Tüm alıştırma soruları

Soru 1861Soru →

In 2018, a forestry service recorded $400$ pine trees in a specific section of a nature reserve. Due to a reforestation initiative, the number of pine trees in this section increased by $15\%$ from 2018 to 2021. From 2021 to 2024, the number of pine trees increased by $20\%$ of the number of pine trees in 2021. What was the number of pine trees in this section of the reserve in 2024?

480

540

542

552

Cevabı ve açıklamayı göster

Cevap: 552

Cevap

The correct number of pine trees in the reserve in 2024 is 552.

To find the number of pine trees in 2024, first calculate the population in 2021. A

15\%

increase on the initial

400

trees results in

400 \times 1.15 = 460

trees. Next, find the population in 2024 by applying a

20\%

increase to the 2021 population of

460

trees, which yields

460 \times 1.20 = 552

trees.

Adım Adım Çözüm

1

Calculate the number of pine trees in 2021 after the

15\%

increase.

400 \times (1 + 0.15) = 460

trees

The initial number of trees in 2018 was

400

, and it increased by

15\%

, so we multiply

400

by

1.15

.

2

Calculate the number of pine trees in 2024 after the

20\%

increase.

460 \times (1 + 0.20) = 552

trees

The percentage increase of

20\%

from 2021 to 2024 is based on the 2021 tree population of

460

, so we multiply

460

by

1.20

.

Anahtar Kavram

Calculating successive percentage increases by applying each percentage change to the correct sequential baseline value.

Alternatif Yöntem

Alternatively, you can compute the final value in a single expression by multiplying the initial value by both growth factors:

400 \times 1.15 \times 1.20 = 552

.

Tahmini Süre:1m 30s

Soru 1862Soru →

At the beginning of an experiment, the number of bacteria in Population X is $50\%$ greater than the number of bacteria in Population Y. During the experiment, Population X increases by $10\%$ and Population Y increases by $r\%$ . At the end of the experiment, the number of bacteria in Population X is $25\%$ greater than the number of bacteria in Population Y. What is the value of $r$ ?

24

32

35

40

Cevabı ve açıklamayı göster

Cevap: 32

Cevap

32

The correct answer is

32

. If the initial population of Y is assumed to be

100

, then the initial population of X is

150

because it is

50\%

greater. A

10\%

increase of X results in a final population of

150 \times 1.10 = 165

. Since the final population of X is

25\%

greater than the final population of Y, we have

165 = 1.25 \times Y_{\text{final}}

, which gives

Y_{\text{final}} = 132

. The growth from

100

to

132

represents a

32\%

increase, so

r = 32

.

Adım Adım Çözüm

1

Set up initial values for both populations.

Let the initial size of Population Y be

100

. Since Population X is

50\%

greater than Y, the initial size of Population X is

150

.

Choosing a concrete initial value like

100

simplifies the percentage calculations.

2

Calculate the final population of X.

The final population of X is

150 \times (1 + 0.10) = 165

.

Population X increased by

10\%

during the experiment.

3

Calculate the final population of Y.

Since the final population of X is

25\%

greater than the final population of Y, we write

165 = 1.25 \times Y_{\text{final}}

, which yields

Y_{\text{final}} = 132

.

We must divide the final population of X by

1.25

to find the final population of Y, because X is

25\%

greater than Y.

4

Find the percentage increase of Population Y.

The percent increase from

100

to

132

is

\frac{132 - 100}{100} \times 100\% = 32\%

. Therefore,

r = 32

.

This determines the value of

r

representing the percentage growth of Population Y.

Anahtar Kavram

Multi-step percentage change and identifying the correct baseline value.

Soru 1863Soru →

At the beginning of the year, a library had $800$ history books. In the first half of the year, the number of history books increased by $15\%$ . In the second half of the year, the library acquired more history books, representing a percent increase of $x\%$ over the number of history books at the midyear point. If the library had a total of $1{}012$ history books at the end of the year, what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The value of

x

is

10

.

To find the second percent increase, we must first calculate the intermediate midyear value. A

15\%

increase on

800

is

800 \times 1.15 = 920

. The subsequent increase of

x\%

is based on this midyear value of

920

. Setting up the equation

920 \times (1 + \frac{x}{100}) = 1{}012

and solving for

x

yields

1 + \frac{x}{100} = 1.10

, which gives

x = 10

.

Adım Adım Çözüm

1

Calculate the number of history books at the midyear point after the first increase of

15\%

920

books

An increase of

15\%

on the initial

800

books is calculated as

800 \times (1 + 0.15) = 920

.

2

Set up an equation for the second percent increase of

x\%

from the midyear value to the final value of

1{}012

920 \times (1 + \frac{x}{100}) = 1{}012

The second increase is

x\%

of the midyear value of

920

, resulting in the final value of

1{}012

.

3

Solve the equation for

x

x = 10

Divide both sides of the equation by

920

to get

1 + \frac{x}{100} = 1.10

, subtract

1

to get

\frac{x}{100} = 0.10

, and multiply by

100

to find

x = 10

.

Anahtar Kavram

Calculating successive percent increases by applying the percent change to the intermediate base value.

Alternatif Yöntem

Instead of calculating the intermediate number of books, we can express the final number of books as

800 \times 1.15 \times (1 + \frac{x}{100}) = 1{}012

. Simplifying

800 \times 1.15

gives

920

, so

920 \times (1 + \frac{x}{100}) = 1{}012

, leading to the same result of

x = 10

.

Tahmini Süre:1m 30s

Soru 1864Soru →

In the $xy$ -plane, the vertex of the parabola with equation $y = -2x^2 + bx + c$ , where $b$ and $c$ are constants, lies on the line $y = 3x + 5$ . If the parabola has its maximum value at $x = -2$ , what is the value of $c$ ?

3

-9

-7

-1

Cevabı ve açıklamayı göster

Cevap: -9

Cevap

-9

The maximum value of the quadratic function

y = -2x^2 + bx + c

occurs at its vertex. Since the maximum occurs at

x = -2

, the

x

-coordinate of the vertex is

-2

. The vertex lies on the line

y = 3x + 5

, so substituting

x = -2

into this equation gives the

y

-coordinate of the vertex:

y = 3(-2) + 5 = -1

. Thus, the vertex of the parabola is

(-2, -1)

. The vertex form of a quadratic function is

y = a(x - h)^2 + k

, where

(h, k)

is the vertex. Substituting

a = -2

,

h = -2

, and

k = -1

gives

y = -2(x + 2)^2 - 1

. Expanding this equation yields

y = -2(x^2 + 4x + 4) - 1 = -2x^2 - 8x - 9

. Comparing this to the standard form

y = -2x^2 + bx + c

shows that

c = -9

.

Adım Adım Çözüm

1

Identify the x-coordinate of the vertex of the parabola

The x-coordinate of the vertex is

h = -2

.

For a quadratic function in standard form with a negative leading coefficient, the maximum value occurs at the vertex. Thus, the x-coordinate of the vertex is

-2

.

2

Determine the y-coordinate of the vertex using the equation of the line

The y-coordinate of the vertex is

k = -1

.

Since the vertex

(h, k)

lies on the line

y = 3x + 5

and

h = -2

, substituting

x = -2

yields

y = 3(-2) + 5 = -1

.

3

Write the quadratic equation in vertex form and expand it to find c

y = -2x^2 - 8x - 9

, which gives

c = -9

.

Using the vertex form

y = a(x - h)^2 + k

with

a = -2

,

h = -2

, and

k = -1

, the equation is

y = -2(x + 2)^2 - 1

. Expanding this yields

y = -2(x^2 + 4x + 4) - 1 = -2x^2 - 8x - 9

. Comparing this with

y = -2x^2 + bx + c

shows that

c = -9

.

Anahtar Kavram

Vertex form of a quadratic function and coordinate geometry.

Tahmini Süre:2m 0s

Soru 1865Soru →

A survey of 80 commuters asked about their primary method of transportation to work and their commute distance. The results are summarized in the table below.

Commute Distance	Bicycle	Walk	Total
Under 1 mile	15	25	40
1 mile or more	35	5	40
Total	50	30	80

Based on the table, what fraction of the commuters who commute a distance of under 1 mile choose walking as their primary transportation method?

Cevabı ve açıklamayı göster

Cevap: 0.625

Cevap

5/8 or 0.625

To find the fraction of commuters who commute a distance of under 1 mile that choose walking, limit the sample space to the row representing 'Under 1 mile', which has a subtotal of 40 commuters. Within this row, 25 commuters walk. Thus, the fraction is 25/40, which simplifies to 5/8 (or 0.625).

Adım Adım Çözüm

1

Identify the conditional group (denominator).

The row 'Under 1 mile' has a total of 40 commuters.

The phrase 'of the commuters who commute a distance of under 1 mile' restricts the group of interest to only those in this row.

2

Identify the target group within the restricted group (numerator).

25 commuters choose walking.

Within the 'Under 1 mile' row, we look at the 'Walk' column.

3

Calculate the conditional fraction.

25/40 = 5/8 (or 0.625).

Divide the target count by the conditional group total to find the fraction.

Anahtar Kavram

Conditional Probability from a Two-Way Table

Soru 1866Soru →

In the $xy$ -plane, a line is represented by the equation $ay - bx = 24$ , where $a$ and $b$ are constants. If the line has a $y$ -intercept of $(0, -3)$ and passes through the point $(4, 5)$ , what is the slope of the line?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The slope of the line is 2.

Substituting the y-intercept

(0, -3)

into the equation

ay - bx = 24

yields

-3a = 24

, which gives

a = -8

. Substituting the point

(4, 5)

and

a = -8

into the equation yields

-8(5) - 4b = 24

, which simplifies to

-40 - 4b = 24

, giving

b = -16

. Re-assembling the equation gives

-8y + 16x = 24

. Solving for

y

in terms of

x

yields

y = 2x - 3

, where the coefficient of

x

is the slope, 2.

Adım Adım Çözüm

1

Substitute the y-intercept

(0, -3)

into the given equation

ay - bx = 24

to find the value of

a

.

a = -8

Since the y-intercept lies on the line, its coordinates satisfy the line's equation.

2

Substitute the point

(4, 5)

and the value of

a = -8

into the equation to find the value of

b

.

b = -16

Since the point

(4, 5)

lies on the line, its coordinates must satisfy the equation.

3

Write the resulting equation

-8y + 16x = 24

in slope-intercept form (

y = mx + c

) to identify the slope.

y = 2x - 3

, which gives a slope of

2

.

The slope of a line in the form

y = mx + c

is represented by the coefficient

m

of

x

.

Anahtar Kavram

Finding the slope of a line from a given equation by determining its constant coefficients using known points.

Tahmini Süre:1m 30s

Soru 1867Soru →

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

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

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

The number of documents the team scans each hour

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

Cevabı ve açıklamayı göster

Cevap: The number of documents the team scans each hour

Cevap

The number of documents the team scans each hour

In the linear model

D = 14{}200 - 350t

, the coefficient

-350

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

350

for each hour that passes, the value

350

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

Adım Adım Çözüm

1

Identify the role of each component in the linear equation

D = 14{}200 - 350t

.

The equation is in the form

y = b + mx

, where the constant

b = 14{}200

is the y-intercept (the starting value of

D

when

t = 0

) and the coefficient

m = -350

is the slope (the constant rate of change of

D

per unit increase in

t

).

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

2

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

The slope of

-350

indicates that for each unit increase in the independent variable

t

(hours), the dependent variable

D

(unscanned documents remaining) decreases by

350

.

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

3

Select the option that matches this rate of change.

A rate of

350

documents per hour means the archiving team scans

350

documents every hour.

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

Anahtar Kavram

Interpreting Linear Relationships in Context

Tahmini Süre:1m 15s

Soru 1868Soru →

The table below shows the initial number of tomato plants and pepper plants in two agricultural fields, Field A and Field B, at the start of a crop monitoring study.

Field	Tomato plants	Pepper plants
Field A	$150$	$100$
Field B	$200$	$300$

During the study:
- The total number of plants in Field A increased by $20\%$ , and the number of tomato plants in Field A increased by $30\%$ .
- The total number of plants in Field B increased by $10\%$ , and the number of pepper plants in Field B increased by $15\%$ .

What was the percent increase in the total number of pepper plants across both fields combined from the start to the end of the study?

10.0%

11.1%

12.5%

16.7%

Cevabı ve açıklamayı göster

Cevap: 12.5%

Cevap

12.5%

The correct option is 12.5%. The initial combined number of pepper plants is

100 + 300 = 400

. In Field A, the total number of plants increases by 20% from 250 to 300, while the tomato plants increase by 30% from 150 to 195. This leaves

300 - 195 = 105

pepper plants in Field A at the end of the study. In Field B, the number of pepper plants increases by 15% from 300 to 345. Combining the final counts gives

105 + 345 = 450

pepper plants. The total increase of 50 pepper plants relative to the initial combined count of 400 represents a

\frac{50}{400} \times 100\% = 12.5\%

increase.

Adım Adım Çözüm

1

Calculate the initial total number of plants in Field A and the initial combined number of pepper plants.

Field A initial total is

150 + 100 = 250

plants. The initial combined number of pepper plants across both fields is

100 + 300 = 400

plants.

Establishing the initial baseline values is necessary to compute final changes.

2

Determine the final number of pepper plants in Field A.

The final total in Field A is

250 \times 1.20 = 300

plants. The final number of tomato plants in Field A is

150 \times 1.30 = 195

plants. Therefore, the final number of pepper plants in Field A is

300 - 195 = 105

plants.

This isolates the final number of pepper plants contributed by Field A since the percent increase was given for the total and for tomato plants.

3

Determine the final number of pepper plants in Field B.

The final number of pepper plants in Field B is

300 \times 1.15 = 345

plants.

Applying the direct percentage increase of 15% to the initial Field B pepper plants gives their final count.

4

Calculate the combined final number of pepper plants and the percent increase.

The final combined number of pepper plants is

105 + 345 = 450

plants. The net increase is

450 - 400 = 50

plants. The percent increase is

\frac{50}{400} \times 100\% = 12.5\%

.

Dividing the change in the target quantity by its original combined base yields the combined percent change.

Anahtar Kavram

Weighted percent change and finding missing values using totals and subcategory percentages

Soru 1869Soru →

For the quadratic function $f(x) = -x^2 + bx + c$ , where $b$ and $c$ are positive constants, the maximum value of $f(x)$ is $k$ . The distance between the two $x$ -intercepts of the graph of $y = f(x)$ in the $xy$ -plane is equal to $\frac{2}{3}k$ . If the graph of $y = f(x)$ passes through the point $(1, 8)$ , what is the value of $b$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The correct answer is 4.

The correct answer is 4. By expressing the maximum value

k

and the distance between the

x

-intercepts

d

in terms of the constants

b

and

c

, we set up the equation

d = \frac{2}{3}k

. This simplifies to

b^2 + 4c = 36

. Using the point

(1, 8)

, we establish

c = 9 - b

. Substituting this into the first equation yields

b^2 - 4b = 0

, giving

b = 4

as the only positive solution.

Adım Adım Çözüm

1

Find the maximum value

k

of the function

f(x) = -x^2 + bx + c

in terms of

b

and

c

.

k = \frac{b^2}{4} + c

The maximum value of a quadratic function with a negative leading coefficient occurs at its vertex, where

x = -\frac{b}{2a} = \frac{b}{2}

.

2

Find the distance

d

between the

x

-intercepts of the graph in terms of

b

and

c

.

d = \sqrt{b^2 + 4c}

The

x

-intercepts are the roots of

-x^2 + bx + c = 0

, which are

x = \frac{b \pm \sqrt{b^2 + 4c}}{2}

. The distance between them is the difference of these roots.

3

Use the relation

d = \frac{2}{3}k

to find the value of the expression

b^2 + 4c

.

b^2 + 4c = 36

Substituting the expressions for

d

and

k

gives

\sqrt{b^2 + 4c} = \frac{1}{6}(b^2 + 4c)

. Solving this radical equation yields

b^2 + 4c = 36

.

4

Use the point

(1, 8)

to express

c

in terms of

b

.

c = 9 - b

Since the graph passes through

(1, 8)

, substituting

x = 1

and

y = 8

into

y = -x^2 + bx + c

yields

8 = -1 + b + c

, which simplifies to

c = 9 - b

.

5

Substitute

c = 9 - b

into

b^2 + 4c = 36

and solve for

b

.

b = 4

Substituting yields

b^2 + 4(9 - b) = 36 \implies b^2 - 4b = 0

. Solving for

b

gives

b = 0

or

b = 4

. Since

b

is a positive constant, we have

b = 4

.

Anahtar Kavram

Quadratic Functions and Graphs

Soru 1870Soru →

A solar panel generates $120$ watt-hours of electricity every $3$ hours. At this rate, how many watt-hours of electricity will the solar panel generate in $8$ hours?

Cevabı ve açıklamayı göster

Cevap: 320

Cevap

The solar panel will generate

320

watt-hours of electricity in

8

hours.

To find the total electricity generated, we determine the unit rate of energy generation:

\frac{120 \text{ watt-hours}}{3 \text{ hours}} = 40

watt-hours per hour. Multiplying this rate by the new duration of

8

hours gives

40 \times 8 = 320

watt-hours. Proportional reasoning also works:

\frac{120}{3} = \frac{x}{8}

, which yields

x = 320

.

Adım Adım Çözüm

1

Calculate the hourly rate of electricity generation.

The rate is

40

watt-hours per hour.

Dividing the energy produced (

120

watt-hours) by the elapsed time (

3

hours) gives the unit rate.

2

Calculate the total energy generated over the new duration of

8

hours.

The total energy is

320

watt-hours.

Multiplying the unit rate (

40

watt-hours per hour) by the target duration (

8

hours) gives the total energy generated.

Anahtar Kavram

Calculating and applying unit rates to solve proportional relationships.

Alternatif Yöntem

Set up a proportion where the ratio of energy to time remains constant:

\frac{120}{3} = \frac{x}{8}

. Cross-multiply to solve for

x

:

3x = 120 \times 8

, which simplifies to

3x = 960

. Divide by

3

to find

x = 320

.

Tahmini Süre:45s

Soru 1871Soru →

A technician at a chemistry lab prepares a solution by mixing $3$ milliliters of acid with every $7$ milliliters of distilled water. If the technician uses $35$ milliliters of distilled water to prepare the solution, how many milliliters of acid are needed?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The correct answer is 15.

To maintain the ratio of

3

milliliters of acid for every

7

milliliters of distilled water, the proportion

\frac{\text{acid}}{\text{water}} = \frac{3}{7} = \frac{x}{35}

is established. Multiplying both sides by

35

gives

x = 35 \times \frac{3}{7} = 15

milliliters of acid.

Adım Adım Çözüm

1

Set up the proportion relating acid to distilled water.

\frac{3}{7} = \frac{x}{35}

, where

x

is the required amount of acid in milliliters.

The ratio of acid to distilled water must remain constant.

2

Solve the proportion for

x

.

x = 15

Multiply both sides of the equation by 35 to isolate

x

.

Anahtar Kavram

Solving proportions to find an unknown quantity when the ratio is constant.

Alternatif Yöntem

Find the scale factor:

35

divided by

7

is

5

. Multiply the acid amount (

3

) by this scale factor of

5

to get

15

.

Tahmini Süre:45s

Soru 1872Soru →

A catering company charges a flat setup fee of $500 plus$ 35 per guest. A company has budgeted at most $2,400 for a banquet. If the company also wants to purchase a celebration cake for$ 150, what is the maximum number of guests they can invite to the banquet without exceeding their budget?

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

The maximum number of guests the company can invite without exceeding their budget is 50.

The total cost of the banquet consists of a flat setup fee (

500), a celebration cake (

150), and a variable cost of

35 per guest (

35g

, where

g

represents the number of guests). Since the company has budgeted at most

2,400, the sum of these expenses must be less than or equal to

2,400. This is represented by the inequality

35g + 500 + 150 \le 2400

. Combining the constants yields

35g + 650 \le 2400

. Subtracting 650 from both sides gives

35g \le 1750

. Dividing both sides by 35 results in

g \le 50$. Therefore, the maximum number of guests they can invite without exceeding their budget is 50.

Adım Adım Çözüm

1

Set up the inequality representing the total expenses.

35g + 500 + 150 \le 2400

The sum of the fixed setup fee, the cake cost, and the variable per-guest cost must not exceed the maximum budget of $2,400.

2

Simplify the inequality by combining the constants.

35g + 650 \le 2400

Combining the fixed expenses (

500 setup fee and

150 cake) simplifies the calculation.

3

Isolate the variable term by subtracting the fixed cost from both sides.

35g \le 1750

This determines the portion of the budget that can be allocated specifically to guests.

4

Solve for the variable by dividing both sides by the cost per guest.

g \le 50

Dividing the remaining guest budget by the rate per guest gives the maximum number of guests allowed.

Anahtar Kavram

Formulating and solving multi-step linear inequalities in one variable to determine a maximum value in context.

Soru 1873Soru →

The polynomial function $p$ is defined by $p(x) = x^3 + bx^2 + cx + d$ , where $b$ , $c$ , and $d$ are constants. In the $xy$ -plane, the graph of $y = p(x)$ has $x$ -intercepts at $(2, 0)$ and $(-3, 0)$ . If the remainder when $p(x)$ is divided by $x - 1$ is $-8$ , what is the value of $d$ ?

-6

6

-2

18

Cevabı ve açıklamayı göster

Cevap: $-6$

Cevap

-6

The correct value is

-6

. Since the graph has

x

-intercepts at

(2, 0)

and

(-3, 0)

, the Factor Theorem dictates that

(x - 2)

and

(x + 3)

are factors of

p(x)

. Since the leading coefficient of the cubic polynomial

p(x)

is

1

, it can be written in factored form as

p(x) = (x - 2)(x + 3)(x - r)

, where

r

is the third root. According to the Remainder Theorem, the remainder when

p(x)

is divided by

x - 1

is equal to

p(1) = -8

. Substituting

x = 1

yields

-4(1 - r) = -8

, which simplifies to

1 - r = 2

, so

r = -1

. The polynomial is thus

p(x) = (x - 2)(x + 3)(x + 1)

. The constant term

d

is equal to the value of the function when

x = 0

, which is

p(0) = (-2)(3)(1) = -6

.

Adım Adım Çözüm

1

Determine the factors corresponding to the given

x

-intercepts of the polynomial.

The factors are

(x - 2)

and

(x + 3)

.

Since the graph of

y = p(x)

has

x

-intercepts at

(2, 0)

and

(-3, 0)

, we know that

p(2) = 0

and

p(-3) = 0

. According to the Factor Theorem,

(x - 2)

and

(x + 3)

must be factors of

p(x)

.

2

Set up the factored form of the cubic polynomial function

p(x)

.

p(x) = (x - 2)(x + 3)(x - r)

Because

p(x)

is a cubic polynomial (degree 3) with a leading coefficient of

1

, it can be written as the product of three linear factors:

(x - 2)

,

(x + 3)

, and

(x - r)

, where

r

is the unknown third root.

3

Apply the Remainder Theorem to solve for the third root

r

.

r = -1

By the Remainder Theorem, the remainder when

p(x)

is divided by

x - 1

is equal to

p(1)

. We are given that this remainder is

-8

, so

p(1) = -8

. Substituting

x = 1

into our factored expression gives:

p(1) = (1 - 2)(1 + 3)(1 - r) = -8 \Rightarrow -1 \cdot 4 \cdot (1 - r) = -8 \Rightarrow -4(1 - r) = -8 \Rightarrow 1 - r = 2 \Rightarrow r = -1

.

4

Determine the value of the constant coefficient

d

.

d = -6

Using the root

r = -1

, the complete factored expression is

p(x) = (x - 2)(x + 3)(x + 1)

. The constant term

d

is equivalent to

p(0)

:

d = p(0) = (0 - 2)(0 + 3)(0 + 1) = (-2)(3)(1) = -6

.

Anahtar Kavram

Using the Factor Theorem and Remainder Theorem to find unknown coefficients in a polynomial function.

Alternatif Yöntem

Alternatively, you can expand the general form

p(x) = (x - 2)(x + 3)(x - r) = (x^2 + x - 6)(x - r) = x^3 + (1 - r)x^2 - (r + 6)x + 6r

. Comparing this to

p(x) = x^3 + bx^2 + cx + d

, we see that

d = 6r

. Since the remainder when

p(x)

is divided by

x - 1

is

-8

, we have

p(1) = -8

. Substituting

x = 1

into our expanded form gives

1^3 + (1 - r)(1)^2 - (r + 6)(1) + 6r = -8 \Rightarrow 1 + 1 - r - r - 6 + 6r = -8 \Rightarrow 4r - 4 = -8 \Rightarrow 4r = -4 \Rightarrow r = -1

. Thus,

d = 6(-1) = -6

.

Tahmini Süre:2m 0s

Soru 1874Soru →

A polynomial function $P(x)$ of degree 4 with real coefficients is symmetric about the line $x = 2$ in the $xy$ -plane. If $P(x)$ is divisible by $x^2 - 4x + 3$ , the remainder when $P(x)$ is divided by $x - 4$ is $36$ , and $P(2) = -8$ , what is the value of $P(5)$ ?

Cevabı ve açıklamayı göster

Cevap: 136

Cevap

The value of

P(5)

is 136.

The correct answer of 136 is found by representing the symmetric fourth-degree polynomial as

P(x) = a(x - 2)^4 + b(x - 2)^2 + c

, solving for the coefficients using the roots at

x=3

, the point

(2, -8)

, and the remainder point

(4, 36)

, and then evaluating the resulting expression

(x-2)^4 + 7(x-2)^2 - 8

at

x=5

.

Adım Adım Çözüm

1

Express the fourth-degree polynomial in a form that reflects its symmetry.

P(x) = a(x - 2)^4 + b(x - 2)^2 + c

Since the graph of

P(x)

is symmetric about the line

x = 2

, the polynomial expression in terms of

(x - 2)

must contain only even powers.

2

Use the divisibility condition to establish an equation for the coefficients.

P(3) = a(3-2)^4 + b(3-2)^2 + c = a + b + c = 0

The divisor

x^2 - 4x + 3

factors into

(x-1)(x-3)

. By the Factor Theorem,

P(3) = 0

and

P(1) = 0

.

3

Use the given value

P(2) = -8

to find the constant term

c

.

c = -8

Substituting

x = 2

into

P(x) = a(x - 2)^4 + b(x - 2)^2 + c

makes the terms with

(x-2)

equal to zero, leaving

P(2) = c

.

4

Formulate a system of linear equations for

a

and

b

.

a + b = 8

and

4a + b = 11

Substituting

c = -8

into

a + b + c = 0

gives

a + b = 8

. By the Remainder Theorem,

P(4) = 36

, which gives

a(4-2)^4 + b(4-2)^2 - 8 = 36 \implies 16a + 4b = 44 \implies 4a + b = 11

.

5

Solve the system of equations for

a

and

b

.

a = 1

and

b = 7

Subtracting

a + b = 8

from

4a + b = 11

yields

3a = 3 \implies a = 1

, which then gives

b = 7

.

6

Evaluate the polynomial at

x = 5

.

P(5) = 136

Substitute

a = 1

,

b = 7

,

c = -8

, and

x = 5

into the symmetric polynomial form:

P(5) = (5-2)^4 + 7(5-2)^2 - 8 = 3^4 + 7(3^2) - 8 = 81 + 63 - 8 = 136

.

Anahtar Kavram

Polynomial Factors and Graphs

Soru 1875Soru →

An agricultural drone sprays liquid fertilizer at a rate of $1.5$ fluid ounces per square yard. The drone travels at a constant speed of $4.5$ miles per hour, spraying a continuous path that is exactly $6$ feet wide. Which of the following is closest to the rate at which the drone sprays the fertilizer, in gallons per minute? (Given that $1\text{ mile} = 5,280\text{ feet}$ , $1\text{ yard} = 3\text{ feet}$ , and $1\text{ gallon} = 128\text{ fluid ounces}$ )

3.09

3.58

9.28

27.84

Cevabı ve açıklamayı göster

Cevap: 3.09

Cevap

The rate of

3.09

gallons per minute is closest to the drone's fertilizer spraying rate.

The correct answer is

3.09

because converting the speed of

4.5

miles per hour gives

396

feet per minute. Multiplying this by the width of

6

feet results in

2,376

square feet per minute. Dividing this by

9

converts the rate to

264

square yards per minute. Multiplying by the spray density of

1.5

fluid ounces per square yard gives

396

fluid ounces per minute. Finally, dividing by

128

fluid ounces per gallon yields approximately

3.09

gallons per minute.

Adım Adım Çözüm

1

Convert the speed of the drone from miles per hour to feet per minute.

4.5\text{ miles/hour} = 396\text{ feet/minute}

To find the rate of area covered per minute, the speed must first be expressed in feet per minute. Multiply

4.5

by

5,280

to convert to feet per hour, then divide by

60

to convert to feet per minute:

\frac{4.5 \times 5,280}{60} = 396

.

2

Calculate the area covered by the drone per minute in square feet.

396\text{ feet/minute} \times 6\text{ feet} = 2,376\text{ square feet/minute}

Multiplying the travel speed in feet per minute by the path width of

6

feet gives the area covered per minute in square feet.

3

Convert the area rate from square feet per minute to square yards per minute.

2,376\text{ square feet/minute} \div 9 = 264\text{ square yards/minute}

Since

1\text{ yard} = 3\text{ feet}

, a square yard is

3 \times 3 = 9

square feet. Therefore, divide the area in square feet by

9

to convert to square yards.

4

Determine the fertilizer spray rate in fluid ounces per minute.

264\text{ square yards/minute} \times 1.5\text{ fluid ounces/square yard} = 396\text{ fluid ounces/minute}

Multiplying the area covered per minute by the spray density yields the volume of fertilizer sprayed per minute.

5

Convert the spray rate from fluid ounces per minute to gallons per minute.

396\text{ fluid ounces/minute} \div 128 \approx 3.09\text{ gallons/minute}

Since

1\text{ gallon} = 128\text{ fluid ounces}

, divide the fluid ounces per minute by

128

to find the rate in gallons per minute.

Anahtar Kavram

Unit conversions involving compound rates and area scaling.

Tahmini Süre:3m 0s

Soru 1876Soru →

Which of the following represents all possible values of $x$ that satisfy the inequality $-3(2x - 5) - 4x \geq 35$ ?

x \geq -2

x \leq -2

x \leq -5

x \leq 2

Cevabı ve açıklamayı göster

Cevap: $x \leq -2$

Cevap

The inequality is satisfied by all values of

x

such that

x \leq -2

.

To solve the inequality, distribute

-3

across the terms in the parentheses to get

-6x + 15 - 4x \geq 35

. Grouping the

x

terms gives

-10x + 15 \geq 35

. Subtracting 15 from both sides yields

-10x \geq 20

. Finally, dividing both sides by

-10

and reversing the inequality symbol gives

x \leq -2

.

Adım Adım Çözüm

1

Distribute the negative coefficient

-3

to both terms inside the parentheses.

-6x + 15 - 4x \geq 35

Applying the distributive property

a(b + c) = ab + ac

allows the removal of parentheses to simplify the expression.

2

Combine the variable terms

-6x

and

-4x

on the left side of the inequality.

-10x + 15 \geq 35

Combining like terms simplifies the left side of the inequality to prepare for isolating the variable.

3

Isolate the variable term by subtracting 15 from both sides of the inequality.

-10x \geq 20

Using the subtraction property of inequality maintains the balance of the inequality while isolating the term with the variable.

4

Divide both sides of the inequality by

-10

and reverse the direction of the inequality symbol.

x \leq -2

Dividing both sides of an inequality by a negative number requires reversing the inequality sign to keep the inequality statement true.

Anahtar Kavram

Solving multi-step linear inequalities in one variable, including distributing terms and reversing the inequality direction when dividing by a negative number.

Tahmini Süre:1m 30s

Soru 1877Soru →

For centuries, researchers have sought to understand the extraordinary durability of ancient Roman concrete, which has survived for millennia in harsh marine environments. Recent analyses reveal that the secret lies in hot mixing, a process that creates lime clasts capable of self-healing ______ water penetrates cracks in the structure, it dissolves these clasts to recrystallize and seal the gaps.

Which choice completes the text so that it conforms to the conventions of Standard English?

, when

when

; when

, but when

Cevabı ve açıklamayı göster

Cevap: ; when

Cevap

The correct answer is the option that uses a semicolon to separate the two independent clauses, stating '; when'.

The correct option uses a semicolon to separate the two independent clauses. The text before the blank ('Recent analyses reveal that the secret lies in hot mixing, a process that creates lime clasts capable of self-healing') is an independent clause, as is the text after the blank ('when water penetrates cracks in the structure, it dissolves these clasts to recrystallize and seal the gaps'). A semicolon is a grammatically correct way to join two independent clauses without a coordinating conjunction.

Adım Adım Çözüm

1

Identify the clause boundary where the blank is situated.

The sentence contains two main parts: 'Recent analyses reveal that the secret lies in hot mixing, a process that creates lime clasts capable of self-healing' and 'when water penetrates cracks in the structure, it dissolves these clasts to recrystallize and seal the gaps.'

To determine the correct punctuation or conjunction, we must analyze the grammatical relationship between these two parts.

2

Analyze the grammatical structure of both parts.

The first part is an independent clause because it contains a subject and a verb and can stand alone as a sentence. The second part is also an independent clause, containing a subordinate clause ('when water penetrates...') and a main clause ('it dissolves...').

Correctly identifying whether the clauses are independent or dependent is necessary because joining two independent clauses requires a semicolon, a period, or a comma coupled with a coordinating conjunction.

3

Evaluate the options to find the one that correctly links the two independent clauses.

The option with the semicolon correctly separates the two independent clauses. The comma-only option creates a comma splice, the option with no punctuation creates a run-on, and the option with 'but' introduces an illogical contrast.

Only the semicolon meets standard English conventions while preserving the logical relationship of the sentence.

Anahtar Kavram

Clause Boundaries and Linking

Soru 1878Soru →

At a certain car dealership, the number of hybrid vehicles sold in 2024 was $40\%$ of the number of gasoline vehicles sold in 2024. In 2025, the number of hybrid vehicles sold increased by $35\%$ compared to 2024, and the number of gasoline vehicles sold decreased by $10\%$ compared to 2024. If the dealership only sold hybrid and gasoline vehicles in both years, what percent of the total vehicles sold in 2025 were hybrid vehicles?

Cevabı ve açıklamayı göster

Cevap: 37.5

Cevap

37.5

To find the percentage of hybrid vehicles sold in 2025, we first express the number of vehicles sold in 2024 in terms of a single variable. Let the number of gasoline vehicles sold in 2024 be

g

. Since the hybrid sales were

40\%

of the gasoline sales, the number of hybrid vehicles sold in 2024 is

0.40g

. In 2025, hybrid sales increased by

35\%

, resulting in

0.40g \times 1.35 = 0.54g

hybrid vehicles sold. Gasoline sales decreased by

10\%

, resulting in

g \times 0.90 = 0.90g

gasoline vehicles sold. The total number of vehicles sold in 2025 is the sum of these two quantities:

0.54g + 0.90g = 1.44g

. The percentage of hybrid vehicles sold in 2025 is the ratio of hybrid sales to total sales:

\frac{0.54g}{1.44g} \times 100 = 37.5\%

.

Adım Adım Çözüm

1

Define variables for the number of vehicles sold in 2024.

Let the number of gasoline vehicles sold in 2024 be

g

. The number of hybrid vehicles sold in 2024 is

0.40g

.

Establishing initial values in terms of a single variable allows for comparison after percentage changes are applied.

2

Determine the number of hybrid and gasoline vehicles sold in 2025 after their respective percentage changes.

Hybrid vehicles sold in 2025:

0.40g \times (1 + 0.35) = 0.54g

. Gasoline vehicles sold in 2025:

g \times (1 - 0.10) = 0.90g

.

Applying the percent increase of

35\%

to the 2024 hybrid vehicles and the percent decrease of

10\%

to the 2024 gasoline vehicles.

3

Calculate the total number of vehicles sold in 2025.

Total vehicles sold in 2025:

0.54g + 0.90g = 1.44g

.

Summing the two types of vehicles sold in 2025 to find the new total.

4

Compute the percentage of hybrid vehicles out of the total vehicles sold in 2025.

\frac{0.54g}{1.44g} \times 100 = 37.5\%

.

Dividing the 2025 hybrid sales by the total 2025 sales and multiplying by 100 to convert to a percentage.

Anahtar Kavram

Calculating percent change and conditional percentages from initial relative ratios.

Soru 1879Soru →

If

x

is a real number that satisfies the equation below, what is the value of

x

?

\frac{x + 2}{x - 4} - \frac{3}{x} = \frac{12}{x^2 - 4x}

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The only valid real solution to the equation is

1

.

Multiplying the equation by the common denominator

x(x - 4)

simplifies the equation to

x^2 - x = 0

. Solving this quadratic equation yields

x = 0

and

x = 1

. Because

x = 0

leads to a division by zero in the original equation, it is extraneous. Therefore, the only valid real solution is

1

.

Adım Adım Çözüm

1

Determine the common denominator and restrictions for the rational equation.

The common denominator is

x(x - 4) = x^2 - 4x

. The restrictions are

x \neq 0

and

x \neq 4

.

Finding the common denominator allows us to eliminate fractions, while the restrictions help us identify potential extraneous solutions.

2

Multiply the entire equation by the common denominator

x(x - 4)

.

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

This step clears the rational expressions, leaving a polynomial equation.

3

Expand and simplify the polynomial equation.

x^2 + 2x - 3x + 12 = 12

, which simplifies to

x^2 - x = 0

.

Expanding the terms allows us to combine like terms and set the quadratic equation to zero.

4

Factor the quadratic equation to solve for

x

.

x(x - 1) = 0

, giving candidate solutions

x = 0

or

x = 1

.

Applying the zero-product property identifies the roots of the quadratic equation.

5

Verify the candidate solutions against the initial restrictions.

Since

x = 0

makes the denominators in the original equation equal to zero, it is extraneous. The candidate solution

x = 1

is valid.

We must verify solutions because multiplying by a variable expression can introduce extraneous roots that make the original expressions undefined.

Anahtar Kavram

Solving rational equations by clearing denominators and checking for extraneous solutions.

Tahmini Süre:1m 30s

Soru 1880Soru →

During a science experiment, a liquid is cooled at a constant rate of $0.5$ degrees Celsius per minute. What is the cooling rate of the liquid, in degrees Celsius per hour?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

The cooling rate of the liquid is

30

degrees Celsius per hour.

To find the cooling rate in degrees Celsius per hour, we multiply the rate per minute by the conversion factor. There are

60

minutes in

1

hour. Multiplying

0.5

degrees Celsius per minute by

60

minutes per hour yields

30

degrees Celsius per hour.

Adım Adım Çözüm

1

Identify the initial cooling rate per minute.

0.5

degrees Celsius per minute

This is the rate given in the problem statement.

2

Convert minutes to hours by multiplying the rate by

60

.

30

degrees Celsius per hour

An hour contains

60

minutes, so the total temperature decrease in one hour will be

60

times the decrease in one minute.

Anahtar Kavram

Unit conversions involving rates

Cevap

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