Tüm alıştırma soruları

Soru 5461Soru →

Although the single-celled slime mold *Physarum polycephalum* lacks a brain or nervous system, it exhibits a form of primitive intelligence. In experiments, researchers have observed the organism navigating complex mazes to locate food sources, choosing the shortest path with remarkable efficiency. To achieve this, the slime mold leaves behind a trail of extracellular slime as it explores. Upon encountering its own previous trails, the mold avoids them, using the slime as a spatial memory system to prevent it from wandering into areas it has already searched. This externalized memory system demonstrates that complex problem-solving behaviors do not necessarily require a highly developed central nervous system; rather, the mold relies on a totally cool externalized trick to get around.

Which choice best maintains the established tone and style of the passage?

NO CHANGE

the organism makes use of a system of spatial navigation that is external to its body and serves to guide it in its movement from one location to another

the organism employs a sophisticated mechanism to guide its movement

the specimen operationalizes a supernal modality of teleological environmental transit

Cevabı ve açıklamayı göster

Cevap: the organism employs a sophisticated mechanism to guide its movement

Cevap

the organism employs a sophisticated mechanism to guide its movement

The correct option, which states that the organism employs a sophisticated mechanism to guide its movement, successfully aligns with the formal, objective, and academic tone established in the rest of the passage. It is also clear, precise, and concise.

Adım Adım Çözüm

1

Analyze the established tone of the passage.

The passage maintains a formal, objective, and scientific tone (using terms like 'Physarum polycephalum,' 'extracellular slime,' and 'spatial memory system').

Stylistic appropriateness requires any replacement to align with this established academic register.

2

Evaluate the underlined portion.

The phrase 'relies on a totally cool externalized trick to get around' is identified as highly informal and colloquial.

Identifying the stylistic clash confirms that a change is necessary.

3

Assess the options to find the one that fits the passage's tone while being concise and grammatically correct.

The phrasing 'the organism employs a sophisticated mechanism to guide its movement' is formal, concise, and stylistically appropriate.

This option restores stylistic consistency without wordiness or overly dramatic language.

Anahtar Kavram

Stylistic Appropriateness

Soru 5462Soru →

A class of $40$ students has a ratio of boys to girls of $3:5$ . How many more girls than boys are in the class?

5

8

10

15

25

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

There are 10 more girls than boys in the class.

To find the difference between the number of girls and boys, we first determine the value of one part of the ratio. Since the ratio of boys to girls is

3:5

, there are

3 + 5 = 8

total parts. With

40

students in total, each part corresponds to

40 \div 8 = 5

students. The difference between the ratio parts is

5 - 3 = 2

parts. Multiplying this difference by the value of one part gives

2 \times 5 = 10

more girls than boys.

Adım Adım Çözüm

1

Find the total number of parts in the ratio by adding the parts for boys and girls.

3 + 5 = 8

parts

To determine how the total number of students is divided, we need the sum of the ratio parts.

2

Calculate how many students represent one part of the ratio by dividing the total number of students by the total parts.

40 \div 8 = 5

students per part

This establishes the scale factor for the ratio to convert parts into actual student counts.

3

Find the difference in ratio parts between girls and boys.

5 - 3 = 2

parts

The question asks for the difference in the number of girls and boys, which corresponds to the difference in their ratio parts.

4

Multiply the difference in parts by the number of students per part.

2 \times 5 = 10

students

This gives the final difference in the number of girls and boys.

Anahtar Kavram

Solving part-to-part ratio problems using total quantities

Alternatif Yöntem

Alternatively, calculate the actual number of boys and girls first. The number of boys is

\frac{3}{8} \times 40 = 15

and the number of girls is

\frac{5}{8} \times 40 = 25

. Subtracting the number of boys from the number of girls yields

25 - 15 = 10

.

Tahmini Süre:45s

Soru 5463Soru →

A portion of the passage below is underlined.

In 1956, Malcom McLean, a former trucking magnate, revolutionized global trade by introducing the modern shipping container. Before this innovation, cargo was loaded piece by piece, a process that was incredibly slow and expensive. McLean’s standardized steel boxes could be easily transferred from trucks to trains to ships, which was an efficient method that served to drastically reduce both loading times and shipping costs by a significant margin. Today, millions of these containers traverse the globe, forming the backbone of international commerce.

Which choice most effectively and concisely expresses the underlined idea?

NO CHANGE

ships, drastically reducing both loading times and shipping costs.

ships. Which drastically reduced both loading times and shipping costs.

ships, this drastically reduced both loading times and shipping costs.

Cevabı ve açıklamayı göster

Cevap: ships, drastically reducing both loading times and shipping costs.

Cevap

ships, drastically reducing both loading times and shipping costs.

The correct option is the most concise and direct phrasing that maintains grammatical correctness. It uses a participle clause to cleanly show the result of the standardized boxes being easily transferred, avoiding redundant words like 'efficient method' or 'significant margin'.

Adım Adım Çözüm

1

Identify redundancy and wordiness in the original sentence.

The phrase 'which was an efficient method that served to... by a significant margin' is highly repetitive and adds unnecessary length.

ACT English questions require the most concise and direct phrasing that is grammatically correct.

2

Evaluate the answer choices for conciseness and grammatical structure.

The option containing the participle 'drastically reducing' expresses the exact same idea in a fraction of the words and fits smoothly into the sentence structure without creating punctuation errors.

Eliminating wordiness while keeping the sentence grammatically sound is the primary goal of this question type.

Anahtar Kavram

Eliminating Wordiness

Tahmini Süre:1m 0s

Soru 5464Soru →

A writer is editing a passage about Mesoamerican history and has drafted the following paragraph:

Tatiana Proskouriakoff, an architect by training, revolutionized Mesoamerican archaeology in the mid-twentieth century through her meticulous drawings of ancient monuments. Working at the Carnegie Institution of Washington, she analyzed the dates inscribed on stelae—carved stone slabs—at the ruins of Piedras Negras. Rather than viewing the carvings as abstract astrological charts or depictions of mythical gods, Proskouriakoff proposed a groundbreaking theory: the monuments recorded the actual historical biographies of Mayan rulers. She noticed that the lifespan patterns of the recorded dates matched the average lifespans of human beings. Consequently, she hypothesized that the glyphs documented births, accessions to the throne, and deaths of real monarchs.

The writer is considering adding the following sentence to the end of the paragraph:

'Hieroglyphs were also carved into smaller jade ornaments and pottery found in Mayan tombs.'

Should the writer make this addition?

Yes, because it provides additional examples of the mediums upon which Mayan writing has been found.

Yes, because it supports the paragraph's assertion that Mayan hieroglyphs were highly detailed.

No, because it departs from the paragraph's focus on Proskouriakoff's breakthrough regarding the historical meaning of stelae inscriptions.

No, because it contradicts the paragraph's claim that stelae were the only records of Mayan historical events.

Cevabı ve açıklamayı göster

Cevap: No, because it departs from the paragraph's focus on Proskouriakoff's breakthrough regarding the historical meaning of stelae inscriptions.

Cevap

No, because it departs from the paragraph's focus on Proskouriakoff's breakthrough regarding the historical meaning of stelae inscriptions.

The correct answer correctly determines that the sentence should not be added because the paragraph's central focus is on Proskouriakoff's specific breakthrough in interpreting Mayan stelae as biographical records. The proposed sentence introduces irrelevant details about other mediums containing hieroglyphs, which diverges from this focus.

Adım Adım Çözüm

1

Identify the primary focus of the paragraph.

The paragraph focuses on Tatiana Proskouriakoff's breakthrough theory that Mayan stelae recorded the actual historical biographies of Mayan rulers, rather than abstract astrological or mythological data.

Understanding the paragraph's main theme is essential for evaluating whether new information is relevant.

2

Analyze the proposed sentence to see if it aligns with this focus.

The proposed sentence discusses Mayan hieroglyphs carved on jade ornaments and pottery in tombs, which is a different topic from stelae and Proskouriakoff's specific historical biography discovery.

This determines whether the proposed sentence is relevant or is a tangential distraction.

3

Evaluate the decision and justifications in the options.

Since the sentence is irrelevant, it should not be added. The correct reason is that it departs from the paragraph's focus on Proskouriakoff's breakthrough regarding stelae inscriptions.

This allows us to select the option that has both the correct decision (No) and the correct logical explanation.

Anahtar Kavram

Evaluating Relevance for Adding or Deleting Content

Soru 5465Soru →

If the least common multiple of a positive integer $n$ and 15 is 90, how many possible values are there for $n$ ?

1

2

3

4

6

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The correct answer is 2.

The correct answer is 2. The prime factorization of 15 is

3^1 \times 5^1

, and the prime factorization of 90 is

2^1 \times 3^2 \times 5^1

. The least common multiple (LCM) of two numbers is found by taking the highest power of each prime factor present in their factorizations. Since 90 contains

2^1

, and 15 does not,

n

must contain exactly

2^1

. Since 90 contains

3^2

, and 15 only contains

3^1

,

n

must contain exactly

3^2

. Since both 90 and 15 contain

5^1

, the power of 5 in

n

can be either 0 or 1. This results in two possible values:

2^1 \times 3^2 \times 5^0 = 18

and

2^1 \times 3^2 \times 5^1 = 90

.

Adım Adım Çözüm

1

Write out the prime factorizations of 15 and 90.

15 = 3^1 \times 5^1

and

90 = 2^1 \times 3^2 \times 5^1

.

To analyze the relationship between

n

, 15, and their least common multiple (LCM), we must look at their prime factor components.

2

Determine the constraints on the exponents of the prime factors of

n

using the definition of LCM.

The prime factorization of

n

must be of the form

2^a \times 3^b \times 5^c

. For the LCM of

n

and 15 to be 90, the exponent of each prime factor in the LCM is the maximum of its exponents in

n

and 15. Thus, we must have

a = 1

,

b = 2

, and

c \in \{0, 1\}

.

The LCM of two numbers takes the highest power of each prime factor present in both numbers.

3

Count the number of possible values for

n

.

Since

a

has 1 choice,

b

has 1 choice, and

c

has 2 choices, there are

1 \times 1 \times 2 = 2

possible values for

n

(

n = 2^1 \times 3^2 \times 5^0 = 18

and

n = 2^1 \times 3^2 \times 5^1 = 90

).

This gives the total number of distinct integers satisfying the condition.

Anahtar Kavram

The least common multiple of two numbers is found by taking the highest power of each prime factor present in the prime factorizations of both numbers.

Alternatif Yöntem

Alternatively, you can test the positive factors of 90 that are not factors of 15. The positive factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90. By computing the least common multiple of each factor with 15, we find that only LCM(18, 15) = 90 and LCM(90, 15) = 90 satisfy the condition.

Tahmini Süre:1m 0s

Soru 5466Soru →

A circle and a line are graphed in the standard $(x, y)$ coordinate plane. The equations of the circle and the line are given by:

(x-3)^2 + (y-3)^2 = 5

y = x + 1

The line intersects the circle at two points, $P$ and $Q$ . What is the distance between $P$ and $Q$ ?

2\sqrt{10}

3\sqrt{2}

\sqrt{22}

7\sqrt{2}

\sqrt{5}

Cevabı ve açıklamayı göster

Cevap: $3\sqrt{2}$

Cevap

The distance between the intersection points is

3\sqrt{2}

.

To find the points of intersection, substitute the linear equation into the circle's equation. This results in the quadratic equation

x^2 - 5x + 4 = 0

, which yields the solutions

x = 1

and

x = 4

. Substituting these values into the linear equation gives the points

(1, 2)

and

(4, 5)

. The distance between these points is computed using the distance formula, which gives

3\sqrt{2}

.

Adım Adım Çözüm

1

Substitute the equation of the line into the equation of the circle.

(x-3)^2 + ((x+1)-3)^2 = 5 \implies (x-3)^2 + (x-2)^2 = 5

This reduces the system of two equations with two variables to a single quadratic equation in terms of

x

.

2

Expand the squared binomials and simplify the quadratic equation.

(x^2 - 6x + 9) + (x^2 - 4x + 4) = 5 \implies 2x^2 - 10x + 13 = 5 \implies 2x^2 - 10x + 8 = 0 \implies x^2 - 5x + 4 = 0

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

3

Factor the quadratic equation to find the

x

-coordinates of the intersection points.

(x-1)(x-4) = 0 \implies x = 1 \text{ or } x = 4

Factoring is the most direct method to solve the simplified quadratic equation.

4

Find the corresponding

y

-coordinates by substituting the

x

-values back into the linear equation

y = x + 1

.

\text{For } x = 1: y = 1 + 1 = 2 \implies P(1, 2)

\text{For } x = 4: y = 4 + 1 = 5 \implies Q(4, 5)

This determines the coordinates of the two intersection points.

5

Use the distance formula to find the distance between the two points

P(1, 2)

and

Q(4, 5)

.

d = \sqrt{(4-1)^2 + (5-2)^2} = \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2}

The distance formula calculates the straight-line distance between the two coordinates.

Anahtar Kavram

Solving systems of linear and quadratic equations by substitution and finding the distance between intersection points.

Alternatif Yöntem

Find the distance geometrically: The center of the circle is

(3, 3)

and the radius is

r = \sqrt{5}

. The distance

d

from the center to the line

x - y + 1 = 0

is

d = \frac{|3 - 3 + 1|}{\sqrt{1^2 + (-1)^2}} = \frac{1}{\sqrt{2}}

. Using a right triangle formed by the radius, the distance from the center, and half of the chord length

h

, we have

h = \sqrt{r^2 - d^2} = \sqrt{5 - \frac{1}{2}} = \sqrt{\frac{9}{2}} = \frac{3}{\sqrt{2}}

. The total distance between the intersection points is the full chord length,

2h = 2 \times \frac{3}{\sqrt{2}} = 3\sqrt{2}

.

Tahmini Süre:3m 0s

Soru 5467Soru →

For the imaginary unit $i$ , where $i^2 = -1$ , which of the following complex numbers is equal to $\frac{8 + i}{3 + 2i}$ ?

\frac{22}{13} - i

\frac{26}{5} - \frac{13}{5}i

2 + i

2 - i

\frac{8}{3} + \frac{1}{2}i

Cevabı ve açıklamayı göster

Cevap: 2 - i

Cevap

The complex number

2 - i

To divide two complex numbers, we multiply both the numerator and denominator by the complex conjugate of the denominator, which is

3 - 2i

. Expanding the numerator gives

(8 + i)(3 - 2i) = 24 - 16i + 3i - 2i^2 = 26 - 13i

because

i^2 = -1

. Expanding the denominator yields

(3 + 2i)(3 - 2i) = 9 - 4i^2 = 9 + 4 = 13

. Dividing the terms of the numerator by the denominator gives

\frac{26 - 13i}{13} = 2 - i

.

Adım Adım Çözüm

1

Multiply the numerator and the denominator of the fraction by the complex conjugate of the denominator.

\frac{8 + i}{3 + 2i} \cdot \frac{3 - 2i}{3 - 2i} = \frac{(8 + i)(3 - 2i)}{(3 + 2i)(3 - 2i)}

Multiplying by the conjugate rationalizes the denominator, converting it into a real number.

2

Expand the numerator and the denominator using binomial multiplication.

Numerator:

(8 + i)(3 - 2i) = 24 - 16i + 3i - 2i^2

Denominator:

(3 + 2i)(3 - 2i) = 9 - 6i + 6i - 4i^2 = 9 - 4i^2

Distribute each term in the first binomial to each term in the second binomial.

3

Substitute

i^2 = -1

and simplify both expressions.

Numerator:

24 - 13i - 2(-1) = 24 - 13i + 2 = 26 - 13i

Denominator:

9 - 4(-1) = 9 + 4 = 13

The definition of the imaginary unit is

i^2 = -1

.

4

Divide each term of the simplified numerator by the simplified denominator.

\frac{26 - 13i}{13} = \frac{26}{13} - \frac{13i}{13} = 2 - i

Separate the real and imaginary parts to write the complex number in standard form

a + bi

.

Anahtar Kavram

Division of complex numbers using the complex conjugate of the denominator.

Alternatif Yöntem

Instead of dividing directly, let the result be

x + yi

. Then

(x + yi)(3 + 2i) = 8 + i

. Expanding this gives

(3x - 2y) + (2x + 3y)i = 8 + i

. Equating the real and imaginary parts gives the system of equations

3x - 2y = 8

and

2x + 3y = 1

. Solving this system yields

x = 2

and

y = -1

, which corresponds to the complex number

2 - i

.

Tahmini Süre:1m 30s

Soru 5468Soru →

A small drone's path in a vertical plane is modeled by the equation $y = 3x^2 - 4x + 2$ , where $x$ is the horizontal distance in meters and $y$ is the height in meters. A laser beam travels along a straight line in the same plane such that the sum of twice its horizontal distance and its height is a constant $c$ , where both are in meters. The laser beam intersects the drone's path at two distinct points. If the distance between these two intersection points is $\frac{5\sqrt{2}}{3}$ meters, what is the value of $2c$ ?

6

10

5

-\frac{1}{3}

\frac{5}{3}

Cevabı ve açıklamayı göster

Cevap: $5$

Cevap

The correct value of

2c

is

5

.

The correct value of

2c

is

5

. Substituting the linear equation

y = c - 2x

into the quadratic equation

y = 3x^2 - 4x + 2

yields

3x^2 - 2x + (2-c) = 0

. The distance between the intersection points is

d = |x_2 - x_1|\sqrt{5} = \frac{5\sqrt{2}}{3}

, which simplifies to

(x_2 - x_1)^2 = \frac{10}{9}

. Using the identity

(x_2 - x_1)^2 = (x_1 + x_2)^2 - 4x_1 x_2

and Vieta's formulas, we find the equation

\frac{4}{9} - \frac{8-4c}{3} = \frac{10}{9}

. Solving this equation yields

c = \frac{5}{2}

, and thus

2c = 5

.

Adım Adım Çözüm

1

Set up the system of equations.

The drone's path is

y = 3x^2 - 4x + 2

and the laser's path is

2x + y = c \implies y = -2x + c

.

This represents the mathematical formulation of both paths in the vertical plane.

2

Equate the equations to find the x-coordinates of the intersection points.

3x^2 - 4x + 2 = -2x + c \implies 3x^2 - 2x + (2-c) = 0

.

The intersection points satisfy both equations, so we can solve for

x

by substitution.

3

Express the distance between the intersection points

P(x_1, y_1)

and

Q(x_2, y_2)

using the slope.

d = |x_2 - x_1|\sqrt{5}

.

Since the points lie on the line with slope

-2

, we have

y_2 - y_1 = -2(x_2 - x_1)

. The distance formula becomes

d = \sqrt{(x_2 - x_1)^2 + (-2(x_2 - x_1))^2} = \sqrt{5(x_2 - x_1)^2} = |x_2 - x_1|\sqrt{5}

.

4

Equate the distance expression to the given distance to find

(x_2 - x_1)^2

.

(x_2 - x_1)^2 = \frac{10}{9}

.

We are given

d = \frac{5\sqrt{2}}{3}

. Setting

|x_2 - x_1|\sqrt{5} = \frac{5\sqrt{2}}{3}

and squaring both sides gives

5(x_2 - x_1)^2 = \frac{50}{9}

, which simplifies to

(x_2 - x_1)^2 = \frac{10}{9}

.

5

Apply Vieta's formulas and the algebraic identity for

(x_2 - x_1)^2

.

\frac{10}{9} = \frac{4}{9} - \frac{8-4c}{3}

.

For the quadratic equation

3x^2 - 2x + (2-c) = 0

, we have

x_1 + x_2 = \frac{2}{3}

and

x_1 x_2 = \frac{2-c}{3}

. We use the identity

(x_2 - x_1)^2 = (x_1 + x_2)^2 - 4x_1 x_2

.

6

Solve for

c

and find

2c

.

c = \frac{5}{2} \implies 2c = 5

.

Multiplying the equation by

9

gives

10 = 4 - 3(8-4c) \implies 10 = -20 + 12c \implies 12c = 30 \implies c = \frac{5}{2}

. Therefore,

2c = 5

.

Anahtar Kavram

Solving systems of linear and non-linear equations using substitution, coordinate geometry distance formula, and quadratic root relationships.

Tahmini Süre:3m 0s

Soru 5469Soru →

The polynomial $4x^2 - 8x - 5$ is subtracted from the polynomial $7x^2 - 3x + 4$ . The simplified difference can be expressed as $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constant integers. What is the value of the coefficient $b$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The coefficient of the

x

term,

b

, is 5.

Subtracting

4x^2 - 8x - 5

from

7x^2 - 3x + 4

yields

(7x^2 - 4x^2) + (-3x - (-8x)) + (4 - (-5)) = 3x^2 + 5x + 9

. The coefficient of the linear term

x

is

5

.

Adım Adım Çözüm

1

Set up the subtraction expression.

(7x^2 - 3x + 4) - (4x^2 - 8x - 5)

Subtracting the second polynomial from the first requires enclosing the second polynomial in parentheses to apply the subtraction to all terms.

2

Distribute the negative sign to all terms inside the parentheses.

7x^2 - 3x + 4 - 4x^2 + 8x + 5

Distributing the subtraction sign flips the sign of each term: positive terms become negative, and negative terms become positive.

3

Group and combine like terms.

3x^2 + 5x + 9

Combine the coefficients of matching variable parts:

(7 - 4)x^2 = 3x^2

,

(-3 + 8)x = 5x

, and

4 - (-5) = 9

.

4

Identify the coefficient

b

corresponding to the

x

term.

b = 5

Comparing the simplified expression

3x^2 + 5x + 9

to

ax^2 + bx + c

shows that the coefficient of

x

is 5.

Anahtar Kavram

Polynomial Subtraction and Combining Like Terms

Soru 5470Soru →

An ellipse in the standard $(x, y)$ coordinate plane is defined by the equation

\frac{(x-2)^2}{9} + \frac{(y+5)^2}{16} = 1

What are the coordinates of the center of this ellipse?

(-2, 5)

(2, 5)

(2, -5)

(-2, -5)

(-5, 2)

Cevabı ve açıklamayı göster

Cevap: $(2, -5)$

Cevap

The center of the ellipse is

(2, -5)

.

The standard form of an ellipse equation is

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1

, where the center of the ellipse is at the coordinate point

(h, k)

. Comparing the given equation

\frac{(x-2)^2}{9} + \frac{(y+5)^2}{16} = 1

to the standard form reveals that

h = 2

and

k = -5

. Therefore, the center coordinates are

(2, -5)

.

Adım Adım Çözüm

1

Recall the standard form equation of an ellipse with horizontal/vertical axes.

The standard form is

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1

, where

(h, k)

represents the coordinates of the center.

This establishes the framework to extract the center coordinate values.

2

Compare the terms in the given equation to the standard form.

Matching

(x-h)^2

with

(x-2)^2

gives

h = 2

. Matching

(y-k)^2

with

(y+5)^2 = (y-(-5))^2

gives

k = -5

.

Comparing terms identifies the offsets

h

and

k

that determine the center.

3

Write the center coordinate pair

(h, k)

.

The center is

(2, -5)

.

Combining the values of

h

and

k

yields the final coordinates.

Anahtar Kavram

Identifying the center of an ellipse from its standard form equation

Tahmini Süre:45s

Soru 5471Soru →

A business owner registers $4$ identical cell phone lines under a group plan. The service provider charges a monthly base fee of $\$15.00$ per line, plus $\$0.05$ per minute of call time. If the total monthly bill for all $4$ lines was $\$96.00$ and each line used the exact same number of minutes, how many minutes did each line use?

180

405

720

31.5

1,620

Cevabı ve açıklamayı göster

Cevap: $180$

Cevap

Each cell phone line used

180

minutes.

The correct answer is

180

minutes. The total bill for the

4

lines is represented by

4(15.00 + 0.05m) = 96.00

. Dividing both sides by

4

gives the monthly charge per line:

15.00 + 0.05m = 24.00

. Subtracting the base fee of

15.00

from both sides leaves the total per-minute cost of

0.05m = 9.00

. Dividing by the per-minute rate of

0.05

gives

m = 180

.

Adım Adım Çözüm

1

Set up the linear equation representing the total monthly bill.

4(15.00 + 0.05m) = 96.00

, where

m

represents the number of minutes used per line.

The total bill is the sum of the charges for all

4

lines, where each line costs

\$15.00

plus

\$0.05

per minute.

2

Divide both sides of the equation by

4

to isolate the single-line cost expression.

15.00 + 0.05m = 24.00

Dividing both sides by

4

simplifies the equation and isolates the cost per line.

3

Subtract the base fee of

15.00

from both sides of the equation.

0.05m = 9.00

This isolates the variable charge term on the left side of the equation.

4

Divide both sides by

0.05

to solve for

m

.

m = 180

Dividing the remaining total of

9.00

by the rate of

0.05

per minute yields the total number of minutes used.

Anahtar Kavram

Solving Linear Equations

Alternatif Yöntem

Instead of dividing by

4

first, distribute the

4

to both terms inside the parentheses:

4(15.00) + 4(0.05m) = 96.00

. This simplifies to

60.00 + 0.20m = 96.00

. Subtract

60.00

from both sides to get

0.20m = 36.00

. Finally, divide by

0.20

to find

m = 180

.

Tahmini Süre:1m 15s

Soru 5472Soru →

If $x$ is a real number such that $\log_5(x) + \log_5(x - 20) = 3$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

The value of

x

is 25.

Applying the logarithmic product rule simplifies the equation to

\log_5(x^2 - 20x) = 3

. Writing this in exponential form yields

x^2 - 20x = 125

. Rearranging into standard form gives

x^2 - 20x - 125 = 0

, which factors into

(x - 25)(x + 5) = 0

. This gives potential solutions of

25

and

-5

. Because the logarithmic arguments must be strictly positive,

x = -5

is extraneous. Therefore, the only correct value is 25.

Adım Adım Çözüm

1

Use the product property of logarithms to combine the terms on the left side.

\log_5(x(x - 20)) = 3

The sum of logarithms with the same base is equal to the logarithm of the product of their arguments:

\log_b(M) + \log_b(N) = \log_b(MN)

.

2

Rewrite the logarithmic equation in exponential form.

x(x - 20) = 5^3 \implies x^2 - 20x = 125

The logarithmic equation

\log_b(y) = c

is equivalent to the exponential equation

b^c = y

.

3

Rearrange the quadratic equation into standard form and solve by factoring.

x^2 - 20x - 125 = 0 \implies (x - 25)(x + 5) = 0 \implies x = 25 \text{ or } x = -5

Subtracting 125 from both sides sets the quadratic equation to 0, which can then be factored into binomials whose product is 0.

4

Verify the potential solutions in the original equation to identify any extraneous roots.

For

x = -5

, the arguments of the original logarithms are negative, which is undefined. For

x = 25

, the arguments are positive. Thus, the only valid solution is

x = 25

.

Logarithmic functions are only defined for positive real numbers. Therefore, we must have

x > 0

and

x - 20 > 0

, which requires

x > 20

.

Anahtar Kavram

Solving logarithmic equations by combining logarithmic terms and checking for extraneous solutions.

Tahmini Süre:1m 30s

Soru 5473Soru →

The measures of the three interior angles of a triangle are in the ratio $2:3:4$ . What is the measure, in degrees, of the smallest angle of the triangle?

20

40

60

72

80

Cevabı ve açıklamayı göster

Cevap: 40

Cevap

40

The interior angles of any triangle sum to

180^\circ

. Given the ratio of the angles is

2:3:4

, the total number of ratio parts is

2 + 3 + 4 = 9

. Dividing the total degree sum by the number of parts gives

\frac{180^\circ}{9} = 20^\circ

per part. Since the smallest angle corresponds to the smallest part of the ratio, we multiply

2

by

20^\circ

to get

40^\circ

.

Adım Adım Çözüm

1

Find the total number of parts in the ratio.

The total number of parts is

2 + 3 + 4 = 9

.

This determines how the

180^\circ

total sum of a triangle's interior angles is partitioned.

2

Calculate the value of a single part of the ratio.

One part is equal to

\frac{180^\circ}{9} = 20^\circ

.

The sum of the interior angles of any triangle is always

180^\circ

.

3

Find the measure of the smallest angle by multiplying by the smallest part of the ratio.

The smallest angle measures

2 \times 20^\circ = 40^\circ

.

The smallest angle corresponds to the smallest number in the ratio, which is 2.

Anahtar Kavram

Ratio-based angle partitioning in triangles

Soru 5474Soru →

If $9^{2x - 1} = 27^{x + 4}$ , what is the value of $x$ ?

5

-13

14

6

-4

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

14

Converting both bases to 3 yields

(3^2)^{2x - 1} = (3^3)^{x + 4}

. Applying the exponent power rule gives

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

. Equating the exponents results in

4x - 2 = 3x + 12

. Subtracting

3x

from both sides gives

x - 2 = 12

, and adding 2 to both sides results in

x = 14

. This matches the correct value of 14.

Adım Adım Çözüm

1

Express both sides of the equation with a common base of 3.

(3^2)^{2x - 1} = (3^3)^{x + 4}

Since

9 = 3^2

and

27 = 3^3

, rewriting the bases allows us to equate the exponents later.

2

Apply the power of a power property,

(a^m)^n = a^{mn}

, to simplify the exponents on both sides.

3^{2(2x - 1)} = 3^{3(x + 4)}

which simplifies to

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

To simplify an exponent raised to another power, multiply the exponents, ensuring the multiplier is distributed to both terms inside each exponent expression.

3

Set the exponents equal to each other and solve the resulting linear equation for

x

.

4x - 2 = 3x + 12 \implies x = 14

If two exponential expressions with the same positive base (other than 1) are equal, their exponents must be equal.

Anahtar Kavram

Solving exponential equations by expressing bases with a common base and equating the exponents.

Tahmini Süre:1m 30s

Soru 5475Soru →

In the standard $(x, y)$ coordinate plane, line $l_1$ passes through the points $(-2, 5)$ and $(4, 1)$ . Line $l_2$ is perpendicular to line $l_1$ . What is the slope of line $l_2$ ?

-\frac{3}{2}

-\frac{2}{3}

\frac{2}{3}

\frac{3}{2}

\frac{1}{2}

Cevabı ve açıklamayı göster

Cevap: $\frac{3}{2}$

Cevap

The slope of the perpendicular line is

\frac{3}{2}

.

To find the slope of a line perpendicular to a given line, first calculate the slope of the original line,

l_1

, using the slope formula

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

. Substituting the points

(-2, 5)

and

(4, 1)

into the formula gives

m_1 = \frac{1 - 5}{4 - (-2)} = \frac{-4}{6} = -\frac{2}{3}

. The slope of a perpendicular line,

l_2

, is the negative reciprocal of the slope of

l_1

. The negative reciprocal of

-\frac{2}{3}

is

\frac{3}{2}

.

Adım Adım Çözüm

1

Calculate the slope of the line

l_1

passing through the points

(-2, 5)

and

(4, 1)

using the slope formula.

m_1 = \frac{1 - 5}{4 - (-2)} = \frac{-4}{6} = -\frac{2}{3}

The slope

m

of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

.

2

Find the slope of line

l_2

, which is perpendicular to

l_1

, by taking the negative reciprocal of

m_1

.

m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{2}{3}} = \frac{3}{2}

The product of the slopes of two perpendicular lines is

-1

, so the slope of a perpendicular line is the negative reciprocal of the original slope.

Anahtar Kavram

Slope of perpendicular lines and finding slope from two points.

Tahmini Süre:1m 0s

Soru 5476Soru →

In $\triangle ABC$ , the measure of angle $A$ is $40^\circ$ . Point $D$ lies on side $AC$ such that segment $BD$ bisects angle $ABC$ . If the measure of angle $BDC$ is $75^\circ$ , what is the measure, in degrees, of angle $C$ ?

Cevabı ve açıklamayı göster

Cevap: 70

Cevap

The measure of angle C is 70 degrees.

The correct measure of angle

C

is found by first identifying that angle

ADB

is supplementary to angle

BDC

, giving a measure of

105^\circ

. Using the triangle angle sum theorem on triangle

ABD

, we find that angle

ABD

is

35^\circ

. Since

BD

bisects angle

ABC

, angle

DBC

is also

35^\circ

. Finally, applying the triangle angle sum theorem to triangle

BCD

, we subtract the measures of angles

DBC

(

35^\circ

) and

BDC

(

75^\circ

) from

180^\circ

to get

70^\circ

.

Adım Adım Çözüm

1

Find the measure of angle

ADB

using the supplementary angle relationship with angle

BDC

.

105^\circ

Angles

ADB

and

BDC

form a linear pair along the line segment

AC

, so their sum is

180^\circ

.

2

Find the measure of angle

ABD

using the sum of interior angles in

\triangle ABD

.

35^\circ

The sum of interior angles in any triangle is

180^\circ

. Therefore, the measure of angle

ABD

is

180^\circ - (40^\circ + 105^\circ) = 35^\circ

.

3

Find the measure of angle

DBC

using the definition of an angle bisector.

35^\circ

Since segment

BD

bisects angle

ABC

, the measures of angles

ABD

and

DBC

must be equal.

4

Find the measure of angle

C

using the sum of interior angles in

\triangle BCD

.

70^\circ

The sum of interior angles in

\triangle BCD

is

180^\circ

. Therefore, the measure of angle

C

is

180^\circ - (35^\circ + 75^\circ) = 70^\circ

.

Anahtar Kavram

Using the triangle angle sum theorem and angle bisector properties to determine unknown angle measures in a geometric figure.

Soru 5477Soru →

In right triangle $ABC$ , the hypotenuse $AC$ has a length of 13 centimeters, and leg $AB$ has a length of 5 centimeters. What is the length, in centimeters, of leg $BC$ ?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The length of leg

BC

is 12 centimeters.

Applying the Pythagorean theorem, we have

5^2 + BC^2 = 13^2

, which simplifies to

25 + BC^2 = 169

. Subtracting 25 from both sides gives

BC^2 = 144

, and taking the square root of both sides gives

BC = 12

centimeters.

Adım Adım Çözüm

1

Set up the Pythagorean Theorem equation for right triangle

ABC

.

AB^2 + BC^2 = AC^2

The Pythagorean Theorem states that in a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse.

2

Substitute the given values for

AB

and

AC

into the formula.

5^2 + BC^2 = 13^2

The length of leg

AB

is given as 5 centimeters, and the length of the hypotenuse

AC

is given as 13 centimeters.

3

Solve for the unknown leg length

BC

.

BC = 12

Squaring the values gives

25 + BC^2 = 169

. Subtracting 25 from both sides yields

BC^2 = 144

. Taking the square root of both sides gives

BC = 12

.

Anahtar Kavram

Pythagorean Theorem

Alternatif Yöntem

Recognize the triangle as a standard 5-12-13 Pythagorean triple, which immediately gives the missing leg length of 12 without needing calculations.

Tahmini Süre:30s

Soru 5478Soru →

The circle $x^2 + y^2 = 25$ and the line $y = 2x - 5$ intersect at two points. What is the sum of the $y$ -coordinates of these two intersection points?

Cevabı ve açıklamayı göster

Cevap: -2

Cevap

The sum of the

y

-coordinates of the intersection points is

-2

.

Substituting

y = 2x - 5

into the circular equation

x^2 + y^2 = 25

yields the quadratic equation

5x^2 - 20x = 0

. Factoring this equation gives

x = 0

and

x = 4

. Evaluating the linear equation at these values gives the

y

-coordinates

-5

and

3

. The sum of these coordinates is

-5 + 3 = -2

.

Adım Adım Çözüm

1

Substitute

y = 2x - 5

into the circle equation

x^2 + y^2 = 25

.

x^2 + (2x - 5)^2 = 25

To find the points of intersection, we solve the system of equations by substitution.

2

Expand and simplify the resulting equation.

5x^2 - 20x = 0

Expanding

(2x - 5)^2

gives

4x^2 - 20x + 25

. Combining like terms and subtracting 25 from both sides simplifies the equation.

3

Factor the quadratic equation to solve for

x

.

x = 0

or

x = 4

Factoring out

5x

gives

5x(x - 4) = 0

, which yields the roots

x = 0

and

x = 4

.

4

Find the corresponding

y

-coordinates by substituting the

x

-values back into

y = 2x - 5

.

The intersection points are

(0, -5)

and

(4, 3)

.

For

x = 0

,

y = 2(0) - 5 = -5

. For

x = 4

,

y = 2(4) - 5 = 3

.

5

Calculate the sum of the

y

-coordinates.

-2

Adding the

y

-coordinates

-5

and

3

gives

-5 + 3 = -2

.

Anahtar Kavram

Systems of Linear and Non-Linear Equations

Soru 5479Soru →

Consider the system of equations consisting of the quadratic function $f(x) = (x - 2)^2 - 3$ and the linear function $g(x) = 3x - 9$ . If the graphs of these functions intersect at two distinct points, $(x_1, y_1)$ and $(x_2, y_2)$ , what is the value of $x_1 y_2 + x_2 y_1$ ?

-15

-8

-3

15

24

Cevabı ve açıklamayı göster

Cevap: -3

Cevap

The correct answer is

-3

. The intersection points of the two functions are

(2, -3)

and

(5, 6)

, and evaluating the expression

x_1 y_2 + x_2 y_1

gives

-3

.

To find the points where the graphs of the functions intersect, we set their expressions equal to each other:

(x - 2)^2 - 3 = 3x - 9

. Expanding the squared term gives

x^2 - 4x + 4 - 3 = 3x - 9

, which simplifies to

x^2 - 4x + 1 = 3x - 9

. Moving all terms to the left side yields

x^2 - 7x + 10 = 0

. Factoring this quadratic equation gives

(x - 2)(x - 5) = 0

, so the

x

-coordinates of the intersection points are

2

and

5

. Substituting these back into the linear equation gives the corresponding

y

-coordinates:

y = 3(2) - 9 = -3

and

y = 3(5) - 9 = 6

, resulting in the intersection points

(2, -3)

and

(5, 6)

. Finally, evaluating the requested expression gives

(2)(6) + (5)(-3) = 12 - 15 = -3

.

Adım Adım Çözüm

1

Set the quadratic and linear functions equal to find the

x

-coordinates of their intersection points.

(x - 2)^2 - 3 = 3x - 9

At the points of intersection, the values of

f(x)

and

g(x)

must be equal.

2

Expand the binomial squared term

(x - 2)^2

.

x^2 - 4x + 4 - 3 = 3x - 9

Expanding the binomial is necessary to combine like terms and write the equation in standard quadratic form.

3

Move all terms to one side to set the equation to zero.

x^2 - 7x + 10 = 0

A quadratic equation must be set to zero before factoring or applying the quadratic formula.

4

Factor the quadratic equation.

(x - 2)(x - 5) = 0 \implies x_1 = 2 \text{ and } x_2 = 5

Factoring determines the

x

-coordinates of the intersection points.

5

Substitute the

x

-values into the linear equation

g(x) = 3x - 9

to find the corresponding

y

-coordinates.

y_1 = 3(2) - 9 = -3

y_2 = 3(5) - 9 = 6

This yields the two intersection points:

(2, -3)

and

(5, 6)

.

6

Calculate the value of the expression

x_1 y_2 + x_2 y_1

.

(2)(6) + (5)(-3) = 12 - 15 = -3

This evaluates the requested secondary value using the intersection coordinates.

Anahtar Kavram

Solving systems of linear and non-linear equations by substitution and algebraic manipulation.

Soru 5480Soru →

In the standard $(x, y)$ coordinate plane, a triangle has vertices at $A(0, 1)$ , $B(2, 3)$ , and $C(8, 11)$ . If point $M$ is the midpoint of side $AB$ and point $N$ is the midpoint of side $AC$ , what is the length of the line segment $MN$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The length of the line segment

MN

is

5

.

The length of the line segment

MN

is

5

. Calculating the coordinates of the midpoint of

AB

, we get

M(1, 2)

. For

AC

, the midpoint is

N(4, 6)

. Applying the distance formula between

M

and

N

yields

\sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9+16} = 5

. Alternatively, by the Midsegment Theorem, the segment connecting the midpoints of two sides of a triangle is half the length of the third side. The length of the third side

BC

is

\sqrt{(8-2)^2 + (11-3)^2} = \sqrt{36 + 64} = 10

, so the length of

MN

is

\frac{10}{2} = 5

.

Adım Adım Çözüm

1

Find the coordinates of

M

, the midpoint of side

AB

with endpoints

A(0, 1)

and

B(2, 3)

.

M(1, 2)

The midpoint formula is

M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

.

2

Find the coordinates of

N

, the midpoint of side

AC

with endpoints

A(0, 1)

and

C(8, 11)

.

N(4, 6)

Applying the midpoint formula gives

\left(\frac{0 + 8}{2}, \frac{1 + 11}{2}\right) = (4, 6)

.

3

Calculate the distance between

M(1, 2)

and

N(4, 6)

using the distance formula.

5

The distance formula is

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

. Here,

d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5

.

Anahtar Kavram

Midpoint and Distance Formulas

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

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Sık Yapılan Hatalar

Alternatif Yöntem

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

Alternatif Yöntem

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Sık Yapılan Hatalar

Alternatif Yöntem

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Sık Yapılan Hatalar

Alternatif Yöntem

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Sık Yapılan Hatalar

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Anahtar Kavram

Sık Yapılan Hatalar

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Anahtar Kavram

Sık Yapılan Hatalar

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Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Alternatif Yöntem

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Sık Yapılan Hatalar

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