Tüm alıştırma soruları

2789 soru

Soru 2681Soru

Urban ecologists studying the behavior of songbirds in metropolitan areas have noted that some species, such as the great tit (*Parus major*), sing at higher frequencies in noisy environments than their rural counterparts do. This acoustic shift is widely assumed to be a direct, real-time adaptation to prevent songs from being masked by low-frequency traffic rumble. However, biologist Kenji Sato and his team discovered that great tits inhabiting quiet city parks also sing at higher frequencies than rural birds, despite facing minimal traffic noise. This finding suggests that the frequency shift may not be a simple response to immediate acoustic interference. Instead, the behavior might be a genetically fixed trait or linked to other urban factors, such as architectural barriers that affect sound propagation.

Which choice best describes the function of the underlined sentence in the text as a whole?

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Cevap: It introduces empirical evidence that calls into question a common explanation for the acoustic behavior of urban songbirds.

Cevap

The choice that best describes the function of the underlined sentence is the one stating that it introduces empirical evidence that calls into question a common explanation for the acoustic behavior of urban songbirds.
The correct option correctly identifies the function of the underlined sentence as introducing empirical evidence (the discovery of high-frequency singing in quiet parks) that challenges the common assumption that this shift is a direct response to traffic noise.

Adım Adım Çözüm

1
Analyze the context before the underlined sentence.
The first two sentences state that urban songbirds sing at higher frequencies, and this behavior is widely assumed to be a real-time adaptation to avoid being masked by traffic noise.
To understand what assumption or idea the underlined sentence is reacting to.
2
Analyze the underlined sentence itself.
The underlined sentence states that great tits in quiet parks (where there is no traffic noise) also sing at higher frequencies.
To identify the direct rhetorical contribution of the underlined portion.
3
Evaluate how the underlined sentence functions in the text as a whole.
Since the birds sing at high frequencies even without traffic noise, this finding directly challenges the assumption that the pitch shift is simply a real-time adaptation to traffic noise. Therefore, the sentence introduces evidence that calls a common explanation into question.
To match the function to the correct option.

Anahtar Kavram

Identifying the rhetorical function of a specific sentence within a passage, particularly how it relates to preceding assumptions and succeeding conclusions.
Soru 2682Soru

In cognitive neuroscience, the mapping of neural pathways involved in semantic processing ______ significantly to our understanding of language acquisition. By observing how these pathways activate during communication tasks, researchers can pinpoint the precise regions of the brain that decode complex linguistic structures.

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

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Cevap: contributes

Cevap

The choice stating 'contributes' is the correct answer.
The correct answer is the singular verb 'contributes' because it agrees with the singular subject 'the mapping'. The intervening prepositional phrase 'of neural pathways involved in semantic processing' contains plural nouns, but these do not change the number of the subject.

Adım Adım Çözüm

1
Identify the subject of the clause containing the blank.
The subject is the singular noun phrase 'the mapping'.
The phrase 'of neural pathways involved in semantic processing' is a prepositional phrase modifying the main noun 'mapping' and does not affect the subject's number.
2
Select the verb form that agrees with the singular subject.
The singular verb 'contributes' is selected.
A singular subject requires a singular verb in the present tense to ensure subject-verb agreement.

Anahtar Kavram

Subject-Verb Agreement
Soru 2683Soru

In the xyxy-plane, triangle ABCABC is similar to triangle DEFDEF, where vertices AA, BB, and CC correspond to vertices DD, EE, and FF, respectively. The vertices of triangle ABCABC are A(0,0)A(0, 0), B(6,0)B(6, 0), and C(0,8)C(0, 8). The vertices of triangle DEFDEF are D(2,1)D(2, 1), E(11,1)E(11, 1), and F(2,y)F(2, y), where y>1y > 1. What is the value of yy?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

13
The correct answer is 13. Since triangle ABCABC is similar to triangle DEFDEF, the ratio of their corresponding sides is constant. Side ABAB has length 66, and the corresponding side DEDE has length 112=911 - 2 = 9. This gives a scale factor of 96=1.5\frac{9}{6} = 1.5. Applying this scale factor to the vertical side ACAC (length 88) yields a length of 1212 for the corresponding side DFDF. Since DD has coordinates (2,1)(2, 1) and FF has coordinates (2,y)(2, y) with y>1y > 1, the length of DFDF is y1=12y - 1 = 12, which solves to y=13y = 13.

Adım Adım Çözüm

1
Determine the lengths of the corresponding sides of triangle ABCABC.
The vertices of triangle ABCABC are A(0,0)A(0, 0), B(6,0)B(6, 0), and C(0,8)C(0, 8). The length of horizontal side ABAB is 60=66 - 0 = 6, and the length of vertical side ACAC is 80=88 - 0 = 8.
To find the similarity ratio between the two triangles, we need the lengths of the sides of the first triangle.
2
Determine the length of the corresponding side DEDE of triangle DEFDEF.
The vertices of side DEDE are D(2,1)D(2, 1) and E(11,1)E(11, 1). Since they share the same yy-coordinate, this is a horizontal segment with a length of 112=911 - 2 = 9.
Since vertex DD corresponds to AA and vertex EE corresponds to BB, side DEDE corresponds to side ABAB.
3
Calculate the scale factor between the similar triangles.
The scale factor from triangle ABCABC to triangle DEFDEF is DEAB=96=1.5\frac{DE}{AB} = \frac{9}{6} = 1.5.
Similar triangles have corresponding side lengths that are proportional.
4
Find the length of side DFDF and the coordinate yy.
Side DFDF corresponds to side ACAC. The length of DFDF is 1.5×AC=1.5×8=121.5 \times AC = 1.5 \times 8 = 12. Since DD is at (2,1)(2, 1) and FF is at (2,y)(2, y), the vertical distance is y1=12y - 1 = 12, which gives y=13y = 13.
We apply the scale factor to the corresponding side length and use the coordinate of DD to solve for the coordinate yy of FF.

Anahtar Kavram

Using the properties of similar triangles and coordinates in the plane to solve for unknown lengths and coordinates.
Soru 2684Soru

A rotating beacon rotates at a constant rate of 160160^\circ per second. Through how many radians does the beacon rotate in 4.54.5 seconds?

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Cevap: 4π4\pi

Cevap

The correct answer is the value of 4π4\pi radians.
The total angle of rotation in degrees is found by multiplying the rate of rotation by the time elapsed: 160/second×4.5 seconds=720160^\circ/\text{second} \times 4.5\text{ seconds} = 720^\circ. To convert this angle from degrees to radians, multiply the degree measure by the conversion factor π180\frac{\pi}{180^\circ}. This yields 720×π180=4π720 \times \frac{\pi}{180} = 4\pi radians.

Adım Adım Çözüm

1
Calculate the total angle of rotation in degrees by multiplying the rate of rotation by the time.
160×4.5=720160^\circ \times 4.5 = 720^\circ
To find the total angular displacement, we multiply the constant angular speed by the duration of the rotation.
2
Convert the total angle from degrees to radians by multiplying by the conversion factor π180\frac{\pi}{180^\circ}.
720×π180=4π720^\circ \times \frac{\pi}{180^\circ} = 4\pi radians
Since 180180^\circ is equivalent to π\pi radians, multiplying by π180\frac{\pi}{180^\circ} converts the angle to radians.

Anahtar Kavram

Radian-degree conversion

Alternatif Yöntem

Convert the speed of rotation to radians per second first: 160×π180=8π9160^\circ \times \frac{\pi}{180^\circ} = \frac{8\pi}{9} radians per second. Then multiply by the time elapsed: 8π9×4.5=4π\frac{8\pi}{9} \times 4.5 = 4\pi radians.
Tahmini Süre:1m 0s
Soru 2685Soru

A solar energy storage system collects electricity from two solar panel arrays, Array XX and Array YY. Array XX produces 33 kilowatt-hours (kWh) of electricity for every 44 hours of direct sunlight. Array YY produces 55 kWh of electricity for every 66 hours of direct sunlight. If both arrays receive direct sunlight simultaneously, how many hours of direct sunlight are required for the two arrays to produce a combined total of 3838 kWh of electricity?

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Cevap: 24

Cevap

24
The correct answer is 2424. First, find the rate of each array in kWh per hour: Array XX has a rate of 34\frac{3}{4} kWh/hour and Array YY has a rate of 56\frac{5}{6} kWh/hour. Since the arrays operate simultaneously, add their individual rates to find the combined rate of 34+56=912+1012=1912\frac{3}{4} + \frac{5}{6} = \frac{9}{12} + \frac{10}{12} = \frac{19}{12} kWh/hour. To find the total hours tt needed to produce 3838 kWh, set up the equation 1912t=38\frac{19}{12}t = 38. Solving for tt gives t=38×1219=24t = 38 \times \frac{12}{19} = 24 hours.

Adım Adım Çözüm

1
Calculate the individual hourly rates of electricity production for Array XX and Array YY.
Array XX produces at a rate of 34\frac{3}{4} kWh per hour, and Array YY produces at a rate of 56\frac{5}{6} kWh per hour.
Converting the given production rates to unit rates (kWh per hour) allows them to be directly compared and combined.
2
Add the individual rates to find the combined rate of both arrays operating simultaneously.
Combined rate is 1912\frac{19}{12} kWh per hour.
Since both arrays operate simultaneously, their rates of production add together. The common denominator for 44 and 66 is 1212, so 34+56=912+1012=1912\frac{3}{4} + \frac{5}{6} = \frac{9}{12} + \frac{10}{12} = \frac{19}{12}.
3
Determine the time required to produce a combined total of 3838 kWh.
2424 hours.
Divide the total target production of 3838 kWh by the combined rate of 1912\frac{19}{12} kWh per hour: 38÷1912=38×1219=2×12=2438 \div \frac{19}{12} = 38 \times \frac{12}{19} = 2 \times 12 = 24.

Anahtar Kavram

Ratios, Rates, and Proportions
Tahmini Süre:1m 30s
Soru 2686Soru

A surveyor maps a triangular plot of land, LMNLMN. A boundary line segment is drawn from point PP on side LNLN to point QQ on side LMLM, creating a smaller triangular section LPQLPQ. The measure of angle LMNLMN is equal to the measure of angle LPQLPQ. The surveyed lengths are LM=18LM = 18 meters, LP=8LP = 8 meters, and LQ=12LQ = 12 meters. What is the length, in meters, of segment PNPN?

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Cevap: 19

Cevap

19
By identifying that triangle LMNLMN and triangle LPQLPQ share the angle at vertex LL and have another pair of congruent angles (LMN=LPQ\angle LMN = \angle LPQ), we establish similarity between the two triangles: LMNLPQ\triangle LMN \sim \triangle LPQ. Using the proportional relationship of their corresponding sides, we write LMLP=LNLQ\frac{LM}{LP} = \frac{LN}{LQ}. Substituting the given values LM=18LM = 18, LP=8LP = 8, and LQ=12LQ = 12 yields 188=LN12\frac{18}{8} = \frac{LN}{12}, which solves to LN=27LN = 27. Finally, subtracting LP=8LP = 8 from the total length of segment LNLN gives the length of segment PNPN as 1919 meters.

Adım Adım Çözüm

1
Identify similar triangles.
Triangle LMNLMN is similar to triangle LPQLPQ (LMNLPQ\triangle LMN \sim \triangle LPQ).
They share angle LL (MLN=PLQ\angle MLN = \angle PLQ) and we are given that LMN=LPQ\angle LMN = \angle LPQ. By the Angle-Angle (AA) similarity criterion, the two triangles are similar.
2
Set up the ratio of corresponding sides.
LMLP=LNLQ\frac{LM}{LP} = \frac{LN}{LQ}
Corresponding sides of similar triangles are proportional.
3
Solve for the length of side LNLN.
LN=27LN = 27 meters
Substituting LM=18LM = 18, LP=8LP = 8, and LQ=12LQ = 12 into the proportion yields 188=LN12\frac{18}{8} = \frac{LN}{12}. Solving for LNLN gives LN=12×188=27LN = 12 \times \frac{18}{8} = 27.
4
Calculate the length of segment PNPN.
PN=19PN = 19 meters
Point PP lies on segment LNLN, so the length of segment PNPN is the difference between LNLN and LPLP, which is 278=1927 - 8 = 19.

Anahtar Kavram

Triangle similarity using the Angle-Angle (AA) criterion and proportional side ratios.
Soru 2687Soru

A pendulum swings through an angle of 4040^\circ, and the tip of the pendulum travels an arc of length 8π8\pi inches. What is the length of the pendulum, in inches?

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Cevap: 36

Cevap

36
To find the length of the pendulum, which represents the radius rr of the circular path it sweeps, we can use the arc length formula s=rθs = r\theta, where ss is the arc length and θ\theta is the central angle in radians. First, convert the given angle from degrees to radians: θ=40×π180=2π9\theta = 40^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{9} radians. Next, substitute the arc length s=8πs = 8\pi and the angle θ=2π9\theta = \frac{2\pi}{9} into the formula: 8π=r(2π9)8\pi = r \left(\frac{2\pi}{9}\right). Solving for rr by multiplying both sides by 92π\frac{9}{2\pi} gives r=36r = 36. Alternatively, you can use the ratio of the sector's central angle to the total angle of a circle: 40360=19\frac{40^\circ}{360^\circ} = \frac{1}{9}. This means the arc length is 19\frac{1}{9} of the circumference of the circle: 8π=19(2πr)8\pi = \frac{1}{9}(2\pi r). Dividing both sides by 2π2\pi yields 4=19r4 = \frac{1}{9}r, so r=36r = 36.

Adım Adım Çözüm

1
Convert the swing angle of the pendulum from degrees to radians.
θ=2π9\theta = \frac{2\pi}{9} radians
The arc length formula s=rθs = r\theta requires the angle θ\theta to be in radians.
2
Set up the arc length equation using s=rθs = r\theta, where s=8πs = 8\pi is the arc length and rr is the length of the pendulum.
8π=r(2π9)8\pi = r \left(\frac{2\pi}{9}\right)
The tip of the pendulum travels along a circular path whose radius is the length of the pendulum.
3
Solve the equation for the radius rr.
r=36r = 36
Isolating rr by multiplying both sides by 92π\frac{9}{2\pi} yields the length of the pendulum.

Anahtar Kavram

Converting angle measures between degrees and radians and applying the arc length formula.
Soru 2688Soru

The analysis of fossilized leaves of the deciduous tree *Ginkgo biloba* from the Cretaceous-Paleogene transition reveals a sharp decrease in stomatal density—the number of pores per unit of leaf area. Because modern *Ginkgo* trees respond to elevated atmospheric carbon dioxide (CO2CO_2) concentrations by reducing their stomatal density to conserve water, some researchers conclude that global CO2CO_2 levels rose significantly during this period. However, skeptics suggest that this reduction was merely a localized adaptation to regional soil moisture levels in the specific basins where the fossils were deposited. This skeptical view is undermined, however, by the discovery of fossilized *Ginkgo* leaves from geographically isolated regions of the same era that also show uniform stomatal reduction, which suggests that

Which choice most logically completes the text?

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Cevap: the stomatal changes were likely a response to a widespread atmospheric shift rather than localized soil factors.

Cevap

the stomatal changes were likely a response to a widespread atmospheric shift rather than localized soil factors.
The correct option correctly infers that because the stomatal density reduction is uniform across geographically separated basins, the cause is a global factor (such as atmospheric changes) rather than isolated local conditions.

Adım Adım Çözüm

1
Identify the main argument and the objection raised in the passage.
The main argument is that decreasing stomatal density in Ginkgo leaves indicates rising global carbon dioxide. The objection is that this change was due to local soil moisture conditions in specific depositional basins.
To evaluate a logical inference, we must first map the opposing perspectives presented in the text.
2
Analyze the counterevidence presented against the objection.
Fossilized leaves from geographically isolated regions from the same era show the same uniform reduction in stomatal density.
This counterevidence directly tests whether the local environment can explain the phenomenon.
3
Draw the logical conclusion that follows from the counterevidence.
Because the reduction is uniform across different, geographically isolated regions, a local factor cannot be the primary cause. Instead, a global factor (like the atmosphere) must be responsible.
A uniform effect across isolated and distinct environments is logically explained by a common, widespread cause rather than multiple identical localized anomalies.

Anahtar Kavram

Identifying a logically necessary conclusion that resolves a conflict between localized and global environmental explanations.
Soru 2689Soru

In right triangle XYZXYZ, the measure of angle XYZXYZ is 9090^\circ. Altitude YWYW is drawn to hypotenuse XZXZ, dividing the triangle into two smaller triangles, XYW\triangle XYW and YZW\triangle YZW. If the area of XYW\triangle XYW is 99 and the area of YZW\triangle YZW is 3636, what is the length of altitude YWYW?

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Cevap: 66

Cevap

The length of altitude YWYW is 66.
Because altitude YWYW is perpendicular to hypotenuse XZXZ in right triangle XYZXYZ, the two smaller triangles XYW\triangle XYW and YZW\triangle YZW are similar. The ratio of the area of XYW\triangle XYW to the area of YZW\triangle YZW is 99 to 3636, which simplifies to 11 to 44. Since the ratio of the areas of similar triangles is the square of their linear scale factor, the ratio of their corresponding side lengths is 1/4=1/2\sqrt{1/4} = 1/2. Side XWXW corresponds to side YWYW, and side YWYW corresponds to side WZWZ, so YW=2XWYW = 2 \cdot XW. The area of XYW\triangle XYW is given by 12XWYW=9\frac{1}{2} \cdot XW \cdot YW = 9. Substituting YW=2XWYW = 2 \cdot XW gives 12XW(2XW)=9\frac{1}{2} \cdot XW \cdot (2 \cdot XW) = 9, which simplifies to XW2=9XW^2 = 9, so XW=3XW = 3. Thus, the length of altitude YWYW is 23=62 \cdot 3 = 6.

Adım Adım Çözüm

1
Set up the similarity relationship between the two smaller triangles.
XYWYZW\triangle XYW \sim \triangle YZW because both are right triangles sharing an acute angle relationship with the main right triangle XYZ\triangle XYZ.
An altitude drawn to the hypotenuse of a right triangle divides it into two triangles that are similar to the original triangle and to each other.
2
Determine the scale factor between the similar triangles using their areas.
The ratio of the area of XYW\triangle XYW to the area of YZW\triangle YZW is 9/36=1/49/36 = 1/4. The linear scale factor is the square root of the area ratio, which is 1/4=1/2\sqrt{1/4} = 1/2.
For similar figures, the ratio of their areas is the square of the ratio of their corresponding linear dimensions.
3
Express the relationship between the corresponding sides of the two triangles.
In XYW\triangle XYW and YZW\triangle YZW, the side XWXW corresponds to YWYW, and the side YWYW corresponds to WZWZ. Therefore, YW=2XWYW = 2 \cdot XW.
The linear scale factor of 1/21/2 means each side of the smaller triangle is half the length of the corresponding side of the larger triangle.
4
Use the area formula for XYW\triangle XYW to solve for the length of XWXW.
Area(XYW)=12XWYW=9    12XW(2XW)=9    XW2=9    XW=3\text{Area}(\triangle XYW) = \frac{1}{2} \cdot XW \cdot YW = 9 \implies \frac{1}{2} \cdot XW \cdot (2 \cdot XW) = 9 \implies XW^2 = 9 \implies XW = 3.
Substituting the relationship between YWYW and XWXW into the area equation isolates the single variable XWXW.
5
Calculate the length of the altitude YWYW.
YW=23=6YW = 2 \cdot 3 = 6.
Multiplying the length of XWXW by the scale factor of 22 yields the length of YWYW.

Anahtar Kavram

Similarity in right triangles and the geometric relationships of altitudes
Tahmini Süre:2m 30s
Soru 2690Soru

In right triangle ABCABC, the measure of angle CC is 9090^\circ. A point DD lies on side ACAC such that BDBD is the angle bisector of angle ABCABC. If BC=28BC = 28 and BD=35BD = 35, what is the length of segment ADAD?

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Cevap: 75

Cevap

The length of segment ADAD is 75.
By applying the Pythagorean theorem to right triangle BCDBCD, we find CD=21CD = 21. Then, by the angle bisector theorem, the ratio of ABAB to BCBC equals the ratio of ADAD to CDCD, which gives AB=43ADAB = \frac{4}{3}AD. Using the Pythagorean theorem on right triangle ABCABC, we solve (43AD)2=282+(21+AD)2(\frac{4}{3}AD)^2 = 28^2 + (21 + AD)^2 to find the positive length AD=75AD = 75.

Adım Adım Çözüm

1
Find the length of CDCD using the Pythagorean theorem on right triangle BCDBCD.
CD=21CD = 21
Since angle CC is a right angle, triangle BCDBCD is a right triangle with hypotenuse BDBD and leg BCBC.
2
Apply the angle bisector theorem to express ABAB in terms of ADAD.
AB=43ADAB = \frac{4}{3}AD
The angle bisector theorem states that ADCD=ABBC\frac{AD}{CD} = \frac{AB}{BC}. Substituting BC=28BC = 28 and CD=21CD = 21 gives AD21=AB28\frac{AD}{21} = \frac{AB}{28}.
3
Set up a quadratic equation using the Pythagorean theorem on right triangle ABCABC.
(43AD)2=282+(21+AD)2(\frac{4}{3}AD)^2 = 28^2 + (21 + AD)^2
In right triangle ABCABC, the hypotenuse is ABAB and the legs are BC=28BC = 28 and AC=CD+AD=21+ADAC = CD + AD = 21 + AD.
4
Solve the quadratic equation for ADAD.
AD=75AD = 75
Expanding and simplifying the equation yields AD254AD1575=0AD^2 - 54AD - 1575 = 0, which factors as (AD75)(AD+21)=0(AD - 75)(AD + 21) = 0. Since length must be positive, AD=75AD = 75.

Anahtar Kavram

Pythagorean Theorem and Angle Bisector Theorem

Alternatif Yöntem

Let θ=DBC\theta = \angle DBC. Since BDBD bisects angle BB, ABC=2θ\angle ABC = 2\theta. In right triangle BCDBCD, cosθ=BCBD=2835=45\cos\theta = \frac{BC}{BD} = \frac{28}{35} = \frac{4}{5}. In right triangle ABCABC, cos(2θ)=BCAB=28AB\cos(2\theta) = \frac{BC}{AB} = \frac{28}{AB}. Using the double-angle identity cos(2θ)=2cos2θ1\cos(2\theta) = 2\cos^2\theta - 1, we get 28AB=2(45)21=725\frac{28}{AB} = 2(\frac{4}{5})^2 - 1 = \frac{7}{25}, which gives AB=100AB = 100. Finally, AC=AB2BC2=1002282=96AC = \sqrt{AB^2 - BC^2} = \sqrt{100^2 - 28^2} = 96, so AD=ACCD=9621=75AD = AC - CD = 96 - 21 = 75.
Tahmini Süre:2m 30s
Soru 2691Soru

A student is preparing a presentation on the history of superconductivity and has taken the following notes:

* In 19111911, Dutch physicist Heike Kamerlingh Onnes discovered superconductivity.
* Superconductivity is the complete absence of electrical resistance in certain materials.
* Onnes first observed this phenomenon while cooling solid mercury to a temperature of 4.24.2 Kelvin.
* For his pioneering low-temperature research, Onnes was awarded the Nobel Prize in Physics in 19131913.

The student wants to write a sentence that emphasizes the year superconductivity was discovered and the temperature at which it was first observed. Complete the student's statement by filling in the blanks.

Aşağıdaki boşlukları doldurun

First observed when mercury was cooled to , superconductivity was discovered in .
Cevabı ve açıklamayı göster

Cevap

First observed when mercury was cooled to 4.2 Kelvin, superconductivity was discovered in 1911.
The correct response completes the sentence by identifying '4.2 Kelvin' as the temperature at which the phenomenon was first observed, and '1911' as the year superconductivity was discovered, directly fulfilling the rhetorical goal.

Adım Adım Çözüm

1
Determine the rhetorical goal.
The statement must emphasize the year superconductivity was discovered (1911) and the temperature of first observation (4.2 Kelvin).
This ensures the sentence meets the student's specific presentation goal.
2
Extract the temperature of first observation from the notes.
The notes state Onnes cooled mercury to a temperature of 4.2 Kelvin.
This fills in the first blank of the statement.
3
Extract the discovery year from the notes.
The notes state Onnes discovered superconductivity in 1911.
This fills in the second blank of the statement.

Anahtar Kavram

Synthesizing specific details to meet a rhetorical goal requires selecting the correct facts from the provided notes and placing them in the appropriate contexts in the sentence.
Tahmini Süre:1m 0s
Soru 2692Soru

A sector of a circle with center OO has an area of 54π54\pi. The perimeter of the sector is 36+6π36 + 6\pi. If the radius of the circle is an integer, what is the radius of the circle?

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

18
The area of a sector is given by A=12rsA = \frac{1}{2}rs, where rr is the radius and ss is the arc length. Given A=54πA = 54\pi, we have 12rs=54π\frac{1}{2}rs = 54\pi, which simplifies to rs=108πrs = 108\pi, or s=108πrs = \frac{108\pi}{r}. The perimeter of a sector is P=2r+sP = 2r + s. Given P=36+6πP = 36 + 6\pi, we can substitute ss to get 2r+108πr=36+6π2r + \frac{108\pi}{r} = 36 + 6\pi. Multiplying the entire equation by rr and rearranging terms yields the quadratic equation 2r2(36+6π)r+108π=02r^2 - (36 + 6\pi)r + 108\pi = 0. Factoring by grouping gives (2r6π)(r18)=0(2r - 6\pi)(r - 18) = 0. This yields two potential solutions: r=3πr = 3\pi and r=18r = 18. Since the radius is specified to be an integer, the correct answer is 1818.

Adım Adım Çözüm

1
Relate sector area and perimeter to radius and arc length
Area = 12rs=54π\frac{1}{2}rs = 54\pi and Perimeter = 2r+s=36+6π2r + s = 36 + 6\pi, where rr is the radius and ss is the arc length of the sector.
The area of a sector with radius rr and arc length ss is given by 12rs\frac{1}{2}rs, and its perimeter consists of the two radii plus the arc length.
2
Express arc length in terms of radius
s=108πrs = \frac{108\pi}{r}
Isolating ss from the area equation allows for substitution into the perimeter equation.
3
Substitute and form a quadratic equation
2r+108πr=36+6π    2r2(36+6π)r+108π=02r + \frac{108\pi}{r} = 36 + 6\pi \implies 2r^2 - (36 + 6\pi)r + 108\pi = 0
Multiplying both sides of the equation by rr eliminates the fraction and forms a standard quadratic equation.
4
Factor the quadratic equation
(2r6π)(r18)=0(2r - 6\pi)(r - 18) = 0
Grouping the terms as (2r236r)(6πr108π)=0(2r^2 - 36r) - (6\pi r - 108\pi) = 0 allows us to factor out 2r(r18)6π(r18)=02r(r - 18) - 6\pi(r - 18) = 0.
5
Identify the integer radius
r=18r = 18
The two solutions to the equation are r=3πr = 3\pi and r=18r = 18. Since the problem specifies that the radius is an integer, we select 1818.

Anahtar Kavram

Calculating sector area and perimeter using relationships between radius, arc length, and angle measures.
Soru 2693Soru

In the early modern era, the rapid proliferation of printing technology did not simply accelerate the spread of ideas; it also had a profound standardizing effect on regional dialects. Before the printing press, local vernaculars varied widely over short distances. However, the commercial necessity of printing books for a broad market forced publishers to adopt more uniform spelling and grammatical conventions. Consequently, this market-driven standardization gradually ______ the previously diverse linguistic landscape, establishing dominant national languages at the expense of regional variations.

Which choice completes the text with the most logical and precise word or phrase?

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Cevap: homogenized

Cevap

The word that completes the text most logically and precisely is 'homogenized'.
The correct option is 'homogenized' because it means to make uniform or similar. The passage describes how the commercial needs of publishers led to standardized conventions, which reduced regional variations and established dominant national languages, thereby making the linguistic landscape uniform.

Adım Adım Çözüm

1
Analyze the context of the blank
The sentence indicates that market-driven standardization did something to the 'previously diverse linguistic landscape,' resulting in the establishment of 'dominant national languages at the expense of regional variations.'
Understanding the overall direction and relationship of the ideas helps determine what kind of word is required in the blank.
2
Determine the meaning of the required word
The word must mean to make uniform, standard, or similar, since it is described as a 'standardization' that replaced a 'diverse' landscape with 'dominant national' languages.
This establishes the semantic requirement for the correct vocabulary word.
3
Evaluate the choices against the context
'Homogenized' means to make uniform or similar, which perfectly fits. 'Delineated' (outlined/described), 'forestalled' (prevented), and 'exacerbated' (worsened) do not fit the context of making something uniform.
Selecting the option that matches the semantic and grammatical requirements of the sentence.

Anahtar Kavram

Selecting words in context by using surrounding textual clues to determine the precise meaning required.
Soru 2694Soru

While preparing a presentation on Viking exploration in North America, a student took the following notes:
- L'Anse aux Meadows is an archaeological site located in Newfoundland, Canada.
- In 1960, Norwegian explorer Helge Ingstad and archaeologist Anne Stine Ingstad discovered the site.
- The site contains the remains of eight timber-framed sod buildings constructed by Norse settlers.
- Radiocarbon dating suggests the Norse settlement was occupied around 1000 CE.
- UNESCO designated L'Anse aux Meadows a World Heritage site in 1978.

The student wants to emphasize the year of the site's discovery and the names of its discoverers. Which choice most effectively uses information from the given notes to accomplish this goal?

Cevabı ve açıklamayı göster

Cevap: In 1960, explorer Helge Ingstad and archaeologist Anne Stine Ingstad discovered the site of L'Anse aux Meadows.

Cevap

In 1960, explorer Helge Ingstad and archaeologist Anne Stine Ingstad discovered the site of L'Anse aux Meadows.
The sentence stating that in 1960, explorer Helge Ingstad and archaeologist Anne Stine Ingstad discovered the site of L'Anse aux Meadows directly accomplishes the writing goal because it includes both the year of discovery (1960) and the names of the discoverers (Helge Ingstad and Anne Stine Ingstad).

Adım Adım Çözüm

1
Identify the target details requested by the prompt.
The target details are the year of the site's discovery (1960) and the names of the discoverers (Helge Ingstad and Anne Stine Ingstad).
This establishes the criteria that the correct choice must satisfy.
2
Evaluate the answer choices against the target details.
Only one choice contains both the year 1960 and the names Helge Ingstad and Anne Stine Ingstad.
This filters out choices that only present partial information or address different goals.

Anahtar Kavram

Rhetorical Synthesis: Emphasizing Specific Details
Soru 2695Soru

In deep-sea marine biology, the migration of bioluminescent organisms living near hydrothermal vents ______ a crucial role in nutrient distribution across oceanic layers. By tracking these vertical movements, researchers have gathered extensive data indicating that these biological processes help sustain complex ecosystems in the otherwise nutrient-poor bathypelagic zone.

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

Cevabı ve açıklamayı göster

Cevap: plays

Cevap

plays
The singular verb 'plays' correctly agrees with the singular subject of the sentence, 'migration'. The plural nouns 'organisms' and 'vents' are part of a prepositional phrase modifying the subject and do not affect the subject-verb agreement.

Adım Adım Çözüm

1
Identify the true subject of the verb in the sentence.
The subject is the singular noun 'migration'. The phrase 'of bioluminescent organisms living near hydrothermal vents' is a prepositional phrase with a participial modifier that describes the migration, but it does not change the subject's number.
Determining the correct subject is necessary to establish subject-verb agreement.
2
Determine the grammatical number and tense required for the verb.
The singular subject 'migration' requires a singular verb. The rest of the passage uses the present tense ('help sustain'), indicating that the simple present tense is appropriate.
The verb must agree in number with the subject and maintain tense consistency with the surrounding context.
3
Select the option that provides the singular, present-tense form of the verb.
The singular, present-tense verb is 'plays'.
Only 'plays' is singular and fits the simple present aspect of the sentence.

Anahtar Kavram

Subject-Verb Agreement
Soru 2696Soru

In the xyxy-plane, the graph of the equation x2+y214x12y+q=0x^2 + y^2 - 14x - 12y + q = 0 is a circle that is tangent to the yy-axis, where qq is a constant. What is the value of qq?

Cevabı ve açıklamayı göster

Cevap: 36

Cevap

36
To find the value of qq, we convert the general form of the circle's equation into standard form (xh)2+(yk)2=r2(x-h)^2 + (y-k)^2 = r^2 by completing the square. Grouping the variables gives (x214x)+(y212y)=q(x^2 - 14x) + (y^2 - 12y) = -q. Adding (14/2)2=49(14/2)^2 = 49 and (12/2)2=36(12/2)^2 = 36 to both sides yields (x7)2+(y6)2=85q(x - 7)^2 + (y - 6)^2 = 85 - q. Thus, the center of the circle is (7,6)(7, 6) and the radius squared is r2=85qr^2 = 85 - q. Since the circle is tangent to the yy-axis, its radius must equal the horizontal distance from its center to the yy-axis, which is 77. Therefore, the radius squared is 72=497^2 = 49. Setting 85q=4985 - q = 49 and solving for qq gives q=36q = 36.

Adım Adım Çözüm

1
Rearrange and group the terms of the equation to prepare for completing the square.
(x214x)+(y212y)=q(x^2 - 14x) + (y^2 - 12y) = -q
Grouping terms allows us to complete the square for the xx and yy variables independently.
2
Complete the square for both the xx and yy expressions by adding the square of half the coefficient of the linear term to both sides.
(x214x+49)+(y212y+36)=q+49+36(x^2 - 14x + 49) + (y^2 - 12y + 36) = -q + 49 + 36
Adding (14/2)2=49(14/2)^2 = 49 and (12/2)2=36(12/2)^2 = 36 transforms the trinomials into perfect squares.
3
Factor the perfect square trinomials and simplify the right side of the equation.
(x7)2+(y6)2=85q(x - 7)^2 + (y - 6)^2 = 85 - q
This puts the equation in the standard form of a circle, (xh)2+(yk)2=r2(x-h)^2 + (y-k)^2 = r^2, identifying the center as (7,6)(7, 6) and the radius squared as r2=85qr^2 = 85 - q.
4
Determine the radius of the circle using the condition of tangency to the yy-axis.
r=7r = 7
A circle tangent to the yy-axis has a radius equal to the absolute value of the x-coordinate of its center, which is 7=7|7| = 7.
5
Solve for the constant qq by equating the two expressions for the radius squared.
85q=49    q=3685 - q = 49 \implies q = 36
Since r=7r = 7, we have r2=49r^2 = 49. Setting 85q=4985 - q = 49 yields q=36q = 36.

Anahtar Kavram

Completing the square to find the standard equation of a circle and applying coordinate geometry tangency conditions.
Soru 2697Soru

In the nutrient-poor soils of temperate forests, many orchid species depend on mycorrhizal fungi to initiate seed germination. While researchers have long established that these symbiotic relationships are essential during the early stages of orchid development, botanical models have traditionally assumed that mature orchids become fully autotrophic, no longer relying on their fungal hosts for carbon. However, a recent study by Dr. Charles Albright revealed that several adult orchid species continue to receive significant amounts of carbon from mycorrhizal networks, particularly during periods of intense shade. This finding suggests that mycoheterotrophy may play a far more persistent role in orchid survival than previously believed, prompting researchers to reevaluate the ecological significance of belowground fungal networks in forest ecosystems.

Which choice best describes the function of the underlined sentence in the text as a whole?

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Cevap: It introduces empirical evidence that challenges a common assumption about mature orchid nutrition.

Cevap

The correct answer is the option that states the underlined sentence introduces empirical evidence that challenges a common assumption about mature orchid nutrition.
The passage begins by describing a widely held assumption in botany: that mature orchids are fully autotrophic and independent of fungal hosts for carbon. The underlined sentence introduces Dr. Albright's research findings, which demonstrate that adult orchids do in fact receive carbon from fungal networks. Therefore, the underlined sentence serves to present empirical evidence that calls the traditional assumption into question.

Adım Adım Çözüm

1
Analyze the context preceding the underlined sentence.
The passage outlines a traditional botanical assumption: mature orchids are fully autotrophic and do not depend on fungal hosts for carbon.
Establishing the background assumption is necessary to understand what the underlined sentence is responding to.
2
Analyze the content and function of the underlined sentence itself.
The underlined sentence begins with the transition 'However' and introduces Dr. Albright's study showing that adult orchids do continue to receive carbon from fungal networks.
Determining the direct meaning of the underlined portion helps identify its immediate rhetorical role.
3
Evaluate the relationship of the underlined sentence to the text as a whole.
The sentence presents empirical findings that contradict the traditional assumption detailed in the preceding sentence, setting up the broader conclusion in the final sentence.
Connecting the sentence's specific findings to the preceding assumption reveals its rhetorical function as a challenge to that assumption.

Anahtar Kavram

Part-to-Whole Rhetorical Function
Soru 2698Soru

A digital designer is scaling two similar triangular logos. The smaller logo has a perimeter of 1818 centimeters and an area of 1212 square centimeters. If the larger logo has a perimeter of 5454 centimeters, what is the area, in square centimeters, of the larger logo?

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Cevap: 108108

Cevap

108 square centimeters
The correct answer is 108108. Since the two triangular logos are similar, the ratio of their areas is the square of the ratio of their perimeters. The ratio of their perimeters is 5418=3\frac{54}{18} = 3, meaning the larger logo has linear dimensions that are 33 times those of the smaller logo. Squaring this scale factor gives the area scaling factor, which is 32=93^2 = 9. Therefore, the area of the larger logo is 12×9=10812 \times 9 = 108 square centimeters.

Adım Adım Çözüm

1
Determine the linear scale factor between the two similar logos.
The linear scale factor kk is 5418=3\frac{54}{18} = 3.
For similar figures, the ratio of any corresponding linear dimensions (such as perimeters) is equal to the linear scale factor.
2
Find the area scaling factor.
The area scaling factor is k2=32=9k^2 = 3^2 = 9.
The ratio of the areas of two similar figures is equal to the square of their linear scale factor.
3
Calculate the area of the larger logo.
The area of the larger logo is 12×9=10812 \times 9 = 108 square centimeters.
Multiplying the area of the smaller logo by the area scaling factor yields the area of the larger logo.

Anahtar Kavram

The ratio of the areas of two similar two-dimensional shapes is equal to the square of their linear scale factor.
Tahmini Süre:1m 30s
Soru 2699Soru

A quality control inspector at an electronics factory found that in a large shipment of smartphone screens, the ratio of the number of damaged screens to the number of undamaged screens was 33 to 2222. If the shipment contained a total of 6,6006,600 smartphone screens, how many of the screens in the shipment were damaged?

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Cevap: 792

Cevap

792
The correct value of 792792 is obtained by first determining the total number of parts in the ratio, which is 3+22=253 + 22 = 25. The fraction of damaged screens in the shipment is therefore 325\frac{3}{25}. Multiplying this fraction by the total number of screens in the shipment (6,6006,600) yields 325×6,600=792\frac{3}{25} \times 6,600 = 792.

Adım Adım Çözüm

1
Determine the total number of parts in the ratio.
3+22=253 + 22 = 25 total parts
To find the fraction of the total shipment that is damaged, we must add the parts of the damaged and undamaged screens.
2
Set up the fraction of the shipment that is damaged.
325\frac{3}{25}
Since there are 33 damaged screens for every 2525 total screens, the proportion of damaged screens is 325\frac{3}{25}.
3
Multiply the fraction by the total number of screens in the shipment.
325×6,600=792\frac{3}{25} \times 6,600 = 792
Multiplying the proportion of damaged screens by the total size of the shipment gives the total number of damaged screens.

Anahtar Kavram

Applying part-to-whole ratios to solve for a specific quantity in a given total.
Tahmini Süre:1m 15s
Soru 2700Soru

In behavioral economics, the systematic evaluation of potential financial losses—often influenced by complex cognitive biases and individual risk tolerance—______ a far more pronounced effect on consumer decision-making than the prospect of equivalent gains. This phenomenon, commonly known as loss aversion, explains why shoppers frequently choose options that prioritize avoiding risk over maximizing their overall financial utility.

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

Cevabı ve açıklamayı göster

Cevap: exerts

Cevap

exerts
The singular, present-tense verb 'exerts' correctly agrees with the singular subject 'evaluation'. Despite the intervening prepositional phrase 'of potential financial losses' and the parenthetical modifier, the true subject remains singular, requiring a singular verb form.

Adım Adım Çözüm

1
Identify the subject of the clause containing the blank.
The subject is the singular noun phrase 'the systematic evaluation'.
Verbs must agree in number with their grammatical subjects.
2
Analyze intervening phrases and modifiers.
The prepositional phrase 'of potential financial losses' and the parenthetical modifier contain plural nouns ('losses', 'biases'), but they do not change the singular nature of the subject 'evaluation'.
Intervening phrases do not affect subject-verb agreement.
3
Select the verb form that is singular and fits the present-tense context of the passage.
The singular, present-tense verb 'exerts' is the only form that maintains subject-verb agreement and matches the surrounding present-tense verbs ('explains', 'choose').
This completes the sentence grammatically and maintains tense consistency.

Anahtar Kavram

Subject-verb agreement requires a singular subject to take a singular verb, even when separated by intervening plural modifiers.
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