Kavram

Maximizing a coordinate value subject to a system of linear inequalities in two variables

Alternatif Yöntem

Instead of graphing or checking boundaries, we can algebraically solve for the boundary. Since

x \geq 2

, multiplying by

-2

and reversing the inequality gives

-2x \leq -4

. Adding 12 to both sides gives

-2x + 12 \leq 8

. Since

y \leq -2x + 12

, we get

y \leq 8

. Checking if

y = 8

and

x = 2

satisfies the second inequality

x - 2y \leq 6

confirms that

2 - 16 = -14 \leq 6

is true, meaning

y = 8

is indeed a valid solution and thus the maximum.

Tahmini Süre:1m 30s

Soru 320Soru →

A shipping container has a maximum weight capacity of $24,150$ kilograms. The container is loaded with $12$ machinery units, each weighing $1,150$ kilograms. The remaining space will be filled with packing crates, each weighing $180$ kilograms. A safety regulation requires that a clearance weight of at least $15\%$ of the total loaded weight (the combined weight of the machinery units and the packing crates) must be left unused. What is the maximum number of packing crates that can be loaded into the container?

Cevabı ve açıklamayı göster

Cevap: 40

Cevap

The maximum number of packing crates that can be loaded is 40.

By setting up the inequality representing the physical constraints, we find that the number of packing crates,

x

, must satisfy

x \le 40

. Since the question asks for the maximum number of packing crates, the maximum value is 40.

Adım Adım Çözüm

1

Calculate the constant weight of the machinery units.

The total weight of the

12

machinery units is

12 \times 1,150 = 13,800

kilograms.

This establishes the base weight that is already loaded in the container.

2

Define the variable and write the expression for the total loaded weight.

Let

x

be the number of packing crates. The total loaded weight is

13,800 + 180x

kilograms.

This represents the combined weight of the machinery and the crates in terms of the variable

x

.

3

Set up the inequality representing the safety clearance requirement.

24,150 - (13,800 + 180x) \ge 0.15(13,800 + 180x)

The unused weight capacity (maximum capacity minus loaded weight) must be at least

15\%

of the loaded weight.

4

Solve the inequality for

x

.

24,150 \ge 1.15(13,800 + 180x) \implies 21,000 \ge 13,800 + 180x \implies 7,200 \ge 180x \implies x \le 40

.

Isolating

x

gives the range of allowable values for the number of packing crates.

Anahtar Kavram

Formulating and solving linear inequalities in one variable based on real-world constraints.

Algebra

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