Mathematical Analysis and Data Trends Alıştırma Soruları — ACT

Table 1 shows the heights of 5 sunflower seedlings grown under identical greenhouse conditions for 14 days.

Seedling	Height ( $\text{cm}$ )
Seedling 1	12
Seedling 2	15
Seedling 3	18
Seedling 4	15
Seedling 5	20

Based on the data in Table 1, match each statistical measure of seedling height to its correct calculated value.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

Mean corresponds to

16\text{ cm}

, Median corresponds to

15\text{ cm}

, and Range corresponds to

8\text{ cm}

.

The mean height is the average value (

16\text{ cm}

), the median height is the middle value when the dataset is ordered (

15\text{ cm}

), and the range is the difference between the highest and lowest values (

8\text{ cm}

).

Adım Adım Çözüm

1

Calculate the mean of the seedling heights.

16\text{ cm}

Sum the heights of all 5 seedlings (

12 + 15 + 18 + 15 + 20 = 80\text{ cm}

) and divide by the total number of seedlings (

5

) to obtain the arithmetic average:

\frac{80}{5} = 16\text{ cm}

.

2

Determine the median of the seedling heights.

15\text{ cm}

Order the heights from least to greatest:

12, 15, 15, 18, 20

. The median is the middle value in this list, which is the third value (

15\text{ cm}

).

3

Calculate the range of the seedling heights.

8\text{ cm}

Identify the maximum height (

20\text{ cm}

) and the minimum height (

12\text{ cm}

). Subtract the minimum from the maximum to find the range:

20 - 12 = 8\text{ cm}

.

Anahtar Kavram

Basic Statistical Calculations

Soru 2Soru →

A student measures the volume of a sample of gas at different temperatures while keeping the pressure constant. The data are shown in the table below:

Temperature ( $K$ )	Volume ( $L$ )
100	2.0
200	4.0
300	6.0
400	8.0

Based on this table, which of the following mathematical relationships best describes the relationship between the temperature and the volume of the gas sample?

The volume is inversely proportional to the temperature.

The volume is independent of the temperature.

The volume is directly proportional to the temperature.

The volume decreases linearly as the temperature increases.

Cevabı ve açıklamayı göster

Cevap: The volume is directly proportional to the temperature.

Cevap

The volume is directly proportional to the temperature.

The correct answer shows that the volume is directly proportional to the temperature because the ratio of volume to temperature (

V/T

) remains constant at

0.02\text{ L/K}

across all measured values. In a direct proportion, when one variable is multiplied by a factor, the other variable is multiplied by the same factor.

Adım Adım Çözüm

1

Examine the direction of change for both variables in the table.

As the temperature increases from

100\text{ K}

to

400\text{ K}

, the volume also increases from

2.0\text{ L}

to

8.0\text{ L}

.

This rules out inverse relationships and independence, indicating a direct relationship.

2

Calculate the ratio of the volume (

V

) to the temperature (

T

) at each data point.

The ratio

V/T

is constant:

\frac{2.0}{100} = 0.02\text{ L/K}

,

\frac{4.0}{200} = 0.02\text{ L/K}

,

\frac{6.0}{300} = 0.02\text{ L/K}

, and

\frac{8.0}{400} = 0.02\text{ L/K}

.

A constant ratio between two variables confirms that they are directly proportional.

Anahtar Kavram

Direct and Inverse Proportionality

Tahmini Süre:45s

Soru 3Soru →

A student conducts a series of measurements on a sample of gas in a container. Match each experimental variable relationship on the left with its correct proportional trend on the right.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The relationship between gas pressure and gas volume at a constant temperature matches with inverse proportionality. The relationship between gas volume and absolute temperature at a constant pressure matches with direct proportionality. The relationship between gas pressure and volume when pressure is controlled to remain constant matches with no proportionality.

The correct pairings align the inverse relationship of pressure and volume to inverse proportionality, the direct relationship of volume and temperature to direct proportionality, and the constant pressure relationship to no proportionality.

Adım Adım Çözüm

1

Examine the relationship between gas pressure and gas volume at constant temperature.

The formula

P = \frac{k}{V}

shows that pressure is inversely proportional to volume.

As volume increases, pressure decreases by the same factor.

2

Examine the relationship between gas volume and absolute temperature at constant pressure.

The formula

V = kT

shows that volume is directly proportional to absolute temperature.

As absolute temperature increases, volume increases by the same factor.

3

Examine the relationship between gas pressure and volume when pressure is kept constant.

The formula

P = c

shows that pressure does not change regardless of volume.

Since pressure remains constant, there is no proportional change.

Anahtar Kavram

Identifying direct and inverse proportional relationships from physical equations.

Tahmini Süre:1m 0s

Soru 4Soru →

A student measures the electric current, $I$ (in amperes, $\text{A}$ ), passing through a resistor in a closed circuit with a constant voltage. The current is inversely proportional to the resistance, $R$ (in ohms, $\Omega$ ). When the resistance is $4.0\text{ }\Omega$ , the current is $3.0\text{ A}$ . What is the current, in amperes, when the resistance is changed to $6.0\text{ }\Omega$ ?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The current is

2.0\text{ A}

when the resistance is changed to

6.0\text{ }\Omega

.

Since current and resistance are inversely proportional, their product remains constant:

I_1 R_1 = I_2 R_2

. Substituting the values gives

(3.0)(4.0) = I_2 (6.0)

, which simplifies to

12.0 = 6.0 I_2

. Solving for

I_2

yields

2.0\text{ A}

.

Adım Adım Çözüm

1

State the inverse proportionality relationship between current and resistance.

I \times R = k

, where

k

is a constant.

Since the voltage is constant, current and resistance share an inverse relationship.

2

Calculate the constant value (

k

) using the initial measurements.

k = 3.0\text{ A} \times 4.0\text{ }\Omega = 12.0

Multiplying the known corresponding current and resistance values yields the proportionality constant.

3

Use the constant to calculate the new current at the new resistance.

I = \frac{12.0}{6.0} = 2.0\text{ A}

Dividing the constant by the new resistance value of

6.0\text{ }\Omega

gives the new current.

Anahtar Kavram

In an inverse proportionality relationship, the product of the two variables remains constant (

y \times x = k

). If one variable increases, the other must decrease proportionally.

Soru 5Soru →

A student measures the frequency ( $f$ , in hertz) and wavelength ( $\lambda$ , in meters) of sound waves propagating through a room at a constant temperature. The results are recorded in the table below:

Frequency ( $Hz$ )	Wavelength ( $m$ )
170	2.00
340	1.00
680	0.50
1360	0.25

Based on these results, which of the following statements best describes the relationship between the frequency and wavelength of the sound waves?

Wavelength is directly proportional to frequency because as frequency increases, wavelength increases.

Wavelength is inversely proportional to frequency because as frequency increases, wavelength decreases.

Wavelength is independent of frequency because wavelength remains constant as frequency increases.

Wavelength is directly proportional to frequency because the product of wavelength and frequency is constant.

Cevabı ve açıklamayı göster

Cevap: Wavelength is inversely proportional to frequency because as frequency increases, wavelength decreases.

Cevap

Wavelength is inversely proportional to frequency because as frequency increases, wavelength decreases.

The correct answer correctly identifies that wavelength and frequency have an inverse relationship. As the frequency increases, the wavelength decreases proportionally. For example, doubling the frequency from

170\text{ Hz}

to

340\text{ Hz}

results in the wavelength being halved from

2.00\text{ m}

to

1.00\text{ m}

, satisfying the mathematical definition of inverse proportionality where the product of the two variables remains constant (

170 \times 2.00 = 340 \times 1.00 = 340

).

Adım Adım Çözüm

1

Analyze the trends of both variables in the provided data table.

As frequency increases from

170\text{ Hz}

to

1360\text{ Hz}

, the wavelength decreases from

2.00\text{ m}

to

0.25\text{ m}

.

Identifying whether variables move in the same or opposite directions is the first step in determining proportionality.

2

Determine the mathematical factor by which the variables change.

When frequency is doubled (from

170\text{ Hz}

to

340\text{ Hz}

), wavelength is halved (from

2.00\text{ m}

to

1.00\text{ m}

). When frequency is quadrupled (to

680\text{ Hz}

), wavelength is divided by four (to

0.50\text{ m}

).

This reciprocal relationship (

x \rightarrow 2x \Rightarrow y \rightarrow \frac{1}{2}y

) confirms that the two variables are inversely proportional.

3

Select the statement that matches this mathematical relationship.

The statement describing the relationship as inversely proportional because wavelength decreases as frequency increases is correct.

This aligns with the observed behavior in the data.

Anahtar Kavram

Inverse proportionality describes a relationship where one variable increases in proportion to the decrease in another variable, such that their product remains constant.

Tahmini Süre:45s

Soru 6Soru →

During a biology lab, a student measures the volume of a liquid sample to be $2.5 \times 10^{-3}\text{ liters}$ . What is the volume of this sample in milliliters ( $\text{mL}$ )? (Enter only the numeric value.)

Cevabı ve açıklamayı göster

Cevap: 2.5

Cevap

The volume of the sample is

2.5\text{ mL}

.

Since

1\text{ liter} = 1000\text{ milliliters}

, we multiply the volume in liters by

1000

. Thus,

2.5 \times 10^{-3} \times 10^3 = 2.5\text{ mL}

.

Adım Adım Çözüm

1

Identify the conversion factor between liters (L) and milliliters (mL).

1\text{ L} = 1000\text{ mL}

To convert from liters to milliliters, we need to know the volumetric ratio between the two units.

2

Convert the volume using scientific notation multiplication.

2.5\text{ mL}

2.5 \times 10^{-3}\text{ L} \times 10^3\text{ mL/L} = 2.5 \times 10^{0}\text{ mL} = 2.5\text{ mL}

.

Anahtar Kavram

Converting scientific notation measurements from liters to milliliters

Soru 7Soru →

In an environmental science study, a student collects a sample of airborne particulate matter and determines its mass to be $8.5 \times 10^{-6}\text{ grams}$ . Which of the following is equivalent to this mass expressed in milligrams ( $\text{mg}$ )?

8.5 \times 10^{-9}\text{ mg}

8.5 \times 10^{-3}\text{ mg}

8.5 \times 10^{-5}\text{ mg}

8.5 \times 10^{3}\text{ mg}

Cevabı ve açıklamayı göster

Cevap: $8.5 \times 10^{-3}\text{ mg}$

Cevap

8.5 \times 10^{-3}\text{ mg}

To convert from grams to milligrams, the given mass in grams must be multiplied by

10^3

(since

1\text{ gram} = 1,000\text{ milligrams}

). Carrying out this operation:

(8.5 \times 10^{-6}) \times 10^3 = 8.5 \times 10^{-6 + 3} = 8.5 \times 10^{-3}\text{ mg}

. This value is the correct equivalent mass.

Adım Adım Çözüm

1

Identify the conversion factor between grams and milligrams.

Since

1\text{ gram} = 10^3\text{ milligrams}

, the conversion factor is

10^3\text{ mg/g}

.

Establishing the relationship between the units is necessary to perform the conversion.

2

Multiply the given mass by the conversion factor.

(8.5 \times 10^{-6}\text{ g}) \times (10^3\text{ mg/g})

Multiplying the value in grams by the number of milligrams per gram converts the unit to milligrams.

3

Simplify the expression by adding the exponents of the base 10 terms.

8.5 \times 10^{-6 + 3} = 8.5 \times 10^{-3}\text{ mg}

Applying the laws of exponents (adding exponents when multiplying powers of the same base) yields the final answer in scientific notation.

Anahtar Kavram

Scientific Notation and Unit Conversions

Tahmini Süre:45s

Soru 8Soru →

During a biology experiment, a student measures the lengths of four different biological specimens. Based on these measurements, arrange the following specimens in order from smallest to largest length.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

Virus (

40\text{ nm}

), Bacterium (

2.0\text{ }\mu\text{m}

), Red blood cell (

8.0\text{ }\mu\text{m}

), Human hair diameter (

0.1\text{ mm}

)

To arrange the specimens from smallest to largest, convert each value to meters using standard scientific notation. The prefix nano- represents

10^{-9}

, micro- represents

10^{-6}

, and milli- represents

10^{-3}

. Converting the values yields: Virus =

4.0 \times 10^{-8}\text{ m}

, Bacterium =

2.0 \times 10^{-6}\text{ m}

, Red blood cell =

8.0 \times 10^{-6}\text{ m}

, and Human hair =

1.0 \times 10^{-4}\text{ m}

. Comparing the exponents shows that the virus is the smallest, followed by the bacterium and the red blood cell (since both have an exponent of

-6

and

2.0 < 8.0

), and the human hair is the largest.

Adım Adım Çözüm

1

Convert all measurements to a common unit, meters (

\text{m}

), using scientific notation.

Virus:

40\text{ nm} = 40 \times 10^{-9}\text{ m} = 4.0 \times 10^{-8}\text{ m}

. Bacterium:

2.0\text{ }\mu\text{m} = 2.0 \times 10^{-6}\text{ m}

. Red blood cell:

8.0\text{ }\mu\text{m} = 8.0 \times 10^{-6}\text{ m}

. Human hair:

0.1\text{ mm} = 0.1 \times 10^{-3}\text{ m} = 1.0 \times 10^{-4}\text{ m}

.

To compare physical sizes, they must be represented in the same base unit and format.

2

Compare the exponents of the values in scientific notation.

The exponents are

-8

for the virus,

-6

for both the bacterium and the red blood cell, and

-4

for the human hair.

A smaller (more negative) exponent in scientific notation indicates a smaller value.

3

Compare values with the same exponent and arrange the entire list from smallest to largest.

Comparing the coefficients for the exponent

-6

:

2.0 < 8.0

. Therefore,

2.0 \times 10^{-6}\text{ m}

is smaller than

8.0 \times 10^{-6}\text{ m}

. The final ordered list is: Virus, Bacterium, Red blood cell, and Human hair.

For values with the same exponent, compare their coefficients directly.

Anahtar Kavram

Converting metric units to scientific notation in base meters and comparing exponent values.

Soru 9Soru →

In an environmental study monitoring air quality near an industrial site, a scientist uses a flat horizontal collector plate with an active surface area of $5.0 \times 10^{-2}\text{ m}^2$ to gather falling dust particles. Over a sampling period of $1.8 \times 10^4\text{ seconds}$ , a total mass of $2.0 \times 10^{-3}\text{ grams}$ of dust is deposited on the plate. Assuming the dust deposition rate is constant, what is the average dust deposition rate in micrograms per square meter per hour ( $\mu\text{g}/(\text{m}^2\cdot\text{hr})$ )?

Cevabı ve açıklamayı göster

Cevap: 8000

Cevap

The average dust deposition rate is 8000 micrograms per square meter per hour.

The correct rate of 8000 micrograms per square meter per hour is found by converting mass (

2.0 \times 10^{-3}\text{ g} = 2,000\text{ }\mu\text{g}

) and time (

1.8 \times 10^4\text{ s} = 5.0\text{ hours}

), then dividing this mass by the product of the area (

5.0 \times 10^{-2}\text{ m}^2

) and the time (

5.0\text{ hours}

).

Adım Adım Çözüm

1

Convert the mass of collected dust from grams (g) to micrograms (\mu g).

2,000 \mu g

The target unit requires mass in micrograms. Since

1\text{ g} = 10^6\text{ }\mu\text{g}

, multiplying

2.0 \times 10^{-3}\text{ g}

by

10^6

gives

2.0 \times 10^3\text{ }\mu\text{g} = 2,000\text{ }\mu\text{g}

.

2

Convert the collection time from seconds (s) to hours (hr).

5.0 hr

The target unit requires time in hours. Since

1\text{ hr} = 3,600\text{ s}

, dividing the total seconds by

3,600

gives

1.8 \times 10^4\text{ s} \div (3.6 \times 10^3\text{ s/hr}) = 5.0\text{ hr}

.

3

Calculate the average deposition rate by dividing mass by the product of area and time.

8,000 \mu g/(m^2\cdot hr)

Deposition rate is given by the formula

\text{Rate} = \frac{\text{Mass}}{\text{Area} \times \text{Time}}

. Substituting the values:

\text{Rate} = \frac{2,000\text{ }\mu\text{g}}{(5.0 \times 10^{-2}\text{ m}^2) \times 5.0\text{ hr}} = \frac{2,000}{0.25} = 8,000\text{ }\mu\text{g}/(\text{m}^2\cdot\text{hr})

.

Anahtar Kavram

Multi-step dimensional analysis and calculation with scientific notation

Tahmini Süre:2m 30s

Soru 10Soru →

A chemist investigated the reaction rate of a reactant, Substance Y, at various initial concentrations. The initial rate of reaction, $R$ , in millimoles per liter per second ( $\text{mmol}\cdot\text{L}^{-1}\cdot\text{s}^{-1}$ ), was recorded for each concentration, $[Y]$ , in millimoles per liter ( $\text{mmol/L}$ ), at a constant temperature of $298\text{ K}$ . The results are presented in the table below:

Initial Concentration $[Y]$ ( $\text{mmol/L}$ )	Initial Rate of Reaction $R$ ( $\text{mmol}\cdot\text{L}^{-1}\cdot\text{s}^{-1}$ )
$1.5$	$4.5$
$3.0$	$18.0$
$4.5$	$40.5$
$6.0$	$72.0$
$7.5$	$112.5$

Based on the trend shown in the table, what would be the expected initial rate of reaction, in $\text{mmol}\cdot\text{L}^{-1}\cdot\text{s}^{-1}$ , if the initial concentration of Substance Y is increased to $12.0\text{ mmol/L}$ ?

Cevabı ve açıklamayı göster

Cevap: 288

Cevap

The expected initial rate of reaction at a concentration of 12.0 mmol/L is 288.0 mmol*L^-1*s^-1.

The rate of reaction scales quadratically with concentration. Calculating the ratio of the rate to the concentration for each data point reveals that the ratio is equal to 2.0 times the concentration, yielding the equation R = 2.0 * [Y]^2. Substituting the target concentration of 12.0 mmol/L gives R = 2.0 * (12.0)^2 = 288.0 mmol*L^-1*s^-1. Alternatively, using the method of finite differences, the second difference between successive values is constant at 9.0, and continuing this sequence to 12.0 mmol/L also results in 288.0.

Adım Adım Çözüm

1

Calculate the ratio of the rate R to the concentration [Y] for each data point.

The ratios are 3.0, 6.0, 9.0, 12.0, and 15.0.

To determine whether a direct proportional or higher-order relationship exists.

2

Formulate the mathematical model that represents this trend.

The ratio R/[Y] increases by 3.0 for every 1.5 mmol/L increase in [Y], which corresponds to R/[Y] = 2.0 * [Y], or R = 2.0 * [Y]^2.

To establish the quadratic relationship governing the dataset.

3

Substitute the target concentration value of 12.0 mmol/L into the derived quadratic equation.

R = 2.0 * (12.0)^2 = 288.0.

To calculate the extrapolated reaction rate.

Anahtar Kavram

Extrapolation of Quadratic Trends

Soru 11Soru →

A student in a materials science lab is analyzing the layers of a multi-junction solar cell. The thicknesses of the four distinct layers are measured using different units, as shown in the table below:

Layer Name	Thickness
Antireflective layer	$6.0 \times 10^4 \text{ pm}$
Perovskite layer	$1.8 \times 10^{-4} \text{ mm}$
Silicon layer	$3.5 \times 10^{-7} \text{ m}$
Contact layer	$4.2 \times 10^2 \text{ nm}$

Based on these measurements, arrange the four solar cell layers by thickness from smallest to largest.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct order of layers from smallest to largest thickness is: Antireflective layer, Perovskite layer, Silicon layer, and Contact layer.

By converting all measurements to standard meters, we get: Antireflective layer =

0.6 \times 10^{-7} \text{ m}

, Perovskite layer =

1.8 \times 10^{-7} \text{ m}

, Silicon layer =

3.5 \times 10^{-7} \text{ m}

, and Contact layer =

4.2 \times 10^{-7} \text{ m}

. Comparing these values yields the correct order from smallest to largest thickness: Antireflective, Perovskite, Silicon, and Contact.

Adım Adım Çözüm

1

Convert the thickness of the Antireflective layer to meters.

6.0 \times 10^4 \text{ pm} = 6.0 \times 10^4 \times 10^{-12} \text{ m} = 6.0 \times 10^{-8} \text{ m} = 0.6 \times 10^{-7} \text{ m}

To compare the thicknesses, all values should be converted to the same standard unit (meters). One picometer (

1 \text{ pm}

) is equal to

10^{-12} \text{ m}

.

2

Convert the thickness of the Perovskite layer to meters.

1.8 \times 10^{-4} \text{ mm} = 1.8 \times 10^{-4} \times 10^{-3} \text{ m} = 1.8 \times 10^{-7} \text{ m}

Convert millimeters to meters. One millimeter (

1 \text{ mm}

) is equal to

10^{-3} \text{ m}

.

3

Ensure the Silicon layer thickness is expressed in the same exponent scale.

3.5 \times 10^{-7} \text{ m}

This value is already in meters with a

10^{-7}

exponent, making comparison straightforward.

4

Convert the thickness of the Contact layer to meters.

4.2 \times 10^2 \text{ nm} = 4.2 \times 10^2 \times 10^{-9} \text{ m} = 4.2 \times 10^{-7} \text{ m}

Convert nanometers to meters. One nanometer (

1 \text{ nm}

) is equal to

10^{-9} \text{ m}

.

5

Order the converted values from smallest to largest.

0.6 \times 10^{-7} \text{ m} < 1.8 \times 10^{-7} \text{ m} < 3.5 \times 10^{-7} \text{ m} < 4.2 \times 10^{-7} \text{ m}

Comparing the coefficient values multiplied by

10^{-7} \text{ m}

allows ordering of the layers.

Anahtar Kavram

Expressing and comparing measurements by converting prefix units to a standard baseline unit using scientific notation.

Soru 12Soru →

During a nuclear physics experiment, a detector measures a neutron flux of $1.5 \times 10^{12}\text{ neutrons per square millimeter per minute } (\text{neutrons}/(\text{mm}^2\cdot\text{min}))$ . What is this neutron flux expressed in units of neutrons per square meter per second (\text{neutrons}/(\text{m}^2\cdot\text{s}))?

2.5 \times 10^{13}

2.5 \times 10^{10}

2.5 \times 10^{16}

9.0 \times 10^{19}

Cevabı ve açıklamayı göster

Cevap: $2.5 \times 10^{16}$

Cevap

The correct neutron flux is

2.5 \times 10^{16}\text{ neutrons}/(\text{m}^2\cdot\text{s})

.

To convert the rate, we substitute the equivalent values for the units in the denominator:

1\text{ mm}^2 = 10^{-6}\text{ m}^2

and

1\text{ min} = 60\text{ s}

. Subsituting these gives

\frac{1.5 \times 10^{12}\text{ neutrons}}{10^{-6}\text{ m}^2 \times 60\text{ s}} = \frac{1.5 \times 10^{12}}{6.0 \times 10^{-5}}\text{ neutrons}/(\text{m}^2\cdot\text{s}) = 0.25 \times 10^{17}\text{ neutrons}/(\text{m}^2\cdot\text{s})

, which simplifies to

2.5 \times 10^{16}\text{ neutrons}/(\text{m}^2\cdot\text{s})

.

Adım Adım Çözüm

1

Write the given rate expression as a ratio of quantities.

Rate =

\frac{1.5 \times 10^{12}\text{ neutrons}}{1\text{ mm}^2 \cdot 1\text{ min}}

This establishes the starting units that need to be converted in the denominator.

2

Convert the unit of area from square millimeters (

\text{mm}^2

) to square meters (

\text{m}^2

).

1\text{ mm}^2 = (10^{-3}\text{ m})^2 = 10^{-6}\text{ m}^2

Since the unit is squared, the linear prefix conversion factor (

10^{-3}

) must also be squared.

3

Convert the unit of time from minutes to seconds.

1\text{ min} = 60\text{ s}

This matches the target time unit in the denominator.

4

Substitute the converted unit values into the denominator of the rate expression.

Rate =

\frac{1.5 \times 10^{12}\text{ neutrons}}{(10^{-6}\text{ m}^2) \cdot (60\text{ s})}

This replaces the original units with their equivalent values in the target units.

5

Simplify the expression to find the final value in scientific notation.

Rate =

\frac{1.5 \times 10^{12}}{6.0 \times 10^{-5}}\text{ neutrons}/(\text{m}^2\cdot\text{s}) = 0.25 \times 10^{17} = 2.5 \times 10^{16}\text{ neutrons}/(\text{m}^2\cdot\text{s})

Dividing the coefficient

1.5

by

6.0

yields

0.25

, and subtracting the exponent in the denominator (

-5

) from the numerator (

12

) yields

17

. Shifting the decimal point to standard scientific notation results in

2.5 \times 10^{16}

.

Anahtar Kavram

Scientific Notation and Unit Conversions

Soru 13Soru →

An agricultural scientist measured the water absorption capacity of three different types of superabsorbent polymer gels (Gel A, Gel B, and Gel C) used in soil conditioning. Six trials were conducted for each gel type, and the mass of water absorbed (in grams per gram of gel) was recorded in the table below:

Gel Type	Trial 1 (g)	Trial 2 (g)	Trial 3 (g)	Trial 4 (g)	Trial 5 (g)	Trial 6 (g)
Gel A	42.1	44.5	41.8	43.2	45.0	42.6
Gel B	51.5	48.2	53.4	49.8	50.6	52.1
Gel C	33.7	35.2	34.0	32.9	36.1	34.5

Based on the table, what is the median water absorption capacity, in grams, for Gel B across the 6 trials?

Cevabı ve açıklamayı göster

Cevap: 51.05

Cevap

The median water absorption capacity for Gel B across the 6 trials is 51.05 grams.

The correct answer is 51.05. To find the median value for Gel B, the 6 water absorption capacities must first be ordered from least to greatest: 48.2, 49.8, 50.6, 51.5, 52.1, and 53.4. Because there is an even number of values, the median is the average of the two middle values (50.6 and 51.5), which is 51.05.

Adım Adım Çözüm

1

Identify the water absorption capacity values for Gel B.

51.5, 48.2, 53.4, 49.8, 50.6, and 52.1

To find the median, we must start with the complete set of data points for the specified group.

2

Sort the data points in ascending order.

48.2, 49.8, 50.6, 51.5, 52.1, 53.4

Calculating a median requires the values to be ordered from least to greatest.

3

Locate the middle values and calculate their average.

Middle values: 50.6 and 51.5. Average: 51.05

For an even number of data points, the median is the arithmetic mean of the two middle values.

Anahtar Kavram

Calculating the median of a dataset with an even number of values.

Soru 14Soru →

During a space weather observation, a satellite detector measures the solar wind proton flux. The detector has an active surface area of $4.0 \times 10^{-4}\text{ m}^2$ . Over a continuous period of $2.5\text{ hours}$ , the detector records a total of $1.8 \times 10^{15}\text{ protons}$ . What is the average proton flux measured by the detector, in units of $\text{protons}\cdot\text{cm}^{-2}\cdot\text{s}^{-1}$ ?

5.0 \times 10^{6}\text{ protons}\cdot\text{cm}^{-2}\cdot\text{s}^{-1}

5.0 \times 10^{10}\text{ protons}\cdot\text{cm}^{-2}\cdot\text{s}^{-1}

5.0 \times 10^{12}\text{ protons}\cdot\text{cm}^{-2}\cdot\text{s}^{-1}

5.0 \times 10^{14}\text{ protons}\cdot\text{cm}^{-2}\cdot\text{s}^{-1}

Cevabı ve açıklamayı göster

Cevap: $5.0 \times 10^{10}\text{ protons}\cdot\text{cm}^{-2}\cdot\text{s}^{-1}$

Cevap

The average proton flux measured by the detector is

5.0 \times 10^{10}\text{ protons}\cdot\text{cm}^{-2}\cdot\text{s}^{-1}

.

The correct option is

5.0 \times 10^{10}\text{ protons}\cdot\text{cm}^{-2}\cdot\text{s}^{-1}

because it represents the total number of recorded protons (

1.8 \times 10^{15}

) divided by the product of the surface area in square centimeters (

4.0\text{ cm}^2

) and the time duration in seconds (

9.0 \times 10^3\text{ s}

).

Adım Adım Çözüm

1

Convert the continuous time interval from hours to seconds.

2.5\text{ hours} \times 3600\text{ s/hour} = 9.0 \times 10^3\text{ s}

The target unit requires time in seconds (

\text{s}^{-1}

).

2

Convert the active detector surface area from square meters (

\text{m}^2

) to square centimeters (

\text{cm}^2

).

4.0 \times 10^{-4}\text{ m}^2 \times 10^4\text{ cm}^2/\text{m}^2 = 4.0\text{ cm}^2

The target unit requires area in square centimeters (

\text{cm}^{-2}

).

3

Calculate the average proton flux by dividing the total number of protons by the product of the converted area and time.

\text{Flux} = \frac{1.8 \times 10^{15}\text{ protons}}{4.0\text{ cm}^2 \times 9.0 \times 10^3\text{ s}} = 5.0 \times 10^{10}\text{ protons}\cdot\text{cm}^{-2}\cdot\text{s}^{-1}

Flux is defined as the quantity of particles passing through a unit area per unit time.

Anahtar Kavram

Scientific Notation and Unit Conversions

Soru 15Soru →

A student investigated the electrical properties of a negative temperature coefficient (NTC) thermistor. The thermistor was placed in a temperature-controlled water bath, and its electrical resistance, $R$ (in kilohms, $\text{k}\Omega$ ), was measured at various temperatures, $T$ (in degrees Celsius, $^\circ\text{C}$ ). The data from this experiment are presented in the table below.

Temperature ( $T$ , $^\circ\text{C}$ )	Resistance ( $R$ , $\text{k}\Omega$ )
10	48.0
30	24.0
50	12.0
70	6.0

Based on the trend shown in the table, what is the predicted electrical resistance, in $\text{k}\Omega$ , of the thermistor at a temperature of $110^\circ\text{C}$ ?

Cevabı ve açıklamayı göster

Cevap: 1.5

Cevap

The predicted electrical resistance of the thermistor at

110^\circ\text{C}

is

1.5\text{ k}\Omega

.

The correct calculation identifies that the resistance decreases by a factor of 2 for every

20^\circ\text{C}

increase in temperature. Following this exponential trend, the resistance at

90^\circ\text{C}

is

3.0\text{ k}\Omega

, and at

110^\circ\text{C}

it is half of that, which equals

1.5\text{ k}\Omega

.

Adım Adım Çözüm

1

Analyze the pattern of temperature changes in the table.

The temperature increments are constant at

\Delta T = 20^\circ\text{C}

(e.g.,

30 - 10 = 20^\circ\text{C}

,

50 - 30 = 20^\circ\text{C}

,

70 - 50 = 20^\circ\text{C}

).

Establishing a constant independent variable interval simplifies trend extrapolation.

2

Analyze the corresponding ratio of resistance values at each temperature step.

At each interval, the resistance value is divided by 2:

\frac{24.0}{48.0} = 0.5

,

\frac{12.0}{24.0} = 0.5

, and

\frac{6.0}{12.0} = 0.5

. This indicates a non-linear, exponential decay trend.

Determining the mathematical relationship allows for precise calculation of values outside the dataset range.

3

Extrapolate the trend to

90^\circ\text{C}

by applying the factor of

0.5

to the resistance at

70^\circ\text{C}

.

R(90^\circ\text{C}) = R(70^\circ\text{C}) \times 0.5 = 6.0 \times 0.5 = 3.0\text{ k}\Omega

.

Since

90^\circ\text{C}

is exactly

20^\circ\text{C}

above

70^\circ\text{C}

, the pattern dictates that the resistance halves.

4

Extrapolate the trend further to

110^\circ\text{C}

by applying the factor of

0.5

to the resistance at

90^\circ\text{C}

.

R(110^\circ\text{C}) = R(90^\circ\text{C}) \times 0.5 = 3.0 \times 0.5 = 1.5\text{ k}\Omega

.

Since

110^\circ\text{C}

is exactly

20^\circ\text{C}

above

90^\circ\text{C}

, the resistance halves once more.

Anahtar Kavram

Extrapolating non-linear (exponential) relationships by identifying constant ratios over equal intervals of the independent variable.

Soru 16Soru →

During an astrophysics study, the density of a stellar nebula's core is determined to be $4.5 \times 10^{-12}$ grams per cubic centimeter ( $\text{g/cm}^3$ ). Which of the following is equivalent to this density expressed in kilograms per cubic meter ( $\text{kg/m}^3$ )?

4.5 \times 10^{-15}\text{ kg/m}^3

4.5 \times 10^{-9}\text{ kg/m}^3

4.5 \times 10^{-12}\text{ kg/m}^3

4.5 \times 10^{-6}\text{ kg/m}^3

Cevabı ve açıklamayı göster

Cevap: $4.5 \times 10^{-9}\text{ kg/m}^3$

Cevap

4.5 \times 10^{-9}\text{ kg/m}^3

The correct answer is

4.5 \times 10^{-9}\text{ kg/m}^3

. To convert density from grams per cubic centimeter (

\text{g/cm}^3

) to kilograms per cubic meter (

\text{kg/m}^3

), we convert the numerator (

1\text{ g} = 10^{-3}\text{ kg}

) and the denominator (

1\text{ cm}^3 = 10^{-6}\text{ m}^3

). This gives the conversion:

1\text{ g/cm}^3 = \frac{10^{-3}\text{ kg}}{10^{-6}\text{ m}^3} = 10^3\text{ kg/m}^3

. Multiplying the initial measurement by this factor gives:

(4.5 \times 10^{-12}) \times 10^3 = 4.5 \times 10^{-9}\text{ kg/m}^3

.

Adım Adım Çözüm

1

Convert the mass unit from grams (

\text{g}

) to kilograms (

\text{kg}

).

1\text{ g} = 10^{-3}\text{ kg}

Since

1\text{ kg} = 10^3\text{ g}

, converting from a smaller unit (g) to a larger unit (kg) requires multiplying by

10^{-3}

.

2

Convert the volume unit in the denominator from cubic centimeters (

\text{cm}^3

) to cubic meters (

\text{m}^3

).

1\text{ cm}^3 = (10^{-2}\text{ m})^3 = 10^{-6}\text{ m}^3

Since

1\text{ m} = 10^2\text{ cm}

, cubing both sides of the conversion factor gives

1\text{ m}^3 = 10^6\text{ cm}^3

, or

1\text{ cm}^3 = 10^{-6}\text{ m}^3

.

3

Combine the conversion factors to find the overall density conversion factor from

\text{g/cm}^3

to

\text{kg/m}^3

.

\frac{1\text{ g}}{1\text{ cm}^3} = \frac{10^{-3}\text{ kg}}{10^{-6}\text{ m}^3} = 10^3\text{ kg/m}^3

Dividing the mass conversion factor by the volume conversion factor yields the combined multiplier for density.

4

Multiply the given core density by the combined conversion factor.

(4.5 \times 10^{-12}) \times 10^3 = 4.5 \times 10^{-9}

Applying the combined factor converts the value to the desired units.

Anahtar Kavram

Unit conversion involving compound units and powers of ten

Alternatif Yöntem

We can set up dimensional analysis:

4.5 \times 10^{-12} \text{ g/cm}^3 \times \left(\frac{1\text{ kg}}{1000\text{ g}}\right) \times \left(\frac{100\text{ cm}}{1\text{ m}}\right)^3 = 4.5 \times 10^{-12} \times 10^{-3} \times 10^6 = 4.5 \times 10^{-9} \text{ kg/m}^3

.

Tahmini Süre:1m 30s

Soru 17Soru →

A physicist studies the rate of heat transfer, $H$ (in watts, $\text{W}$ ), through cylindrical metal rods. The researcher determines that $H$ is directly proportional to both the cross-sectional area of the rod and the temperature difference ( $\Delta T$ , in kelvins, $\text{K}$ ) between its two ends, and inversely proportional to the length of the rod ( $L$ , in meters, $\text{m}$ ). For Rod 1, the radius is $0.020\text{ m}$ , the length is $0.80\text{ m}$ , the temperature difference is $50.0\text{ K}$ , and the rate of heat transfer is $100.0\text{ W}$ . Rod 2 is made of the same metal and has a radius of $0.040\text{ m}$ , a length of $0.40\text{ m}$ , and a temperature difference of $30.0\text{ K}$ . What is the rate of heat transfer, in watts, for Rod 2?

Cevabı ve açıklamayı göster

Cevap: 480

Cevap

The rate of heat transfer for Rod 2 is 480 W.

The correct answer is 480 W because the rate of heat transfer is proportional to the square of the radius and the temperature difference, and inversely proportional to the length. The radius is doubled (scaling factor of

2^2 = 4

), the temperature difference is multiplied by 0.6, and the length is halved (scaling factor of

\frac{1}{0.5} = 2

). This yields a combined factor of

4 \times 0.6 \times 2 = 4.8

, and multiplying 100.0 W by 4.8 results in 480 W.

Adım Adım Çözüm

1

Relate the cross-sectional area to the radius of the rod.

The area

A

is directly proportional to the square of the radius

r

:

A \propto r^2

.

The cross-section of a cylinder is a circle with area

A = \pi r^2

.

2

Formulate the complete proportionality expression for the rate of heat transfer.

H \propto \frac{r^2 \cdot \Delta T}{L}

.

Heat transfer is directly proportional to area (

r^2

) and temperature difference (

\Delta T

), and inversely proportional to length (

L

).

3

Set up a ratio to compare Rod 2's heat transfer rate to Rod 1's rate.

\frac{H_2}{H_1} = \left(\frac{r_2}{r_1}\right)^2 \cdot \left(\frac{\Delta T_2}{\Delta T_1}\right) \cdot \left(\frac{L_1}{L_2}\right)

.

Using a ratio cancels out the constant of proportionality.

4

Calculate the ratio multiplier by inserting the known values.

Multiplier =

2^2 \cdot 0.6 \cdot 2 = 4 \cdot 0.6 \cdot 2 = 4.8

.

The radius doubles (factor of 4), the temperature difference is multiplied by 0.6, and the length is halved (factor of 2).

5

Multiply the original heat transfer rate by the calculated factor.

H_2 = 100.0 \text{ W} \times 4.8 = 480 \text{ W}

.

To find the final heat transfer rate of Rod 2.

Anahtar Kavram

Combining direct and inverse proportionalities to calculate a new value using scaling factors.

Soru 18Soru →

During an environmental monitoring study of urban air quality, scientists use a cascade impactor to collect and analyze different types of airborne particulate matter. The average diameters of four distinct particulate samples were recorded as follows:

* Particle W: $3.5 \times 10^2 \text{ nm}$
* Particle X: $1.2 \times 10^{-6} \text{ m}$
* Particle Y: $7.5 \times 10^{-2} \text{ }\mu\text{m}$
* Particle Z: $4.0 \times 10^{-5} \text{ cm}$

Based on these measurements, arrange the four particulate samples in order of their average diameters from smallest to largest.

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct order from smallest to largest is Particle Y, Particle W, Particle Z, and Particle X.

By converting all measurements to meters, we find: Particle Y is

7.5 \times 10^{-8} \text{ m}

, Particle W is

3.5 \times 10^{-7} \text{ m}

, Particle Z is

4.0 \times 10^{-7} \text{ m}

, and Particle X is

1.2 \times 10^{-6} \text{ m}

. Comparing these values shows that Particle Y is the smallest, followed by Particle W, Particle Z, and Particle X as the largest.

Adım Adım Çözüm

1

Convert all measurements to meters (

{\text{m}}

) using standard conversion factors (

1 \text{ nm} = 10^{-9} \text{ m}

,

1 \text{ }\mu\text{m} = 10^{-6} \text{ m}

, and

1 \text{ cm} = 10^{-2} \text{ m}

).

Particle W:

3.5 \times 10^2 \text{ nm} = 3.5 \times 10^2 \times 10^{-9} \text{ m} = 3.5 \times 10^{-7} \text{ m}

.
Particle X:

1.2 \times 10^{-6} \text{ m}

.
Particle Y:

7.5 \times 10^{-2} \text{ }\mu\text{m} = 7.5 \times 10^{-2} \times 10^{-6} \text{ m} = 7.5 \times 10^{-8} \text{ m}

.
Particle Z:

4.0 \times 10^{-5} \text{ cm} = 4.0 \times 10^{-5} \times 10^{-2} \text{ m} = 4.0 \times 10^{-7} \text{ m}

.

Expressing all values in the same base metric unit (meters) allows for a direct comparison of their scales.

2

Express all values in terms of the same exponent of

10

(such as

10^{-7}

) to easily compare the coefficients.

Particle Y:

0.75 \times 10^{-7} \text{ m}

.
Particle W:

3.5 \times 10^{-7} \text{ m}

.
Particle Z:

4.0 \times 10^{-7} \text{ m}

.
Particle X:

12.0 \times 10^{-7} \text{ m}

.

Aligning the exponents simplifies the comparison to just ordering the coefficient numbers.

3

Compare the coefficients from smallest to largest.

Since

0.75 < 3.5 < 4.0 < 12.0

, the order from smallest to largest is Particle Y, Particle W, Particle Z, then Particle X.

The coefficients directly scale the common base exponent, yielding the final ordered list.

Anahtar Kavram

To compare values with different metric prefixes, convert each value to a common base unit (such as meters) and write them in scientific notation with a matching exponent.

Tahmini Süre:1m 30s

Soru 19Soru →

A group of students investigated the flow of a viscous liquid through capillary tubes. They measured the volumetric flow rate, $Q$ (in $\text{cm}^3/\text{s}$ ), under different conditions by varying the pressure difference ( $\Delta P$ , in $\text{kPa}$ ) across the tube, the tube length ( $L$ , in $\text{cm}$ ), and the tube radius ( $r$ , in $\text{mm}$ ). The results of their trials are recorded in the table below:

Trial	Pressure Difference ( $\Delta P$ , $\text{kPa}$ )	Tube Length ( $L$ , $\text{cm}$ )	Tube Radius ( $r$ , $\text{mm}$ )	Flow Rate ( $Q$ , $\text{cm}^3/\text{s}$ )
1	100	10	1.0	0.20
2	100	20	1.0	0.10
3	200	10	1.0	0.40
4	100	10	2.0	3.20

Based on the trends shown in the table, if the students were to conduct a fifth trial using a pressure difference of $150\text{ kPa}$ , a tube length of $5\text{ cm}$ , and a tube radius of $3.0\text{ mm}$ , what would be the expected flow rate of the liquid?

48.60\text{ cm}^3/\text{s}

12.15\text{ cm}^3/\text{s}

5.40\text{ cm}^3/\text{s}

1.80\text{ cm}^3/\text{s}

Cevabı ve açıklamayı göster

Cevap: $48.60\text{ cm}^3/\text{s}$

Cevap

The expected flow rate of the liquid is

48.60\text{ cm}^3/\text{s}

.

The correct answer is

48.60\text{ cm}^3/\text{s}

. Comparing the trials shows that the flow rate (

Q

) is directly proportional to the pressure difference (

\Delta P

), inversely proportional to the tube length (

L

), and directly proportional to the fourth power of the radius (

r^4

). When compared to Trial 1, Trial 5 has

1.5

times the pressure difference, half the length, and

3

times the radius. Therefore, the new flow rate is

0.20\text{ cm}^3/\text{s} \times 1.5 \times 2 \times 3^4 = 48.60\text{ cm}^3/\text{s}

.

Adım Adım Çözüm

1

Determine the relationship between flow rate (

Q

) and pressure difference (

\Delta P

) using Trial 1 and Trial 3.

Q \propto \Delta P

(direct proportionality). When the pressure difference is doubled from

100\text{ kPa}

to

200\text{ kPa}

while other variables are kept constant, the flow rate doubles from

0.20\text{ cm}^3/\text{s}

to

0.40\text{ cm}^3/\text{s}

.

To identify how changes in pressure affect flow rate.

2

Determine the relationship between flow rate (

Q

) and tube length (

L

) using Trial 1 and Trial 2.

Q \propto \frac{1}{L}

(inverse proportionality). When the tube length is doubled from

10\text{ cm}

to

20\text{ cm}

while other variables are kept constant, the flow rate is halved from

0.20\text{ cm}^3/\text{s}

to

0.10\text{ cm}^3/\text{s}

.

To identify how changes in tube length affect flow rate.

3

Determine the relationship between flow rate (

Q

) and tube radius (

r

) using Trial 1 and Trial 4.

Q \propto r^4

(fourth-power direct proportionality). When the tube radius is doubled from

1.0\text{ mm}

to

2.0\text{ mm}

while other variables are kept constant, the flow rate increases by a factor of

16

(

3.20 / 0.20 = 16

), which corresponds to

2^4

.

To identify how changes in tube radius affect flow rate.

4

Calculate the expected flow rate for Trial 5 by comparing its parameters to Trial 1.

The expected flow rate is

48.60\text{ cm}^3/\text{s}

. The pressure difference increases by a factor of

1.5

(

150 / 100

), the tube length is halved (

5 / 10 = 0.5

, which doubles

Q

due to inverse proportionality), and the tube radius is tripled (

3.0 / 1.0 = 3.0

, which increases

Q

by a factor of

3^4 = 81

). Thus,

Q_5 = 0.20 \times 1.5 \times 2 \times 81 = 48.60\text{ cm}^3/\text{s}

.

To compute the final flow rate based on all combined proportional relationships.

Anahtar Kavram

Direct and Inverse Proportionality

Soru 20Soru →

An environmental microbiologist studying soil ecology estimates the average volume of a single soil bacterium to be $2.0 \times 10^{-12}$ cubic centimeters ( $\text{cm}^3$ ). What is the average volume of this bacterium expressed in cubic micrometers ( $\mu\text{m}^3$ )?

2.0 \times 10^{-24} \text{ } \mu\text{m}^3

2.0 \times 10^{-8} \text{ } \mu\text{m}^3

2.0 \times 10^{-6} \text{ } \mu\text{m}^3

2.0 \times 10^0 \text{ } \mu\text{m}^3

Cevabı ve açıklamayı göster

Cevap: $2.0 \times 10^0 \text{ } \mu\text{m}^3$

Cevap

2.0 \times 10^0 \text{ } \mu\text{m}^3

The correct answer is

2.0 \times 10^0 \text{ } \mu\text{m}^3

. To convert cubic centimeters to cubic micrometers, we first find that

1 \text{ cm} = 10^4 \text{ } \mu\text{m}

. Cubing both sides of this linear relationship yields

(1 \text{ cm})^3 = (10^4 \text{ } \mu\text{m})^3 = 10^{12} \text{ } \mu\text{m}^3

. Multiplying the initial bacterium volume by this factor gives

(2.0 \times 10^{-12} \text{ cm}^3) \times (10^{12} \text{ } \mu\text{m}^3/\text{cm}^3) = 2.0 \times 10^0 \text{ } \mu\text{m}^3

.

Adım Adım Çözüm

1

Determine the linear conversion factor between centimeters (

\text{cm}

) and micrometers (

\mu\text{m}

).

1 \text{ cm} = 10^{-2} \text{ m}

and

1 \text{ } \mu\text{m} = 10^{-6} \text{ m}

, so

1 \text{ cm} = 10^4 \text{ } \mu\text{m}

.

Before converting units of volume, the relationship between the linear units must be established.

2

Convert the linear conversion factor to a volume conversion factor by cubing it.

(1 \text{ cm})^3 = (10^4 \text{ } \mu\text{m})^3 = 10^{12} \text{ } \mu\text{m}^3

.

Volume scales cubically with respect to linear dimensions.

3

Multiply the volume of the bacterium in cubic centimeters by the volume conversion factor.

(2.0 \times 10^{-12} \text{ cm}^3) \times (10^{12} \text{ } \mu\text{m}^3/\text{cm}^3) = 2.0 \times 10^0 \text{ } \mu\text{m}^3

.

Multiplying by the conversion factor cancels the cubic centimeter units and yields the volume in cubic micrometers.

Anahtar Kavram

Volume Unit Conversion with Scientific Notation

Tahmini Süre:1m 30s

Mathematical Analysis and Data Trends

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm