Tüm alıştırma soruları

Soru 1921Soru →

A quadratic function $f$ has its vertex at $(4, -3)$ in the coordinate plane. The point $(2, 5)$ is on the graph of $y = f(x)$ . The function $g$ is defined by $g(x) = -f(x - 3) + 8$ . What is the value of $g(8)$ ?

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The correct answer is 9.

The correct answer is 9. To find this, we first establish the vertex form of the quadratic function

f(x) = a(x - 4)^2 - 3

. Substituting the point

(2, 5)

gives

5 = a(2 - 4)^2 - 3

, which simplifies to

8 = 4a

, so

a = 2

. Thus,

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

. To find the value of

g(8)

, we substitute

8

into the definition of

g(x)

, yielding

g(8) = -f(8 - 3) + 8 = -f(5) + 8

. Evaluating

f(5)

gives

2(5 - 4)^2 - 3 = -1

. Finally, substituting

-1

back into the expression for

g(8)

gives

-(-1) + 8 = 9

.

Adım Adım Çözüm

1

Write the quadratic function

f(x)

in vertex form using its vertex

(4, -3)

.

f(x) = a(x - 4)^2 - 3

The vertex form of a quadratic function with vertex

(h, k)

is

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

.

2

Substitute the coordinates of the point

(2, 5)

into the vertex form to find the value of the constant

a

.

a = 2

, so

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

Since the point

(2, 5)

lies on the graph of

f

, substituting

x = 2

and

f(2) = 5

allows us to solve for the vertical stretch factor

a

.

3

Use the definition of

g(x)

to express

g(8)

in terms of

f

.

g(8) = -f(5) + 8

Substituting

x = 8

into the equation

g(x) = -f(x - 3) + 8

yields

g(8) = -f(8 - 3) + 8 = -f(5) + 8

.

4

Evaluate

f(5)

using the equation for

f(x)

found in Step 2.

f(5) = -1

Substituting

x = 5

into

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

gives

2(5 - 4)^2 - 3 = 2(1) - 3 = -1

.

5

Substitute the value of

f(5)

into the expression for

g(8)

to find the final result.

g(8) = 9

Substituting

f(5) = -1

into

g(8) = -f(5) + 8

yields

-(-1) + 8 = 1 + 8 = 9

.

Anahtar Kavram

Finding the equation of a quadratic function from its vertex and a point, and evaluating transformed functions using function notation.

Soru 1922Soru →

A municipal swimming pool is being filled with water at a constant rate. After $2$ hours of filling, the pool contains $14,500$ gallons of water. After $5$ hours of filling, the pool contains $20,200$ gallons of water. If the relationship between the time the pool has been filling, $t$ , in hours, and the volume of water in the pool, $V$ , in gallons, is linear, which of the following equations represents this relationship?

V = 1,900t + 14,500

V = 5,700t + 3,100

V = 1,900t + 10,700

V = 1,900t + 18,300

Cevabı ve açıklamayı göster

Cevap: $V = 1,900t + 10,700$

Cevap

The equation representing the relationship is

V = 1,900t + 10,700

.

The correct equation is

V = 1,900t + 10,700

. The rate of change of the water volume is the change in volume divided by the change in time:

\frac{20,200 - 14,500}{5 - 2} = \frac{5,700}{3} = 1,900

gallons per hour. Using the point-slope form with the coordinate point

(2, 14,500)

yields

V - 14,500 = 1,900(t - 2)

. Simplifying this expression gives

V - 14,500 = 1,900t - 3,800

, which results in

V = 1,900t + 10,700

.

Adım Adım Çözüm

1

Identify two data points from the problem context.

The two coordinate pairs representing

(t, V)

are

(2, 14,500)

and

(5, 20,200)

.

These points will allow us to calculate the slope and the y-intercept of the linear equation.

2

Calculate the slope (

m

) using the slope formula

m = \frac{V_2 - V_1}{t_2 - t_1}

.

m = \frac{20,200 - 14,500}{5 - 2} = \frac{5,700}{3} = 1,900

.

The slope represents the constant rate, in gallons per hour, at which the pool is being filled.

3

Substitute the slope

m = 1,900

and the point

(2, 14,500)

into the point-slope form equation

V - V_1 = m(t - t_1)

to solve for

V

.

V - 14,500 = 1,900(t - 2) \implies V - 14,500 = 1,900t - 3,800 \implies V = 1,900t + 10,700

.

This yields the equation representing the volume of water

V

in the pool at any time

t

.

Anahtar Kavram

Determining a linear equation in two variables given two points from a word problem context.

Alternatif Yöntem

Instead of solving the linear equation algebraically, you can test the coordinates of the two given points

(2, 14,500)

and

(5, 20,200)

in the answer choices. Substituting

t = 2

and

t = 5

into the correct equation

V = 1,900t + 10,700

satisfies both conditions:

1,900(2) + 10,700 = 14,500

and

1,900(5) + 10,700 = 20,200

. None of the other options satisfy both points.

Tahmini Süre:1m 30s

Soru 1923Soru →

An agricultural researcher categorized a sample of $150$ tomato plants by their watering schedule (daily or weekly) and their fruit yield (high yield or average yield). The results are summarized in the table below.

Watering Schedule	High Yield	Average Yield	Total
Daily	$62$	$18$	$80$
Weekly	$28$	$42$	$70$
Total	$90$	$60$	$150$

Given that a tomato plant selected at random from the sample produced a high yield of fruit, what is the probability that the plant had a weekly watering schedule?

\frac{14}{75}

\frac{2}{5}

\frac{14}{45}

\frac{31}{45}

Cevabı ve açıklamayı göster

Cevap: $\frac{14}{45}$

Cevap

The correct answer is fourteen-forty-fifths (14/45). Given that the plant has a high yield, the sample space is restricted to the 90 plants in the High Yield column. Out of these 90 plants, 28 were on a weekly watering schedule, resulting in a probability of 28/90, which simplifies to 14/45.

To find the conditional probability that a tomato plant had a weekly watering schedule given that it produced a high yield of fruit, restrict the denominator to the total number of high-yield plants, which is 90. The numerator is the number of plants that meet both conditions (weekly watering and high yield), which is 28. The probability is 28/90, which simplifies to 14/45.

Adım Adım Çözüm

1

Identify the condition specified in the question stem to determine the subset of data to consider.

The condition is 'given that a tomato plant ... produced a high yield of fruit'. Thus, we restrict our focus to the 'High Yield' column.

This establishes the denominator for the conditional probability calculation.

2

Find the total number of plants in the 'High Yield' column (denominator) and the number of those plants that had a weekly watering schedule (numerator).

Total high-yield plants = 90. High-yield plants with a weekly watering schedule = 28.

These represent the total possible outcomes and the favorable outcomes within the restricted sample space.

3

Calculate and simplify the fraction representing the conditional probability.

Probability = \frac{28}{90} = \frac{14}{45}.

Expressing the final ratio in its simplest fractional form to match the standard exam options.

Anahtar Kavram

Two-Way Tables and Conditional Probability

Tahmini Süre:1m 30s

Soru 1924Soru →

A study of $180$ college students categorized them by their major field of study (STEM or Humanities) and whether they participate in undergraduate research. The table below shows the partial results of the study, where $a$ , $b$ , $c$ , and $d$ represent the number of students in each category.

Major	Participates in Research	Does Not Participate in Research	Total
STEM	$a$	$b$	$110$
Humanities	$c$	$d$	$70$
Total	$80$	$100$	$180$

Given that a student selected at random is a Humanities major, the probability that the student participates in undergraduate research is $\frac{3}{7}$ . If a student who participates in undergraduate research is selected at random, what is the probability that the student is a STEM major?

\frac{5}{8}

\frac{5}{18}

\frac{5}{11}

\frac{1}{2}

Cevabı ve açıklamayı göster

Cevap: $\frac{5}{8}$

Cevap

The correct probability is

\frac{5}{8}

.

To find the probability that a student is a STEM major given they participate in undergraduate research, we first determine the missing values in the table. The total number of Humanities majors is

70

. Since the probability that a Humanities major participates in research is

\frac{3}{7}

, the number of Humanities majors in research is

c = \frac{3}{7} \times 70 = 30

. Using the research column total, the number of STEM majors in research is

a = 80 - c = 80 - 30 = 50

. The probability that a student is a STEM major given that they participate in research is the ratio of STEM research participants to total research participants:

\frac{a}{a+c} = \frac{50}{80} = \frac{5}{8}

.

Adım Adım Çözüm

1

Find the number of Humanities majors who participate in research (

c

).

c = 30

The table shows there are

70

Humanities majors in total. We are given that the probability of a Humanities major participating in research is

\frac{3}{7}

. Therefore,

\frac{c}{70} = \frac{3}{7}

, which simplifies to

c = 30

.

2

Find the number of STEM majors who participate in research (

a

).

a = 50

The total number of students who participate in research is

80

. Since

a + c = 80

and

c = 30

, we have

a + 30 = 80

, which means

a = 50

.

3

Calculate the conditional probability that a student who participates in research is a STEM major.

\frac{5}{8}

We want to find the probability of selecting a STEM major given that the student participates in research. This is the ratio of STEM majors who participate in research (

a = 50

) to the total number of students who participate in research (

80

). The probability is

\frac{50}{80} = \frac{5}{8}

.

Anahtar Kavram

Conditional Probability in Two-Way Tables

Soru 1925Soru →

In the equation $\frac{25^x}{5^{x-3}} = 125^2$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The correct answer is 3.

By writing all terms with a base of 5, the equation

\frac{25^x}{5^{x-3}} = 125^2

becomes

\frac{5^{2x}}{5^{x-3}} = 5^6

. Applying the quotient rule of exponents, the left side simplifies to

5^{2x - (x-3)} = 5^{x+3}

. Setting the exponents equal gives

x+3 = 6

, which yields

x = 3

.

Adım Adım Çözüm

1

Express all terms with a common base of 5.

25^x = (5^2)^x = 5^{2x}

and

125^2 = (5^3)^2 = 5^6

.

Expressing all exponential terms with the same base allows the exponents to be equated directly once simplified.

2

Substitute these expressions back into the original equation and simplify the left side using the quotient rule of exponents.

\frac{5^{2x}}{5^{x-3}} = 5^{2x - (x-3)} = 5^{x+3}

. The equation becomes

5^{x+3} = 5^6

.

The quotient rule states that dividing exponential terms with the same base requires subtracting the exponent of the denominator from the exponent of the numerator:

\frac{b^m}{b^n} = b^{m-n}

.

3

Equate the exponents and solve for

x

.

x + 3 = 6 \implies x = 3

.

Since the bases on both sides of the equation are equal, their exponents must also be equal: if

b^y = b^z

where

b > 0

and

b \neq 1

, then

y = z

.

Anahtar Kavram

Solving exponential equations by expressing terms with a common base and applying exponent rules.

Tahmini Süre:1m 30s

Soru 1926Soru →

A residential solar power system stores electricity in a battery. The total energy stored in the battery, $E$ , in kilowatt-hours (kWh), is modeled by a linear function of the time $t$ , in hours, since sunrise. After $4$ hours of sunlight, the energy stored is $18$ kWh. After $7$ hours of sunlight, the energy stored is $27$ kWh. Which of the following is the best interpretation of the slope of the graph of this function in the $tE$ -plane?

The energy stored in the battery increases by

3

kilowatt-hours per hour of sunlight.

The initial energy stored in the battery at sunrise is

6

kilowatt-hours.

It takes

3

hours of sunlight to increase the energy stored in the battery by

1

kilowatt-hour.

The energy stored in the battery increases by

9

kilowatt-hours per hour of sunlight.

Cevabı ve açıklamayı göster

Cevap: The energy stored in the battery increases by $3$ kilowatt-hours per hour of sunlight.

Cevap

The energy stored in the battery increases by

3

kilowatt-hours per hour of sunlight.

To find the slope of the linear relationship, we use the formula for the rate of change:

\text{slope} = \frac{\Delta E}{\Delta t}

. Using the two points

(4, 18)

and

(7, 27)

, the slope is

\frac{27 - 18}{7 - 4} = \frac{9}{3} = 3

kilowatt-hours per hour. In the context of the problem, the slope represents the rate at which the stored energy increases per hour of sunlight. Therefore, the energy stored in the battery increases by

3

kilowatt-hours per hour of sunlight.

Adım Adım Çözüm

1

Identify the coordinates

(t, E)

from the given values.

The two data points are

(4, 18)

and

(7, 27)

.

These points represent the time in hours and the corresponding energy stored in the battery.

2

Calculate the slope of the linear relationship using the slope formula.

The slope is

\frac{27 - 18}{7 - 4} = \frac{9}{3} = 3

.

The slope of a linear relationship represents the constant rate of change of the dependent variable (

E

) per unit change of the independent variable (

t

).

3

Interpret the meaning of the calculated slope in the context of the problem.

A slope of

3

means that the energy stored in the battery,

E

, increases by

3

kilowatt-hours (kWh) for every

1

hour of sunlight,

t

.

The units of the slope are units of

E

divided by units of

t

, which is kilowatt-hours per hour.

Anahtar Kavram

Interpreting the rate of change (slope) of a linear relationship in context.

Tahmini Süre:1m 30s

Soru 1927Soru →

The value of a financial asset, $V(t)$ , in dollars, is modeled as a function of time $t$ , in years, for $0 \le t \le 4$ . The growth of the asset's value is described by two different models over this period:

- For $0 \le t \le 2$ , the value of the asset increases by a constant percentage of $r\%$ per year.
- For $2 \le t \le 4$ , the value of the asset increases by a constant amount of $d$ dollars per year.

At $t = 0$ , the value of the asset is $100$ dollars, and at $t = 1$ , the value of the asset is $120$ dollars. If the average rate of change of the asset's value from $t = 0$ to $t = 4$ is $25$ dollars per year, what is the value of the asset, in dollars, at $t = 3$ ?

169

170

172

175

Cevabı ve açıklamayı göster

Cevap: 172

Cevap

172

To find the value of the asset at

t = 3

, we first determine its value at

t = 2

using the exponential growth model. Since the value increases by a constant percentage from

t=0

to

t=2

, the value is modeled by

V(t) = V(0) \times (1 + \frac{r}{100})^t

. Given

V(0) = 100

and

V(1) = 120

, the annual growth factor is

1.2

. Thus,

V(2) = 100 \times (1.2)^2 = 144

dollars.

Next, the linear growth from

t = 2

to

t = 4

is modeled by

V(t) = V(2) + d(t - 2) = 144 + d(t - 2)

, which gives

V(4) = 144 + 2d

dollars. The average rate of change of the asset's value from

t = 0

to

t = 4

is

25

dollars per year, so we set up the equation:

\frac{V(4) - V(0)}{4} = 25 \implies \frac{(144 + 2d) - 100}{4} = 25

. Solving for

d

gives

d = 28

dollars per year. Finally, we evaluate the value at

t = 3

:

V(3) = 144 + 28(3 - 2) = 172

dollars.

Adım Adım Çözüm

1

Identify the growth model for

0 \le t \le 2

and write the corresponding equation using the initial values.

Since the value increases by a constant percentage each year, the growth is exponential:

V(t) = V(0) \times (1 + \frac{r}{100})^t

. Given

V(0) = 100

and

V(1) = 120

, we have

120 = 100(1 + \frac{r}{100})^1 \implies 1 + \frac{r}{100} = 1.2

.

Establishing the exponential model allows us to find the growth factor and calculate the value of the asset at the boundary of the two models (

t = 2

).

2

Calculate the value of the asset at

t = 2

using the exponential growth model.

V(2) = 100 \times (1.2)^2 = 144

dollars.

The value at

t = 2

serves as the initial value for the linear growth model in the second interval.

3

Write the growth model for the interval

2 \le t \le 4

.

The growth is linear, so

V(t) = V(2) + d(t - 2) = 144 + d(t - 2)

for

t \ge 2

. At

t = 4

, the value is

V(4) = 144 + 2d

dollars.

Expressing the value at

t = 4

in terms of the constant rate

d

allows us to use the average rate of change information to solve for

d

.

4

Set up the equation for the average rate of change over the entire interval from

t = 0

to

t = 4

and solve for

d

.

The average rate of change is given by

\frac{V(4) - V(0)}{4 - 0} = 25

. Substituting the values, we get

\frac{(144 + 2d) - 100}{4} = 25 \implies \frac{44 + 2d}{4} = 25 \implies 44 + 2d = 100 \implies 2d = 56 \implies d = 28

dollars per year.

The average rate of change relates the final value at

t=4

to the initial value at

t=0

, enabling us to determine the linear rate of change

d

.

5

Calculate the value of the asset at

t = 3

using the linear model.

V(3) = V(2) + d(3 - 2) = 144 + 28(1) = 172

dollars.

Substituting

t = 3

into the linear growth model yields the final requested value.

Anahtar Kavram

Linear and Exponential Growth

Alternatif Yöntem

Instead of setting up the linear equation from

t = 2

to

t = 4

, we can write

V(4)

directly as

V(0) + 4 \times 25 = 200

dollars using the definition of average rate of change. Since

V(2) = 144

dollars, the total change over the linear interval

[2, 4]

is

200 - 144 = 56

dollars. Because the growth is linear during this interval, the change over one year (from

t = 2

to

t = 3

) must be exactly half of the change over two years (from

t = 2

to

t = 4

). Thus, the increase from

t = 2

to

t = 3

is

56 / 2 = 28

dollars, yielding

V(3) = 144 + 28 = 172

dollars.

Tahmini Süre:3m 0s

Soru 1928Soru →

An equation is shown below.

\frac{x+1}{x-2} + \frac{1}{x} = \frac{6}{x^2-2x}

If

x

satisfies the equation above, what is the value of

x

?

-2

2

-4

4

Cevabı ve açıklamayı göster

Cevap: -4

Cevap

-4

The correct answer is

-4

. To solve the rational equation, we first identify the least common denominator as

x(x-2)

. Multiplying both sides by this expression eliminates the denominators, yielding

x(x+1) + (x-2) = 6

. Expanding and simplifying gives the quadratic equation

x^2 + 2x - 8 = 0

, which factors as

(x+4)(x-2) = 0

. This yields potential solutions of

x = -4

and

x = 2

. However, substituting

x = 2

back into the original equation results in division by zero, making

x = 2

extraneous. The only valid solution is

-4

.

Adım Adım Çözüm

1

Find the least common denominator (LCD) of the rational expressions in the equation.

The denominators are

x-2

,

x

, and

x^2-2x

. Since

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

, the LCD is

x(x-2)

, with the restriction that

x \neq 0

and

x \neq 2

.

Finding a common denominator allows us to clear the fractions by multiplying both sides of the equation.

2

Multiply each term of the equation by the LCD to eliminate the denominators.

x(x+1) + 1(x-2) = 6

This clears the denominators and converts the rational equation into a polynomial equation.

3

Expand and simplify the resulting equation.

x^2 + x + x - 2 = 6

, which simplifies to

x^2 + 2x - 2 = 6

.

Combining like terms prepares the equation to be written in standard quadratic form.

4

Rearrange the terms into standard quadratic form:

ax^2 + bx + c = 0

.

x^2 + 2x - 8 = 0

Subtracting

6

from both sides sets the quadratic equation to zero so it can be solved by factoring.

5

Factor the quadratic equation.

(x+4)(x-2) = 0

Finding two numbers that multiply to

-8

and add to

2

(

4

and

-2

) allows us to solve for

x

.

6

Find the potential solutions by setting each factor to zero.

x = -4

or

x = 2

Applying the zero product property gives the potential roots of the quadratic equation.

7

Check the potential solutions in the original equation to identify any extraneous solutions.

For

x = -4

, the equation is defined and valid. For

x = 2

, the denominators

x-2

and

x^2-2x

become zero, which is undefined.

Rational equations can produce extraneous solutions that make the original denominators zero, so they must be checked and discarded.

Anahtar Kavram

Solving rational equations by finding a common denominator and checking for extraneous solutions.

Soru 1929Soru →

A quadratic function $g$ is defined by $g(x) = 2x^2 - 12x + k$ , where $k$ is a constant. In the $xy$ -plane, the graph of $g$ has its vertex on the line $y = -5$ . What is the value of $k$ ?

-59

-23

-5

13

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

13

The

x

-coordinate of the vertex for the quadratic function

g(x) = ax^2 + bx + c

is found using the formula

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

. Substituting

a = 2

and

b = -12

yields

x = -\frac{-12}{2(2)} = 3

. Evaluating

g(3)

gives the

y

-coordinate of the vertex:

g(3) = 2(3)^2 - 12(3) + k = -18 + k

. Since the vertex lies on the line

y = -5

, setting

-18 + k = -5

and solving for

k

gives

k = 13

.

Adım Adım Çözüm

1

Find the

x

-coordinate of the vertex of the parabola.

The

x

-coordinate is

3

.

For a quadratic function

g(x) = ax^2 + bx + c

, the

x

-coordinate of the vertex is given by

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

. Here,

a = 2

and

b = -12

, so

x = -\frac{-12}{2(2)} = 3

.

2

Evaluate the function at

x = 3

to find the

y

-coordinate of the vertex in terms of

k

.

The

y

-coordinate is

-18 + k

.

Substituting

x = 3

into the function gives

g(3) = 2(3)^2 - 12(3) + k = 18 - 36 + k = -18 + k

.

3

Set the

y

-coordinate of the vertex equal to

-5

and solve for

k

.

k = 13

.

Since the vertex lies on the line

y = -5

, the

y

-coordinate of the vertex must be

-5

. Thus,

-18 + k = -5 \implies k = 13

.

Anahtar Kavram

Determining the vertex of a quadratic function and using its coordinates to solve for unknown parameters.

Soru 1930Soru →

A biologist measures the heights, in centimeters, of five seedlings in a laboratory. The heights are $12$ , $15$ , $8$ , $21$ , and $14$ . What is the range, in centimeters, of these heights?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

13

To find the range of a dataset, subtract the minimum value from the maximum value. In the given dataset of seedling heights (

12

,

15

,

8

,

21

, and

14

), the maximum height is

21

centimeters and the minimum height is

8

centimeters. Calculating the difference gives

21 - 8 = 13

centimeters.

Adım Adım Çözüm

1

Identify the maximum and minimum values in the dataset.

The maximum value is

21

and the minimum value is

8

.

The range is defined as the difference between the largest (maximum) and smallest (minimum) values in a data distribution.

2

Subtract the minimum value from the maximum value to calculate the range.

21 - 8 = 13

Subtracting the minimum value of

8

from the maximum value of

21

yields the range of the dataset.

Anahtar Kavram

The range of a dataset measures the spread of the data and is calculated by subtracting the minimum value from the maximum value.

Soru 1931Soru →

The list below shows the number of packages delivered to a local business on each of $8$ business days:

1, 2, 2, 3, 5, 6, 7, 10

What is the median number of packages delivered on these days?

2

3

4

4.5

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4

The correct answer is the median value of

4

. The list contains

8

values already sorted in ascending order:

1, 2, 2, 3, 5, 6, 7, 10

. Because the number of values is even, the median is the average of the two middle values (the

4

th and

5

th values, which are

3

and

5

). The average is calculated as

\frac{3 + 5}{2} = 4

.

Adım Adım Çözüm

1

Identify the two middle terms of the ordered dataset.

The two middle values in the ordered list of 8 numbers are the 4th term, which is

3

, and the 5th term, which is

5

.

Since the dataset has an even number of values (

8

), the median is the average of the two middle terms.

2

Calculate the average of the two middle terms.

\frac{3 + 5}{2} = \frac{8}{2} = 4

Finding the mean of these two middle values yields the median of the entire dataset.

Anahtar Kavram

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

Tahmini Süre:45s

Soru 1932Soru →

An administrator installs a new security update on a computer server. The update is designed to identify and quarantine system errors such that the total number of errors on the server is reduced by $12\%$ each hour. Which of the following best describes the relationship between the time, in hours, since the update was installed and the number of system errors remaining on the server?

Decreasing exponential, because the number of errors decreases by a constant percentage each hour.

Decreasing linear, because the number of errors decreases by a constant quantity each hour.

Increasing exponential, because the number of errors increases by

12\%

each hour.

Decreasing linear, because the number of errors decreases by exactly

12

errors each hour.

Cevabı ve açıklamayı göster

Cevap: Decreasing exponential, because the number of errors decreases by a constant percentage each hour.

Cevap

Decreasing exponential, because the number of errors decreases by a constant percentage each hour.

The correct answer is the option stating that the relationship is decreasing exponential because the number of errors decreases by a constant percentage each hour. In mathematical modeling, when a quantity changes by a fixed percentage over equal intervals of time, it is modeled by an exponential function. Since the quantity is being reduced, it is a decreasing exponential relationship.

Adım Adım Çözüm

1

Analyze the rate of change described in the problem.

The number of system errors is reduced by

12\%

each hour.

Identifying how the quantity changes per unit of time is key to determining the model type.

2

Distinguish between linear and exponential relationships based on the rate of change.

Since the change is a constant percent decrease (

12\%

) rather than a constant numerical amount, the relationship is exponential rather than linear.

Exponential models represent situations with constant percentage changes, while linear models represent constant absolute changes.

3

Determine the direction of the change (growth or decay).

Because the number of errors is reduced, the quantity is decreasing, representing exponential decay (decreasing exponential).

A reduction in quantity indicates decay (decreasing) rather than growth (increasing).

Anahtar Kavram

Linear and Exponential Growth

Soru 1933Soru →

A solar power station consists of solar panels that generate electricity at a constant rate of $15\text{ watts}$ per square foot of panel surface area (where $1\text{ watt} = 1\text{ joule per second}$ ). The panels have a total surface area of $12,000\text{ square yards}$ . How many megajoules (MJ) of energy do the panels generate in $2\text{ hours}$ ? (Given that $1\text{ yard} = 3\text{ feet}$ and $1\text{ MJ} = 1,000,000\text{ joules}$ )

144

194.4

3,888

11,664

Cevabı ve açıklamayı göster

Cevap: $11,664$

Cevap

The solar panels generate

11,664

megajoules of energy in

2\text{ hours}

.

To find the energy generated, first convert the area from square yards to square feet. Since

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

,

1\text{ square yard} = (3\text{ feet})^2 = 9\text{ square feet}

. Multiplying

12,000\text{ square yards}

by

9

gives

108,000\text{ square feet}

. The power generated is

108,000\text{ square feet} \times 15\text{ watts/square foot} = 1,620,000\text{ watts}

, which is

1,620,000\text{ joules per second}

. In

2\text{ hours}

, which is

2 \times 3,600 = 7,200\text{ seconds}

, the total energy generated is

1,620,000\text{ joules/second} \times 7,200\text{ seconds} = 11,664,000,000\text{ joules}

. Since

1\text{ MJ} = 1,000,000\text{ joules}

, this is equivalent to

11,664\text{ MJ}

.

Adım Adım Çözüm

1

Convert the surface area of the solar panels from square yards to square feet.

Since

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

,

1\text{ square yard} = 3^2 = 9\text{ square feet}

. Therefore, the total area in square feet is

12,000 \times 9 = 108,000\text{ square feet}

.

To apply the generation rate given in watts per square foot, the area must be in square feet.

2

Determine the total rate of energy generation in joules per second.

The total power generated is

108,000\text{ square feet} \times 15\text{ watts per square foot} = 1,620,000\text{ watts}

. Since

1\text{ watt} = 1\text{ joule per second}

, this equals

1,620,000\text{ joules per second}

.

This calculates the total energy produced by the entire solar farm every second.

3

Convert the total time from hours to seconds.

2\text{ hours} \times 60\text{ minutes/hour} \times 60\text{ seconds/minute} = 7,200\text{ seconds}

.

To find the total energy generated, the rate (joules per second) must be multiplied by the duration in seconds.

4

Calculate the total energy generated in joules, then convert it to megajoules (MJ).

Total energy in joules is

1,620,000\text{ joules/second} \times 7,200\text{ seconds} = 11,664,000,000\text{ joules}

. Converting to megajoules:

11,664,000,000\text{ joules} \div 1,000,000\text{ joules/MJ} = 11,664\text{ MJ}

.

This yields the final quantity of energy in the requested units.

Anahtar Kavram

Multi-step unit conversions involving area scaling, compound rates, and time conversions.

Tahmini Süre:2m 30s

Soru 1934Soru →

An express delivery truck travels from Warehouse A to Warehouse B at a constant speed, and then returns from Warehouse B to Warehouse A along the same route at a different constant speed. On the first trip, the ratio of the outbound travel time to the inbound travel time is $3$ to $5$ . On a second trip along the same route, the truck's outbound speed is $50\%$ greater than its outbound speed on the first trip, and its inbound speed is $25\%$ less than its inbound speed on the first trip. If the total round-trip travel time for the second trip is $13$ hours, what is the total round-trip travel time, in hours, for the first trip?

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The total round-trip travel time for the first trip is 12 hours.

The correct answer is

12

. By setting the outbound and inbound travel times of the first trip as

t_1

and

t_2

with

t_1 = 0.6t_2

, we apply the inverse relationships of speed and time to express the second trip's times as

t_1' = \frac{2}{3}t_1

and

t_2' = \frac{4}{3}t_2

. Substituting these into the second trip's total time equation

\frac{2}{3}t_1 + \frac{4}{3}t_2 = 13

yields

t_2 = 7.5

and

t_1 = 4.5

, summing to

12

hours.

Adım Adım Çözüm

1

Set up the ratio of outbound to inbound travel times for the first trip.

t_1 = 0.6t_2

(or

t_1 = \frac{3}{5}t_2

)

The ratio of the outbound travel time (

t_1

) to the inbound travel time (

t_2

) is given as

3

to

5

.

2

Express the new travel times for the second trip (

t_1'

and

t_2'

) in terms of the original travel times.

t_1' = \frac{2}{3}t_1

and

t_2' = \frac{4}{3}t_2

For a constant distance, travel time is inversely proportional to speed. A

50\%

speed increase multiplies time by

\frac{1}{1.5} = \frac{2}{3}

, and a

25\%

speed decrease multiplies time by

\frac{1}{0.75} = \frac{4}{3}

.

3

Create an equation for the total travel time of the second trip.

\frac{2}{3}t_1 + \frac{4}{3}t_2 = 13

The sum of the outbound and inbound travel times on the second trip is given as

13

hours.

4

Substitute the first trip's relationship into the equation and solve for

t_2

.

t_2 = 7.5

Substituting

t_1 = \frac{3}{5}t_2

gives

\frac{2}{3}\left(\frac{3}{5}t_2\right) + \frac{4}{3}t_2 = 13 \implies \frac{26}{15}t_2 = 13 \implies t_2 = 13 \times \frac{15}{26} = 7.5

.

5

Determine

t_1

and calculate the total travel time for the first trip.

t_1 + t_2 = 12

Since

t_2 = 7.5

, then

t_1 = 0.6(7.5) = 4.5

. The total round-trip time is

4.5 + 7.5 = 12

.

Anahtar Kavram

Inverse proportionality of speed and time, and solving linear systems involving ratios.

Soru 1935Soru →

A customer service representative recorded the call durations, in minutes, for $5$ phone calls on a Tuesday morning. The call durations are $1$ , $2$ , $3$ , $5$ , and $9$ minutes. What is the mean duration, in minutes, of these $5$ phone calls?

3

4

5

8

Cevabı ve açıklamayı göster

Cevap: $4$

Cevap

The correct answer is

4

minutes, which represents the mean duration of the phone calls.

The mean duration of the calls is found by dividing the sum of the durations by the total number of calls. The sum of the durations is

1 + 2 + 3 + 5 + 9 = 20

minutes. Dividing this sum by the

5

calls yields

20 \div 5 = 4

minutes.

Adım Adım Çözüm

1

Calculate the sum of all the call durations.

Sum =

1 + 2 + 3 + 5 + 9 = 20

To find the mean, the sum of all values in the dataset must first be calculated.

2

Divide the sum by the total number of call durations.

20 \div 5 = 4

The mean is found by dividing the sum of the values by the number of values in the dataset, which is

5

.

Anahtar Kavram

Calculating the mean (average) of a data set by dividing the sum of all values by the count of values.

Soru 1936Soru →

An environmental study monitored the water volume of two reservoirs, Reservoir A and Reservoir B. In June, the water volume of Reservoir A was $20\%$ greater than the water volume of Reservoir B. By August, the water volume of Reservoir A had decreased by $15\%$ , while the water volume of Reservoir B had increased by $10\%$ . If the water volume of Reservoir A in August was $10.2$ million gallons, what was the water volume of Reservoir B in June, in millions of gallons?

8.5

9.6

10.0

14.4

Cevabı ve açıklamayı göster

Cevap: 10.0

Cevap

The water volume of Reservoir B in June was 10.0 million gallons.

The correct answer is the value that represents Reservoir B's volume in June. Let

B

represent Reservoir B's water volume in June. Since Reservoir A's June volume was

20\%

greater than Reservoir B's, it is represented as

1.20B

. Between June and August, Reservoir A's volume decreased by

15\%

, so its August volume is

85\%

of its June volume, which is

0.85 \times 1.20B = 1.02B

. Setting this equal to the given August volume of

10.2

million gallons yields

1.02B = 10.2

. Solving for

B

gives

B = 10.0

million gallons.

Adım Adım Çözüm

1

Define a variable for the unknown quantity and express Reservoir A's volume in June.

Let

B

represent the water volume of Reservoir B in June. Since Reservoir A's June volume was

20\%

greater than Reservoir B's, the volume of Reservoir A in June is

1.20B

.

This establishes the relationship between the initial volumes of the two reservoirs.

2

Determine Reservoir A's volume in August in terms of the variable

B

.

Since Reservoir A's volume decreased by

15\%

by August, its new volume is

85\%

of its June volume. Thus, the volume in August is

0.85 \times 1.20B = 1.02B

.

This represents the two-stage percentage change for Reservoir A.

3

Solve for the variable

B

using the given August volume.

Set the expression for Reservoir A's August volume equal to

10.2

million gallons:

1.02B = 10.2

. Dividing both sides by

1.02

gives

B = 10.0

million gallons.

This finds the initial volume of Reservoir B.

Anahtar Kavram

Multi-step percentage change and setting up equations to solve for initial values.

Tahmini Süre:1m 30s

Soru 1937Soru →

A manufacturing plant uses a machine lubricant at a constant rate of $0.08$ fluid ounces per second of operation. The plant operates the machinery for $8$ hours per day, $5$ days a week. The lubricant is purchased in drums, where each drum contains $15$ gallons of lubricant. How many drums of lubricant does the plant use in a period of $5$ weeks? (Note: $1\text{ gallon} = 128\text{ fluid ounces}$ )

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

The plant uses 30 drums of lubricant in 5 weeks.

To find the number of drums, calculate the total seconds of operation: 5 weeks * 5 days/week * 8 hours/day * 3,600 seconds/hour = 720,000 seconds. Multiply this by the consumption rate of 0.08 fluid ounces/second to get 57,600 fluid ounces. Convert this to gallons by dividing by 128 fluid ounces/gallon to get 450 gallons. Finally, divide by 15 gallons/drum to find that 30 drums are used.

Adım Adım Çözüm

1

Calculate the total number of operational seconds in the 5-week period.

720{}000\text{ seconds}

Multiply 5 weeks by 5 days per week, 8 hours per day, and 3,600 seconds per hour:

5 \times 5 \times 8 \times 3{}600 = 720{}000

.

2

Calculate the total volume of lubricant used in fluid ounces.

57{}600\text{ fluid ounces}

Multiply the rate of

0.08\text{ fluid ounces per second}

by the total operational seconds:

720{}000 \times 0.08 = 57{}600

.

3

Convert the total fluid ounces to gallons.

450\text{ gallons}

Divide the total fluid ounces by the conversion factor of

128\text{ fluid ounces per gallon}

:

57{}600 / 128 = 450

.

4

Calculate the total number of drums needed.

30\text{ drums}

Divide the total gallons by the capacity of a single drum (

15\text{ gallons}

):

450 / 15 = 30

.

Anahtar Kavram

Multi-step dimensional analysis and unit conversion under compound rates

Soru 1938Soru →

Two environmental cleanup projects, Project A and Project B, begin treating separate, identical bodies of water that each contain $V_0$ gallons of a certain chemical. The volume of the chemical remaining in the water treated by Project A is modeled by the exponential decay function $A(t) = V_0(k)^t$ , where $t$ is the number of days since treatment began and $k$ is a constant. The volume of the chemical remaining in the water treated by Project B is modeled by the linear decay function $B(t) = V_0 - c \cdot t$ , where $c$ is a positive constant. After $1$ day of treatment, the volume of the chemical remaining in both bodies of water is the same. After $2$ days of treatment, the volume of the chemical remaining in the water treated by Project A is exactly $\frac{16}{15}$ times the volume of the chemical remaining in the water treated by Project B. If the treatment for Project B continues at this constant rate, after how many days will the chemical in Project B's water be completely removed?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

Equating both models at

t = 1

gives

\frac{c}{V_0} = 1 - k

. At

t = 2

, Project A's remaining volume is

V_0 k^2

and Project B's is

V_0(2k - 1)

. Setting

k^2 = \frac{16}{15}(2k - 1)

results in the quadratic equation

15k^2 - 32k + 16 = 0

. Factoring this yields

k = 0.8

(discarding

1.33

because the scenario represents decay). The chemical in Project B's water is completely removed when

B(t) = 0 \implies t = \frac{V_0}{c} = \frac{1}{1-k} = \frac{1}{0.2} = 5

days.

Adım Adım Çözüm

1

Relate parameters

k

and

c

using the condition

A(1) = B(1)

.

\frac{c}{V_0} = 1 - k

Establishing a link between the rates of linear decay and exponential decay.

2

Express the remaining volumes at

t = 2

in terms of

k

.

A(2) = V_0 k^2

and

B(2) = V_0(2k - 1)

Preparing equations to apply the ratio condition at

t = 2

using a single variable.

3

Apply the condition

A(2) = \frac{16}{15}B(2)

to solve for

k

.

k = 0.8

Setting up the quadratic equation

15k^2 - 32k + 16 = 0

, factoring it to find roots

0.8

and

1.33

, and selecting

0.8

because

k < 1

for a decay process.

4

Calculate the time

t

when

B(t) = 0

.

t = 5

Solving

V_0 - c \cdot t = 0

yields

t = \frac{V_0}{c} = \frac{1}{1-k} = \frac{1}{0.2} = 5

days.

Anahtar Kavram

Linear and exponential decay modeling

Soru 1939Soru →

A bicycle factory produces road bikes and mountain bikes at a constant ratio of $2$ to $3$ . If the factory produces $120$ road bikes in one day, how many mountain bikes does the factory produce in the same day?

80

180

300

360

Cevabı ve açıklamayı göster

Cevap: 180

Cevap

180

The ratio of road bikes to mountain bikes is

2

to

3

. Since the factory produces

120

road bikes, we can set up the proportion

\frac{2}{3} = \frac{120}{x}

, where

x

represents the number of mountain bikes. Solving for

x

gives

2x = 360

, which simplifies to

x = 180

.

Adım Adım Çözüm

1

Identify the given ratio and set up a proportion comparing road bikes to mountain bikes.

The ratio of road bikes to mountain bikes is

\frac{2}{3}

, so the proportion is

\frac{\text{Road Bikes}}{\text{Mountain Bikes}} = \frac{2}{3}

.

To find the unknown number of mountain bikes using the constant ratio.

2

Substitute

120

for the number of road bikes and solve for the number of mountain bikes,

x

.

\frac{120}{x} = \frac{2}{3} \implies 2x = 360 \implies x = 180

.

To calculate the exact number of mountain bikes produced.

Anahtar Kavram

Using ratios and proportions to solve word problems

Soru 1940Soru →

What is the complete set of solutions for the inequality $7 - 3(2x - 5) \leq -4x + 8$ ?

x \leq 7

x \geq -8

x \geq 7

x \geq -7

Cevabı ve açıklamayı göster

Cevap: $x \geq 7$

Cevap

The complete set of solutions is represented by the inequality

x \geq 7

.

To solve the inequality

7 - 3(2x - 5) \leq -4x + 8

, we first distribute

-3

to get

7 - 6x + 15 \leq -4x + 8

. Combining constant terms on the left side gives

22 - 6x \leq -4x + 8

. Adding

4x

and subtracting

22

from both sides isolates the variable, resulting in

-2x \leq -14

. Dividing both sides by

-2

and flipping the inequality symbol yields the solution

x \geq 7

.

Adım Adım Çözüm

1

Distribute the

-3

to both terms inside the parentheses.

7 - 6x + 15 \leq -4x + 8

Applying the distributive property,

-3(2x) = -6x

and

-3(-5) = 15

.

2

Combine the constant terms on the left side of the inequality.

22 - 6x \leq -4x + 8

Adding

7

and

15

yields

22

.

3

Isolate the variable term on one side and the constant term on the other side.

-2x \leq -14

Adding

4x

to both sides yields

-2x

, and subtracting

22

from both sides yields

-14

.

4

Divide both sides by

-2

and flip the inequality sign.

x \geq 7

Dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol.

Anahtar Kavram

Solving multi-step linear inequalities in one variable including negative coefficient division and the distributive property.

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