Tüm alıştırma soruları

Soru 2721Soru →

While preparing a presentation on Late Bronze Age trade, a student took the following notes:
* The Uluburun shipwreck is a Late Bronze Age vessel dating to the late 14th century BCE.
* It was discovered off the coast of Kas, Turkey, in 1982.
* Excavations conducted between 1984 and 1994 recovered a wide range of goods from across the Mediterranean.
* The ship's cargo included ten tons of copper ingots and one ton of tin ingots, which were the raw materials used to make bronze.
* Other items found on board included Canaanite jars, Mycenaean pottery, and raw glass ingots.

The student wants to emphasize the ship's cargo of raw metal ingots. Which choice most effectively uses information from the given notes to accomplish this goal?

Among the items recovered from the Uluburun shipwreck were ten tons of copper ingots and one ton of tin ingots, the raw materials needed to produce bronze.

Discovered in 1982 off the coast of Kas, Turkey, the Uluburun shipwreck is a Late Bronze Age vessel dating to the late 14th century BCE.

The Uluburun shipwreck carried finished bronze statues that were manufactured from ten tons of copper and one ton of tin.

The presence of Canaanite jars and Mycenaean pottery on the Uluburun shipwreck suggests that the crew traded extensively with various civilizations throughout the Mediterranean.

Cevabı ve açıklamayı göster

Cevap: Among the items recovered from the Uluburun shipwreck were ten tons of copper ingots and one ton of tin ingots, the raw materials needed to produce bronze.

Cevap

Among the items recovered from the Uluburun shipwreck were ten tons of copper ingots and one ton of tin ingots, the raw materials needed to produce bronze.

The option highlighting the ten tons of copper ingots and one ton of tin ingots is correct because it directly presents the raw metal cargo details from the notes, fulfilling the student's rhetorical goal.

Adım Adım Çözüm

1

Analyze the prompt to identify the student's rhetorical goal.

The goal is to emphasize the ship's cargo of raw metal ingots.

Understanding the specific target detail is necessary to evaluate the suitability of each choice.

2

Examine the notes to find details corresponding to the cargo of raw metal ingots.

The notes specify that the cargo included ten tons of copper ingots and one ton of tin ingots, which served as raw materials for bronze.

The correct choice must contain these specific details.

3

Evaluate the options against the rhetorical goal.

The option highlighting the ten tons of copper ingots and one ton of tin ingots aligns perfectly with the goal, while the other options either focus on unrelated details, misread the notes, or make speculative inferences.

Selecting the option that matches the required details and goal avoids distractor errors.

Anahtar Kavram

Rhetorical Synthesis: Emphasizing Specific Details

Soru 2722Soru →

Archaeologists analyzing obsidian artifacts from the ancient Maya city of Kaminaljuyu found that while domestic tools were made from local obsidian sources, ceremonial blades were crafted from Ixtepeque obsidian, located over eighty kilometers away. Because Ixtepeque obsidian is chemically distinct and required traveling through rugged terrain to acquire, researchers long assumed its use was strictly limited to the ruling elites as a marker of high socio-political status. However, recent excavations of humble residential structures occupied by commoners revealed abundant Ixtepeque obsidian ceremonial blade fragments, which suggests that

Which choice most logically completes the text?

the use of Ixtepeque obsidian for ceremonial blades was not restricted solely to the ruling elite class in Kaminaljuyu.

commoners at Kaminaljuyu relied on Ixtepeque obsidian to manufacture their daily domestic tools.

the rugged terrain between Kaminaljuyu and Ixtepeque did not pose a significant obstacle to ancient travelers.

ancient Maya trade networks were decentralized and operated entirely independently of elite regulation.

Cevabı ve açıklamayı göster

Cevap: the use of Ixtepeque obsidian for ceremonial blades was not restricted solely to the ruling elite class in Kaminaljuyu.

Cevap

The use of Ixtepeque obsidian for ceremonial blades was not restricted solely to the ruling elite class in Kaminaljuyu.

The correct option is supported because the passage notes that researchers originally assumed Ixtepeque obsidian was strictly limited to ruling elites. The discovery of abundant Ixtepeque obsidian ceremonial blade fragments in commoner residences directly undermines this assumption, proving that the material's use was not exclusive to the elite class.

Adım Adım Çözüm

1

Identify the researchers' initial hypothesis.

Researchers assumed Ixtepeque obsidian was strictly limited to ruling elites due to its distance, distinctness, and difficult acquisition.

This establishes the premise that is being challenged by the new discovery.

2

Analyze the new evidence.

Excavations of commoner residences revealed abundant fragments of ceremonial blades made of Ixtepeque obsidian.

This provides the key factual constraint that must be incorporated into the final inference.

3

Synthesize the premise and the new evidence to form a logical conclusion.

Since commoner houses contained Ixtepeque obsidian ceremonial blades, the initial assumption of exclusive elite access is false, meaning access was not restricted only to the elites.

This completes the logical chain directly supported by the text.

Anahtar Kavram

Drawing logical conclusions based on new evidence that challenges previous assumptions.

Tahmini Süre:1m 30s

Soru 2723Soru →

In paleoclimatology, the chemical analysis of deep-sea sediment cores retrieved from the North Atlantic Ocean ______ critical evidence regarding historic shifts in oceanic currents. By measuring stable isotope ratios preserved in microfossils, researchers can reconstruct past seawater temperatures and trace changes in circulation patterns that occurred over millions of years.

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

provides

provide

are providing

have provided

Cevabı ve açıklamayı göster

Cevap: provides

Cevap

provides

The true subject of the sentence is the singular noun phrase 'the chemical analysis'. The prepositional phrase 'of deep-sea sediment cores retrieved from the North Atlantic Ocean' intervenes between the subject and the verb. Because the subject is singular, the verb must also be singular. The singular verb form 'provides' correctly completes the sentence.

Adım Adım Çözüm

1

Identify the subject of the sentence.

The subject is the noun phrase 'the chemical analysis', which is singular.

The verb must agree in number with the main subject of the sentence, not with any nouns in intervening prepositional phrases.

2

Identify and isolate the intervening phrase between the subject and the verb.

The phrase 'of deep-sea sediment cores retrieved from the North Atlantic Ocean' is a prepositional phrase that modifies 'analysis'.

Isolating this phrase helps prevent mistaking the plural nouns inside it, like 'cores', as the subject of the verb.

3

Select the verb form that agrees with the singular subject.

The singular verb form 'provides' is selected.

A singular subject requires a singular verb, whereas the other options ('provide', 'are providing', 'have provided') are plural.

Anahtar Kavram

Subject-verb agreement requires the verb to match the number (singular or plural) of its true subject, regardless of any intervening words or phrases.

Soru 2724Soru →

A student is preparing a presentation on the Burgess Shale. She has taken the following notes:

* The Burgess Shale is a fossil-bearing geological formation in the Canadian Rockies.
* It was discovered in 1909 by paleontologist Charles Walcott.
* The site is famous for the exceptional preservation of soft-bodied organisms from the Cambrian Period, approximately 508 million years ago.
* Among the key fossils found there is *Opabinia*, a creature with five eyes.

The student wants to complete a sentence that emphasizes the name of the discoverer of the Burgess Shale and the specific geological period of the fossils found there. Based on the notes, which words or phrases best complete the sentence?

Aşağıdaki boşlukları doldurun

The Burgess Shale, a geological formation containing exceptionally preserved fossils from the Period, was first discovered in 1909 by paleontologist .

Cevabı ve açıklamayı göster

Cevap

The first blank must be filled with 'Cambrian' and the second blank must be filled with 'Charles Walcott' (or 'Walcott').

The completed sentence correctly synthesizes the specific details requested: the Cambrian Period (the geological period of the fossils) and Charles Walcott (the discoverer).

Adım Adım Çözüm

1

Analyze the prompt to identify the specific details that must be emphasized.

The student needs to emphasize the name of the discoverer of the Burgess Shale and the specific geological period of the fossils found there.

This establishes the rhetorical goal for completing the sentence.

2

Locate the relevant details in the provided notes.

The notes state that the fossils are from the 'Cambrian Period' and that the site was discovered by paleontologist 'Charles Walcott'.

This provides the factual information required to fill in the blanks.

3

Complete the sentence with the identified details.

Inserting 'Cambrian' into the first blank and 'Charles Walcott' into the second blank satisfies both the sentence structure and the rhetorical goal.

This achieves the desired emphasis while maintaining factual accuracy and grammatical correctness.

Anahtar Kavram

Identifying and synthesizing specific details from notes to achieve a target rhetorical goal.

Tahmini Süre:1m 30s

Soru 2725Soru →

In a circle, chords $AB$ and $CD$ intersect perpendicularly at point $E$ . If $AE = 35$ , $EB = 5$ , and $CE = 5$ , what is the length of the diameter of the circle?

Cevabı ve açıklamayı göster

Cevap: 50

Cevap

50

The correct answer is 50. By applying the intersecting chords theorem, the segment

ED

is found to be 35. Since the chords are perpendicular and intersect at point

E

, we can find the distance from the center of the circle to each chord by analyzing the distances from the intersection point to the midpoints of the chords. The midpoints of both chords are 20 units from their endpoints. The distance from

E

to the midpoint of

AB

is

35 - 20 = 15

. This distance is equal to the perpendicular distance from the center of the circle to the other chord,

CD

. Using the Pythagorean theorem with a chord half-length of 20 and a distance from the center of 15, the radius of the circle is

\sqrt{15^2 + 20^2} = 25

. Therefore, the diameter of the circle is

2 \times 25 = 50

.

Adım Adım Çözüm

1

Find the length of segment

ED

using the intersecting chords theorem.

ED = 35

For any two intersecting chords

AB

and

CD

intersecting at point

E

, the product of the segments of one chord equals the product of the segments of the other:

AE \cdot EB = CE \cdot ED

. Substituting the given values:

35 \cdot 5 = 5 \cdot ED

, which simplifies to

ED = 35

.

2

Calculate the total lengths of chords

AB

and

CD

and determine their midpoints.

Chord lengths

AB = 40

and

CD = 40

. Midpoint distances

MB = 20

and

ND = 20

.

The total length of chord

AB

is

AE + EB = 35 + 5 = 40

. The perpendicular line from the center

O

to

AB

bisects the chord at midpoint

M

, so

MB = 40 / 2 = 20

. Similarly, the total length of chord

CD

is

CE + ED = 5 + 35 = 40

, and its midpoint

N

bisects it, so

ND = 20

.

3

Find the perpendicular distances from the center

O

to the chords

AB

and

CD

.

OM = 15

and

ON = 15

The distance from the intersection point

E

to the midpoint

M

along chord

AB

is

AE - AM = 35 - 20 = 15

. Because the chords are perpendicular, the perpendicular distance from the center

O

to chord

AB

is equal to the distance

EN

along the other chord, so

OM = EN = 15

. Similarly,

ON = EM = 15

.

4

Calculate the radius of the circle using the Pythagorean theorem.

Radius

R = 25

In the right triangle

OMB

, the hypotenuse is the radius

R = OB

, and the legs are the perpendicular distance

OM = 15

and half the chord length

MB = 20

. By the Pythagorean theorem,

R^2 = OM^2 + MB^2 = 15^2 + 20^2 = 225 + 400 = 625

. Taking the square root gives

R = 25

.

5

Calculate the diameter of the circle.

Diameter =

50

The diameter of a circle is twice its radius:

2R = 2 \cdot 25 = 50

.

Anahtar Kavram

Using perpendicular chords, the intersecting chords theorem, and the Pythagorean theorem to determine the radius and diameter of a circle.

Soru 2726Soru →

A circular dial on a vintage radio is rotated by $\frac{5\pi}{8}$ radians to tune to a specific station. If the dial is then rotated by an additional $45^\circ$ in the same direction, what is the total angle of rotation, in radians, of the dial?

\frac{3\pi}{4}

\frac{7\pi}{8}

\frac{9\pi}{8}

\frac{\pi}{2}

Cevabı ve açıklamayı göster

Cevap: $\frac{7\pi}{8}$

Cevap

The correct answer is

\frac{7\pi}{8}

radians.

To find the total angle of rotation in radians, the rotation of

45^\circ

must first be converted to radians by multiplying by

\frac{\pi}{180^\circ}

, which yields

\frac{\pi}{4}

radians. Expressed with a common denominator of

8

, this is equivalent to

\frac{2\pi}{8}

radians. Adding this to the initial rotation of

\frac{5\pi}{8}

radians gives a total rotation of

\frac{5\pi}{8} + \frac{2\pi}{8} = \frac{7\pi}{8}

radians.

Adım Adım Çözüm

1

Convert the additional rotation angle of

45^\circ

into radians.

Since

180^\circ = \pi

radians, the conversion is

45^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{4}

radians.

To find the total rotation angle in radians, both individual angles must be in the same unit of measure.

2

Add the first rotation angle of

\frac{5\pi}{8}

radians to the converted second rotation angle of

\frac{\pi}{4}

radians.

First, express

\frac{\pi}{4}

with a common denominator of

8

:

\frac{\pi}{4} = \frac{2\pi}{8}

. Then, add the two fractions:

\frac{5\pi}{8} + \frac{2\pi}{8} = \frac{7\pi}{8}

radians.

The total angle of rotation is the sum of the two sequential rotations in the same direction.

Anahtar Kavram

To find the sum of angles given in different units, convert the angle measured in degrees to radians using the conversion factor

\frac{\pi \text{ radians}}{180^\circ}

, and then find the sum of the two radian values using a common denominator.

Tahmini Süre:1m 0s

Soru 2727Soru →

In the field of sociolinguistics, the detailed structural analysis of regional dialects—which often incorporates extensive phonetic recordings and demographic data collected from local communities—______ crucial insights into how pronunciation shifts propagate across geographical boundaries. By tracking these subtle variations, researchers can map the historical migrations of different populations and understand how social networks influence speech patterns.

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

provide

providing

provides

have provided

Cevabı ve açıklamayı göster

Cevap: provides

Cevap

provides

The singular verb 'provides' correctly agrees with the singular subject 'analysis'. The intervening prepositional phrase and relative clause contain plural nouns, but they do not alter the singular grammatical number of the subject.

Adım Adım Çözüm

1

Identify the subject of the sentence.

The subject is 'the detailed structural analysis', which is singular.

Determining the correct grammatical number of the subject is required to apply the rule of subject-verb agreement.

2

Identify intervening phrases or clauses and isolate them.

The prepositional phrase 'of regional dialects' and the relative clause starting with 'which' modify the subject but do not change its singular number.

Isolating intervening elements prevents the common error of agreeing the verb with nearby plural nouns like 'dialects' or 'communities'.

3

Select the verb form that agrees with the singular subject in the present tense.

The verb 'provides' is singular and present tense, matching the rest of the passage.

The singular verb correctly pairs with the singular subject to complete the main clause.

Anahtar Kavram

Subject-Verb Agreement

Soru 2728Soru →

A landscape architect is designing a courtyard in the shape of a right trapezoid. The parallel sides of the courtyard have lengths of $24$ yards and $40$ yards. The side perpendicular to the parallel sides has a length of $15$ yards. A straight path is built from the midpoint of the longer parallel side to the vertex of the shorter parallel side that is adjacent to the perpendicular side, dividing the courtyard into two regions. What is the area, in square yards, of the smaller region?

Cevabı ve açıklamayı göster

Cevap: 150

Cevap

The area of the smaller region is 150 square yards.

The straight path divides the right trapezoid into two regions: a right triangle and a quadrilateral. The right triangle has a vertical leg of 15 yards (the height of the trapezoid) and a horizontal leg of 20 yards (half of the longer parallel side of 40 yards). The area of this right triangle is 0.5 * 20 * 15 = 150 square yards. The total area of the trapezoid is 0.5 * (24 + 40) * 15 = 480 square yards, making the area of the quadrilateral region 480 - 150 = 330 square yards. Comparing the two regions, the smaller region has an area of 150 square yards.

Adım Adım Çözüm

1

Find the length of the segment from the perpendicular corner to the midpoint of the longer parallel side.

20 yards

The midpoint divides the 40-yard side into two equal parts of 20 yards each.

2

Determine the shape and dimensions of the region containing the perpendicular side.

A right triangle with legs of 15 yards and 20 yards.

Since the path goes from the midpoint of the base to the opposite vertex of the perpendicular height, it forms a right triangle with the height and half of the longer base.

3

Calculate the area of this right triangle.

150 square yards

Using the area formula for a triangle, Area = 0.5 * base * height = 0.5 * 20 * 15 = 150.

4

Calculate the total area of the trapezoid and the area of the remaining region to confirm which is smaller.

Total area is 480 square yards; the other region's area is 330 square yards.

The total area is 0.5 * (24 + 40) * 15 = 480. The other region has an area of 480 - 150 = 330. Comparing 150 and 330, 150 is the smaller area.

Anahtar Kavram

Area of composite shapes and trapezoids

Soru 2729Soru →

In temperate forests, common mycorrhizal networks (CMNs) of fungal mycelia connect the roots of adjacent trees, enabling the subterranean transfer of carbon, water, and warning signals. When a host plant is attacked by herbivores, it releases volatile organic compounds into the atmosphere and transmits chemical warnings through the CMN, triggering defense responses in neighboring plants. However, ecologist Dr. Clara Vance demonstrated that if neighboring trees are already experiencing mild drought stress, the defense signals received via the CMN instead prompt them to conserve water by closing their stomata rather than synthesizing chemical defenses. Which choice most logically completes the text?

atmospheric volatile organic compounds are less effective at initiating defensive behaviors in neighboring trees than subterranean signals are.

under drought conditions, receiving warnings about herbivory via mycorrhizal networks leads neighboring trees to prioritize water conservation over defensive chemical synthesis.

mycorrhizal networks fail to transmit chemical warning signals when the host plant is undergoing drought stress.

neighboring trees experiencing drought stress are more vulnerable to herbivore damage than trees that are completely disconnected from the mycorrhizal network.

Cevabı ve açıklamayı göster

Cevap: under drought conditions, receiving warnings about herbivory via mycorrhizal networks leads neighboring trees to prioritize water conservation over defensive chemical synthesis.

Cevap

under drought conditions, receiving warnings about herbivory via mycorrhizal networks leads neighboring trees to prioritize water conservation over defensive chemical synthesis.

The correct option is supported because the passage states that warning signals received via the mycorrhizal network prompt neighboring trees to close their stomata and conserve water rather than synthesize defenses if those neighbors are experiencing drought stress. This directly supports the inference that drought conditions lead to prioritizing water conservation over defense synthesis when warned of herbivore presence.

Adım Adım Çözüm

1

Identify the primary mechanism of plant communication described in the passage.

Host plants attacked by herbivores warn neighboring plants via chemical signals sent through mycorrhizal networks, triggering defenses in those neighbors.

This establishes the default relationship and baseline behavior under normal conditions.

2

Analyze how the introduction of drought stress alters this communication outcome.

Drought-stressed neighbors receiving the signal prioritize water conservation (closing stomata) instead of synthesizing chemical defenses.

This identifies the specific response shift described in the researcher's findings.

3

Evaluate the choices to find the one that directly matches this shift without making external assumptions.

The correct answer accurately states that under drought conditions, herbivore warnings lead neighboring trees to prioritize water conservation over chemical defense synthesis.

This matches the premises directly and avoids speculation.

Anahtar Kavram

Logical Inferences

Tahmini Süre:1m 30s

Soru 2730Soru →

In a hydroponic farming system, a nutrient solution is created by mixing a liquid fertilizer concentrate with water in a ratio of $3$ to $50$ by volume. To fill a reservoir, a technician needs to prepare a total of $318$ liters of this nutrient solution. How many liters of the liquid fertilizer concentrate are required to prepare the solution?

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

18

To find the amount of liquid fertilizer concentrate needed, we first express the ratio in terms of the total mixture. The ratio of concentrate to water is

3

to

50

, meaning there are

3

parts of concentrate for every

50

parts of water, making a total of

3 + 50 = 53

parts. Since the total volume of the solution is

318

liters, each part represents

\frac{318}{53} = 6

liters. The amount of concentrate required is

3

parts, which corresponds to

3 \times 6 = 18

liters.

Adım Adım Çözüm

1

Calculate the total parts represented in the ratio of liquid fertilizer concentrate to water.

The total number of parts is

3 + 50 = 53

.

Since the ratio of concentrate to water is

3

to

50

, the entire nutrient solution is divided into

53

equal parts.

2

Find the volume of solution that corresponds to one part.

One part is equal to

\frac{318}{53} = 6

liters.

Dividing the total volume of the nutrient solution (

318

liters) by the total number of parts (

53

) gives the volume of a single part.

3

Calculate the total volume of liquid fertilizer concentrate needed.

The volume of concentrate required is

3 \times 6 = 18

liters.

Since the concentrate accounts for

3

parts of the total mixture, multiplying the volume of one part by

3

gives the total amount of concentrate needed.

Anahtar Kavram

Solving part-to-whole ratio problems by determining the value of a single unit or part from a given total quantity.

Alternatif Yöntem

Let the volume of the fertilizer concentrate be

3x

and the volume of water be

50x

. The total volume of the nutrient solution is

3x + 50x = 53x

. Given that the total volume is

318

liters, we can set up the equation

53x = 318

. Solving for

x

gives

x = 6

. The volume of the concentrate is

3x = 3(6) = 18

liters.

Tahmini Süre:1m 15s

Soru 2731Soru →

A chef prepares a spice mix by combining paprika, cumin, and coriander in a ratio of $8 : 5 : 2$ by weight. If the chef wants to make a total of $60$ ounces of the spice mix, how many ounces of cumin are needed?

15

20

30

37.5

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

The correct answer is 20 ounces.

The correct answer is 20 ounces. To find the amount of cumin needed, first sum the parts of the ratio to find the total parts of the mixture:

8 + 5 + 2 = 15

parts. Since cumin represents 5 of these 15 parts, the fraction of the spice mix that is cumin is

\frac{5}{15}

, which simplifies to

\frac{1}{3}

. Multiplying this fraction by the total desired weight of the spice mix gives

\frac{1}{3} \times 60 = 20

ounces.

Adım Adım Çözüm

1

Calculate the total number of parts in the ratio.

8 + 5 + 2 = 15

parts.

Determining the total parts is necessary to find what fraction of the whole mixture is represented by cumin.

2

Determine the fractional portion of the mixture that is cumin.

The fraction is

\frac{5}{15}

, which simplifies to

\frac{1}{3}

.

This fraction represents the ratio of the cumin parts to the total parts of the spice mix.

3

Multiply the cumin fraction by the total weight of the mix to find the weight of cumin.

\frac{1}{3} \times 60 = 20

ounces.

Multiplying the proportion of cumin by the total desired weight of the spice mix yields the required ounces of cumin.

Anahtar Kavram

Solving part-to-whole ratio problems by finding the total number of parts and applying the fraction to the overall amount.

Tahmini Süre:1m 15s

Soru 2732Soru →

A circular fountain with a diameter of $8$ feet is positioned in the center of a square lawn. The lawn has a side length of $20$ feet. A concrete walkway of width $2$ feet is built directly around the fountain. What is the area, in square feet, of the remaining grass region of the lawn?

400 - 36\pi

400 - 16\pi

400 - 24\pi

400 - 100\pi

Cevabı ve açıklamayı göster

Cevap: $400 - 36\pi$

Cevap

400 - 36\pi

The area of the square lawn is

20^2 = 400

square feet. The fountain has a radius of

4

feet (half of the

8

-foot diameter), and the walkway adds another

2

feet to the radius, making the total radius of the combined circular area

6

feet. The area of this circular region is

\pi \times 6^2 = 36\pi

square feet. Subtracting this from the total area of the square lawn gives the remaining grass area of

400 - 36\pi

square feet.

Adım Adım Çözüm

1

Calculate the total area of the square lawn.

Area of the lawn is

20 \times 20 = 400

square feet.

The area of a square is calculated by squaring its side length.

2

Find the combined radius of the fountain and the walkway.

The radius of the fountain is

8 \div 2 = 4

feet. Adding the

2

-foot width of the walkway gives a combined radius of

4 + 2 = 6

feet.

The radius is half of the diameter. The walkway surrounds the fountain, so its width must be added to the fountain's radius to find the outer boundary's radius.

3

Calculate the combined area of the fountain and the walkway.

Area of the combined circular region is

\pi \times 6^2 = 36\pi

square feet.

The area of a circle is given by

\pi r^2

, where

r

is the radius.

4

Subtract the combined circular area from the total area of the lawn to find the remaining grass area.

The remaining area is

400 - 36\pi

square feet.

The remaining grass area is the total square area minus the area of the inner circular region that contains the fountain and the walkway.

Anahtar Kavram

Area of composite shapes involving squares and circles, and scaling properties.

Tahmini Süre:1m 30s

Soru 2733Soru →

In subalpine forests, wood-boring beetle larvae excavate tunnels that facilitate fungal colonization and accelerate log decomposition. However, ecologist Elena Rostova hypothesized that in arid environments, low humidity desiccates tunnel interiors, inhibiting fungal growth and negating this decomposition effect. Rostova's team compared decomposition rates of logs with and without tunnels across humid and arid subalpine zones. They found that in the arid zone, logs with tunnels decomposed at the same slow rate as those without tunnels, whereas in the humid zone, logs with tunnels decomposed significantly faster. Which choice most logically completes the text?

beetle larvae are unable to excavate tunnels in logs that are located in arid subalpine environments.

wood-boring beetle larvae are the primary regulators of organic decomposition rates across all terrestrial ecosystems.

beetle larvae in arid environments alter their tunneling depth to seek out higher levels of moisture within the logs.

the accelerating effect of beetle larvae on log decomposition depends on environmental humidity to prevent the desiccation of the tunnels.

Cevabı ve açıklamayı göster

Cevap: the accelerating effect of beetle larvae on log decomposition depends on environmental humidity to prevent the desiccation of the tunnels.

Cevap

The accelerating effect of beetle larvae on log decomposition depends on environmental humidity to prevent the desiccation of the tunnels.

The passage explains that wood-boring beetle larvae accelerate log decomposition by creating tunnels that facilitate fungal colonization. The researchers hypothesized that this effect would be negated in arid environments due to low humidity drying out the tunnels and preventing fungal growth. Their experiment confirmed this, showing that logs with tunnels decomposed faster only in the humid zone, but not in the arid zone. Therefore, the logical conclusion is that the larvae's ability to accelerate decomposition relies on sufficient environmental humidity to prevent tunnel desiccation.

Adım Adım Çözüm

1

Identify the primary mechanism of log decomposition described in the passage.

The passage establishes that wood-boring beetle larvae create tunnels that help fungi colonize logs, which in turn accelerates decomposition.

Understanding this baseline relationship is necessary to trace how climate factors interfere with the process.

2

Analyze the effect of the experimental variable (humidity) on this mechanism.

Low humidity in arid zones is hypothesized to desiccate tunnels and inhibit fungal growth, thereby neutralizing the decomposition-promoting effect of the larvae.

This step determines the predicted consequence of changing the environmental conditions.

3

Evaluate the experimental results to find the most direct logical conclusion.

Logs with tunnels decayed faster only in the humid zone, but decayed at the same slow rate as tunnel-less logs in the arid zone, supporting the hypothesis.

This step synthesizes the hypothesis and results to draw a conclusion supported by the text without introducing external assumptions.

Anahtar Kavram

Logical Inferences

Soru 2734Soru →

In the $xy$ -plane, the graph of the equation $x^2 + y^2 - 12x + 8y - k = 0$ , where $k$ is a positive constant, is a circle. If the line $y = 6$ is tangent to the circle, what is the value of $k$ ?

Cevabı ve açıklamayı göster

Cevap: 48

Cevap

48

To find the value of

k

, we convert the given circle equation into its standard form,

(x - h)^2 + (y - j)^2 = r^2

. Grouping the terms gives

(x^2 - 12x) + (y^2 + 8y) = k

. Completing the square for

x

and

y

gives

(x - 6)^2 - 36 + (y + 4)^2 - 16 = k

, which simplifies to

(x - 6)^2 + (y + 4)^2 = k + 52

. This tells us that the center of the circle is

(6, -4)

and the radius squared is

r^2 = k + 52

. A horizontal line

y = 6

is tangent to the circle, meaning the perpendicular distance from the center

(6, -4)

to the line

y = 6

is equal to the radius. This distance is

|6 - (-4)| = 10

. Therefore, the radius is

10

, which means

r^2 = 100

. Equating the two expressions for the radius squared gives

k + 52 = 100

. Solving this equation yields

k = 48

.

Adım Adım Çözüm

1

Group the variables and complete the square for the

x

and

y

terms.

(x - 6)^2 + (y + 4)^2 = k + 52

Completing the square converts the equation from general form to standard form, which reveals the center and radius.

2

Identify the center of the circle and the algebraic representation of the radius.

Center is

(6, -4)

and radius

r = \sqrt{k + 52}

.

The standard equation of a circle is

(x - h)^2 + (y - j)^2 = r^2

, where

(h, j)

is the center and

r

is the radius.

3

Find the radius of the circle using the given tangent line.

Radius

r = 10

The distance from the center's

y

-coordinate,

-4

, to the horizontal tangent line

y = 6

is

|6 - (-4)| = 10

, which represents the radius of the circle.

4

Equate the radius squared value to the algebraic expression for the radius squared and solve for

k

.

k = 48

Since the radius is

10

, the radius squared is

100

. Setting

k + 52 = 100

and solving for

k

yields

48

.

Anahtar Kavram

Converting a circle's equation from general to standard form by completing the square, and using the distance from the center to a tangent line to find the radius.

Soru 2735Soru →

The equation $x^2 + y^2 + 10x - 6y - 15 = 0$ represents a circle in the coordinate plane. If this circle is translated $2$ units to the right and $4$ units down, which of the following equations represents the translated circle?

(x - 3)^2 + (y - 1)^2 = 49

(x + 3)^2 + (y + 1)^2 = 49

(x + 3)^2 + (y + 1)^2 = 7

(x + 7)^2 + (y - 7)^2 = 49

Cevabı ve açıklamayı göster

Cevap: $(x + 3)^2 + (y + 1)^2 = 49$

Cevap

The equation of the translated circle is

(x + 3)^2 + (y + 1)^2 = 49

.

The correct answer shows

(x + 3)^2 + (y + 1)^2 = 49

. Completing the square on the original equation reveals the original center is

(-5, 3)

and

r^2 = 49

. Translating the center

2

units right and

4

units down shifts the

x

-coordinate from

-5

to

-3

and the

y

-coordinate from

3

to

-1

. The equation for a circle with center

(-3, -1)

and radius squared of

49

is

(x + 3)^2 + (y + 1)^2 = 49

.

Adım Adım Çözüm

1

Group the

x

-terms and

y

-terms together, and move the constant term to the right side of the equation.

(x^2 + 10x) + (y^2 - 6y) = 15

This groups terms by variable to prepare for completing the square.

2

Complete the square for the

x

-terms by adding

(\frac{10}{2})^2 = 25

and for the

y

-terms by adding

(\frac{-6}{2})^2 = 9

to both sides of the equation.

(x^2 + 10x + 25) + (y^2 - 6y + 9) = 15 + 25 + 9

, which simplifies to

(x + 5)^2 + (y - 3)^2 = 49

.

This converts the circle's equation into standard form

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

.

3

Identify the center and the radius squared of the original circle from the standard form equation.

The center is

(h, k) = (-5, 3)

and the radius squared is

r^2 = 49

.

The standard form reveals the center coordinates as the opposite signs of the constants inside the parentheses, and the right side is

r^2

.

4

Calculate the center of the translated circle by shifting the original center

(-5, 3)

by

2

units to the right and

4

units down.

The new center coordinates are

(-5 + 2, 3 - 4) = (-3, -1)

.

Translating a point right increases its

x

-coordinate, and translating it down decreases its

y

-coordinate.

5

Write the standard form equation for the new circle with center

(-3, -1)

and the unchanged radius squared

r^2 = 49

.

(x - (-3))^2 + (y - (-1))^2 = 49

, which simplifies to

(x + 3)^2 + (y + 1)^2 = 49

.

Substituting

h = -3

and

k = -1

into the standard circle equation form

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

yields the final equation.

Anahtar Kavram

Completing the square to find standard circle equations and translating circle centers in the coordinate plane.

Soru 2736Soru →

To preserve historic silk textiles, conservators often regulate relative humidity (RH) to prevent fiber degradation. However, research by textile chemist Clara Alvarez indicates that maintaining a constant, low RH (below $30\%$ ) can cause micro-fissures in silk fibers that have already undergone historical aging, making them highly susceptible to mechanical stress during handling. Conversely, keeping the RH above $50\%$ promotes fungal growth that enzymatically breaks down the silk's fibroin structure. Consequently, for museums planning to actively exhibit and periodically rotate aged silk garments, Alvarez suggests that

Which choice most logically completes the text?

micro-fissures in aged silk can be successfully repaired by temporarily raising the relative humidity above

50\%

before handling the garments.

maintaining the display environment's relative humidity between

30\%

and

50\%

is necessary to minimize both mechanical damage and fungal degradation.

fungal growth on silk garments is primarily triggered by the mechanical stress that occurs when the garments are handled at low relative humidity levels.

all historical organic textiles require relative humidity levels strictly between

30\%

and

50\%

to prevent any form of structural deterioration.

Cevabı ve açıklamayı göster

Cevap: maintaining the display environment's relative humidity between $30\%$ and $50\%$ is necessary to minimize both mechanical damage and fungal degradation.

Cevap

maintaining the display environment's relative humidity between

30\%

and

50\%

is necessary to minimize both mechanical damage and fungal degradation.

The correct option states that the relative humidity should be maintained between

30\%

and

50\%

. This is directly supported by the passage, which outlines that relative humidity below

30\%

causes micro-fissures that make the silk susceptible to stress during handling, while relative humidity above

50\%

promotes fungal growth. Since the museum plans to actively exhibit and rotate (handle) the garments, maintaining the humidity within this intermediate range is necessary to avoid both forms of degradation.

Adım Adım Çözüm

1

Analyze the consequences of low relative humidity (below

30\%

) mentioned in the text.

Low humidity makes aged silk garments vulnerable to mechanical stress, which is a key risk since the museum plans to handle them during active exhibition and rotation.

To identify constraints on the lower bound of relative humidity.

2

Analyze the consequences of high relative humidity (above

50\%

) mentioned in the text.

High humidity (above

50\%

) leads to fungal growth that enzymatically breaks down the silk fibers.

To identify constraints on the upper bound of relative humidity.

3

Synthesize the two findings to determine the safe operating range for active exhibition and rotation.

To prevent both mechanical damage from handling (which occurs below

30\%

) and fungal breakdown (which occurs above

50\%

), the relative humidity must be maintained between

30\%

and

50\%

.

To draw a logically valid, non-speculative conclusion that completes the passage.

Anahtar Kavram

Logical Inferences

Soru 2737Soru →

In the $xy$ -plane, the equation $2x^2 + 2y^2 - 12x + 16y - 22 = 0$ represents a circle. What is the diameter of this circle?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The diameter of the circle is 12.

Dividing the given equation

2x^2 + 2y^2 - 12x + 16y - 22 = 0

by 2 gives

x^2 + y^2 - 6x + 8y - 11 = 0

. Grouping the

x

and

y

terms and moving the constant to the right side gives

(x^2 - 6x) + (y^2 + 8y) = 11

. To complete the square, add

(\frac{-6}{2})^2 = 9

and

(\frac{8}{2})^2 = 16

to both sides, yielding

(x^2 - 6x + 9) + (y^2 + 8y + 16) = 11 + 9 + 16

, which simplifies to

(x-3)^2 + (y+4)^2 = 36

. Since the standard equation of a circle is

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

, the radius squared

r^2

is 36, which means the radius

r

is 6. The diameter is twice the radius, so

2 \times 6 = 12

.

Adım Adım Çözüm

1

Divide the entire equation by 2.

x^2 + y^2 - 6x + 8y - 11 = 0

To simplify the coefficients of

x^2

and

y^2

to 1, which is necessary before completing the square.

2

Group the variables and move the constant term.

(x^2 - 6x) + (y^2 + 8y) = 11

To isolate the quadratic and linear terms for both

x

and

y

on one side of the equation.

3

Complete the square for both variables by adding the appropriate values to both sides.

(x - 3)^2 + (y + 4)^2 = 36

Adding 9 (which is

(\frac{-6}{2})^2

) and 16 (which is

(\frac{8}{2})^2

) to both sides allows us to rewrite the trinomials as perfect squares:

(x-3)^2

and

(y+4)^2

. The right side becomes

11 + 9 + 16 = 36

.

4

Find the radius and calculate the diameter.

Diameter = 12

Comparing the equation to the standard form

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

shows that

r^2 = 36

, so the radius

r

is 6. The diameter is

2r = 2(6) = 12

.

Anahtar Kavram

Converting a circle's equation from general form to standard form by completing the square to identify its geometric properties.

Soru 2738Soru →

In triangle $ABC$ , point $D$ lies on side $BC$ . Line segment $AD$ is drawn such that $AB = AD$ and $AD = CD$ . If the measure of angle $BAC$ is $75^\circ$ , what is the measure, in degrees, of angle $B$ ?

Cevabı ve açıklamayı göster

Cevap: 70

Cevap

The measure of angle

B

is

70^\circ

.

By representing the angles using the properties of the isosceles triangles

ABD

and

ADC

, we set up a system of equations where

\angle B = x

and

\angle C = y

such that

x = 2y

. Using the angle sum theorem and angle addition, we find that

(180^\circ - 2x) + y = 75^\circ

. Substituting

x = 2y

yields

y = 35^\circ

, and thus

\angle B = x = 70^\circ

.

Adım Adım Çözüm

1

Set up base angles for the isosceles triangle

ABD

.

Let

\angle B = \angle ADB = x

.

Because

AB = AD

, triangle

ABD

is an isosceles triangle, making its base angles equal.

2

Express the measure of angle

ADC

in terms of

x

.

\angle ADC = 180^\circ - x

.

Angles

\angle ADB

and

\angle ADC

form a linear pair along the line segment

BC

.

3

Establish the relationship between

x

and

y

using triangle

ADC

.

x = 2y

, where

y = \angle DAC = \angle C

.

Since

AD = CD

, triangle

ADC

is isosceles. The sum of angles in

\triangle ADC

is

(180^\circ - x) + y + y = 180^\circ

, which simplifies to

x = 2y

.

4

Write the equation for the total measure of angle

BAC

.

(180^\circ - 2x) + y = 75^\circ

.

The angle addition postulate states that

\angle BAC = \angle BAD + \angle DAC

. Since

\angle BAD = 180^\circ - 2x

(from the sum of angles in

\triangle ABD

) and

\angle DAC = y

, this equals

75^\circ

.

5

Solve the system of equations for

y

.

y = 35^\circ

.

Substituting

x = 2y

into

(180^\circ - 2x) + y = 75^\circ

yields

180^\circ - 3y = 75^\circ

, which gives

3y = 105^\circ

, so

y = 35^\circ

.

6

Find the measure of angle

B

.

\angle B = 70^\circ

.

Since

\angle B = x

and

x = 2y

, we have

x = 2(35^\circ) = 70^\circ

.

Anahtar Kavram

Using properties of isosceles triangles and angle sum theorems to solve for angle measures.

Soru 2739Soru →

An L-shaped region is created by removing a smaller square from the corner of a larger square. The perimeter of the L-shaped region is $48$ inches, and the area of the smaller square that was removed is $16$ square inches. What is the area, in square inches, of the L-shaped region?

Cevabı ve açıklamayı göster

Cevap: 128

Cevap

128

The area of the L-shaped region is the difference between the area of the original larger square and the area of the removed corner square. The perimeter of an L-shaped region formed by removing a corner square is identical to the perimeter of the original square, which is

48

inches. This means the side length of the larger square is

12

inches, and its area is

144

square units. Subtracting the area of the removed square (

16

square units) from the area of the larger square gives

128

square units.

Adım Adım Çözüm

1

Calculate the side length of the smaller square from its area.

The side length of the smaller square is

4

inches.

The area of a square is the square of its side length (

A = s^2

). Since the area is

16

, we have

s = \sqrt{16} = 4

.

2

Find the side length of the larger square using the perimeter of the L-shaped region.

The side length of the larger square is

12

inches.

When a corner square is removed from a larger square, the perimeter remains unchanged because the two cut-out edges going inward have the same lengths as the two outer edges that were removed. Thus, the perimeter of the L-shaped region is equal to

4S

. With a perimeter of

48

, the side length

S

is

48 \div 4 = 12

.

3

Compute the area of the L-shaped region.

The area of the L-shaped region is

128

square inches.

The area of the L-shaped region is the area of the larger square minus the area of the removed smaller square:

12^2 - 16 = 144 - 16 = 128

.

Anahtar Kavram

The area of a composite shape can be calculated by subtracting the area of a removed sub-region from the area of the outer boundary. The perimeter of a rectangle or square remains unchanged when a corner square is removed.

Soru 2740Soru →

In a circle with center $O$ , the radius is $6$ . Points $A$ and $B$ lie on the circle such that the area of the sector $AOB$ is $12$ . What is the length of the minor arc $AB$ ?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The length of the minor arc

AB

is 4.

The area of a sector with central angle

\theta

in radians is

A = \frac{1}{2}r^2\theta

. Setting

A = 12

and

r = 6

, we get

12 = \frac{1}{2}(6)^2\theta

, which simplifies to

12 = 18\theta

, and thus

\theta = \frac{2}{3}

radians. The length of the arc is

s = r\theta = 6 \left(\frac{2}{3}\right) = 4

.

Adım Adım Çözüm

1

Find the central angle

\theta

in radians using the sector area formula.

\theta = \frac{2}{3}

The area of a sector is given by

A = \frac{1}{2}r^2\theta

, so substituting

A = 12

and

r = 6

gives

12 = 18\theta

, which yields

\theta = \frac{2}{3}

.

2

Calculate the arc length

s

using the formula

s = r\theta

.

s = 4

Substituting

r = 6

and

\theta = \frac{2}{3}

into the arc length formula gives

s = 6 \left(\frac{2}{3}\right) = 4

.

Anahtar Kavram

Calculating arc length from sector area and radius using radian measures

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