Tüm alıştırma soruları

Soru 421Soru →

A hybrid car's fuel tank has a capacity of $12$ gallons. When driving on a highway, the amount of fuel in the tank decreases at a constant rate. After driving for $1.5$ hours, $9.6$ gallons of fuel remain in the tank. If the amount of fuel in the tank, $F$ , in gallons, after driving for $t$ hours is modeled by a linear equation, what is the value of $F$ when $t = 4$ ?

Cevabı ve açıklamayı göster

Cevap: 5.6

Cevap

The amount of fuel remaining in the tank after driving for

4

hours is

5.6

gallons.

The amount of fuel in the tank,

F

, and the driving time,

t

, share a linear relationship. The initial amount of fuel at

t = 0

is

12

gallons, representing the vertical intercept. The fuel decreases at a constant rate, which is the slope of the line. Over a period of

1.5

hours, the amount of fuel decreases by

12 - 9.6 = 2.4

gallons. The rate of decrease is

\frac{2.4}{1.5} = 1.6

gallons per hour, so the slope is

-1.6

. The linear model is

F = -1.6t + 12

. Substituting

t = 4

into this equation yields

F = -1.6(4) + 12 = -6.4 + 12 = 5.6

gallons.

Adım Adım Çözüm

1

Calculate the constant rate of fuel consumption (the slope of the linear relationship).

The rate of consumption is

1.6

gallons per hour.

Since the fuel decreases at a constant rate, the change in fuel divided by the change in time gives the rate of consumption. In

1.5

hours, the fuel decreases from

12

gallons to

9.6

gallons, which is a decrease of

12 - 9.6 = 2.4

gallons. Thus, the rate of consumption is

\frac{2.4\text{ gallons}}{1.5\text{ hours}} = 1.6

gallons per hour.

2

Write the linear equation modeling the fuel remaining in the tank,

F

, as a function of time,

t

.

F = -1.6t + 12

The initial amount of fuel when

t = 0

is

12

gallons, which represents the vertical intercept (

b = 12

). The fuel decreases at a constant rate of

1.6

gallons per hour, which represents a slope of

m = -1.6

.

3

Substitute

t = 4

into the linear equation to find the value of

F

.

F = 5.6

Evaluating the equation at

t = 4

yields

F = -1.6(4) + 12 = -6.4 + 12 = 5.6

.

Anahtar Kavram

Linear Equations in Two Variables

Soru 422Soru →

The table below shows some values for a cubic polynomial function $g$ .

$x$	$g(x)$
$-1$	$0$
$1$	$16$
$3$	$0$

In the $xy$ -plane, the graph of $y = g(x)$ is tangent to the $x$ -axis at $x = 3$ . What is the value of $g(0)$ ?

Cevabı ve açıklamayı göster

Cevap: 18

Cevap

18

The correct value of

g(0)

is 18. By using the fact that

g(-1) = 0

, we establish

(x + 1)

as a factor. The tangency at

x = 3

tells us that

(x - 3)^2

is a factor. Writing the function as

g(x) = a(x + 1)(x - 3)^2

and substituting

g(1) = 16

gives

8a = 16 \implies a = 2

. Evaluating

g(0)

yields

2(1)(9) = 18

.

Adım Adım Çözüm

1

Determine the factors of the cubic polynomial

g(x)

using the given zeros and the tangency condition.

The factors are

(x + 1)

and

(x - 3)^2

, so the function is of the form

g(x) = a(x + 1)(x - 3)^2

.

Since

g(-1) = 0

,

x = -1

is a root of the polynomial. The graph being tangent to the

x

-axis at

x = 3

indicates that

x = 3

is a root with a multiplicity of at least 2. Since

g(x)

is a cubic polynomial (degree 3), the multiplicity of the root at

x = 3

must be exactly 2.

2

Use the table value

g(1) = 16

to solve for the constant coefficient

a

.

a = 2

Substituting

x = 1

into

g(x) = a(x + 1)(x - 3)^2

gives

g(1) = a(1 + 1)(1 - 3)^2 = 8a

. Setting this equal to the table value of 16 yields

8a = 16

, which simplifies to

a = 2

.

3

Evaluate the polynomial at

x = 0

using the fully determined function

g(x) = 2(x + 1)(x - 3)^2

.

18

To find

g(0)

, substitute

x = 0

into the expression:

g(0) = 2(0 + 1)(0 - 3)^2 = 2(1)(9) = 18

.

Anahtar Kavram

Identifying polynomial factors from graphs and tables, and analyzing root multiplicity (tangency vs. crossing).

Soru 423Soru →

The quadratic equation $x^2 - 12x + 4 = 0$ has solutions $x_1$ and $x_2$ . What is the value of $\frac{1}{x_1} + \frac{1}{x_2}$ ?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The correct answer is 3.

By writing the expression

\frac{1}{x_1} + \frac{1}{x_2}

with a common denominator, we get

\frac{x_1 + x_2}{x_1 x_2}

. For the quadratic equation

x^2 - 12x + 4 = 0

, Vieta's formulas give the sum of the roots

x_1 + x_2 = 12

and the product of the roots

x_1 x_2 = 4

. Substituting these values into the fraction yields

\frac{12}{4} = 3

.

Adım Adım Çözüm

1

Find a common denominator to combine the terms in the given expression.

\frac{1}{x_1} + \frac{1}{x_2} = \frac{x_1 + x_2}{x_1 x_2}

To express the target quantity in terms of the sum and product of the quadratic solutions.

2

Apply Vieta's formulas to find the sum and product of the solutions from the quadratic equation

x^2 - 12x + 4 = 0

.

x_1 + x_2 = 12

and

x_1 x_2 = 4

For any quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-\frac{b}{a}

and the product is

\frac{c}{a}

.

3

Substitute the sum and product values into the combined fraction.

\frac{12}{4} = 3

To compute the numerical value of the expression.

Anahtar Kavram

Sum and product of solutions of a quadratic equation

Soru 424Soru →

The graph of the quadratic function $f$ in the $xy$ -plane has $x$ -intercepts at $(-2, 0)$ and $(8, 0)$ . If the maximum value of $f(x)$ is $25$ , what is the value of $f(0)$ ?

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

16

The axis of symmetry of the quadratic function

f

lies halfway between the

x

-intercepts

x = -2

and

x = 8

, which is at

x = \frac{-2 + 8}{2} = 3

. Since the function has a maximum value of

25

, this maximum must occur at the vertex, giving the vertex coordinates

(3, 25)

. In vertex form, the function is

f(x) = a(x - 3)^2 + 25

. Substituting the

x

-intercept

(8, 0)

into the function yields

0 = a(8 - 3)^2 + 25

, which simplifies to

25a = -25

, or

a = -1

. Therefore, the equation of the function is

f(x) = -(x - 3)^2 + 25

. Evaluating this at

x = 0

gives

f(0) = -(0 - 3)^2 + 25 = -9 + 25 = 16

.

Adım Adım Çözüm

1

Find the

x

-coordinate of the vertex (axis of symmetry)

x = 3

The axis of symmetry of a parabola is located exactly halfway between its

x

-intercepts:

x = \frac{-2 + 8}{2} = 3

.

2

Determine the vertex coordinates

(3, 25)

The maximum value of the quadratic function occurs at its vertex, so the

y

-coordinate of the vertex is the maximum value

25

.

3

Write the vertex form of the quadratic function

f(x) = a(x - 3)^2 + 25

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex.

4

Solve for the leading coefficient

a

a = -1

Substitute the

x

-intercept

(8, 0)

into the vertex form:

0 = a(8 - 3)^2 + 25 \implies 25a = -25 \implies a = -1

.

5

Find the value of

f(0)

f(0) = 16

Substitute

x = 0

into the function:

f(0) = -(0 - 3)^2 + 25 = -9 + 25 = 16

.

Anahtar Kavram

Using

x

-intercepts and the maximum value to determine the vertex and equation of a quadratic function.

Soru 425Soru →

If $x$ satisfies the equation below, what is the value of $x + 5$ ?

x - 4 = \sqrt{4x + 5}

Cevabı ve açıklamayı göster

Cevap: 16

Cevap

The correct answer is 16.

Squaring both sides of the equation

x - 4 = \sqrt{4x + 5}

results in

(x - 4)^2 = 4x + 5

, which expands to

x^2 - 8x + 16 = 4x + 5

. Subtracting

4x + 5

from both sides gives the quadratic equation

x^2 - 12x + 11 = 0

. Factoring this quadratic equation yields

(x - 11)(x - 1) = 0

, giving candidate solutions of

x = 11

and

x = 1

. Checking these solutions in the original equation shows that

x = 11

is valid (

7 = 7

), whereas

x = 1

is extraneous (

-3 = 3

is false). Therefore, the only real solution is

x = 11

, and the value of the expression

x + 5

is

11 + 5 = 16

.

Adım Adım Çözüm

1

Square both sides of the equation to remove the radical.

(x - 4)^2 = 4x + 5

Squaring a square root isolates the expression under the radical.

2

Expand the squared binomial on the left side.

x^2 - 8x + 16 = 4x + 5

Applying the distributive property to

(x - 4)(x - 4)

yields

x^2 - 8x + 16

.

3

Rearrange the equation to set it equal to zero.

x^2 - 12x + 11 = 0

Subtracting

4x + 5

from both sides simplifies the equation into standard quadratic form.

4

Factor the quadratic equation.

(x - 11)(x - 1) = 0

Finding two numbers that multiply to 11 and add to -12 gives -11 and -1.

5

Solve for the candidate values of x.

x = 11

and

x = 1

Setting each factor equal to zero yields the possible solutions.

6

Check the candidate values in the original equation to identify any extraneous solutions.

x = 11

is valid;

x = 1

is extraneous.

Substituting

x = 1

results in

-3 = 3

, which is false because the principal square root is always non-negative.

7

Evaluate the target expression using the valid solution.

11 + 5 = 16

The question asks for the value of

x + 5

, so we substitute the only valid solution,

x = 11

.

Anahtar Kavram

Solving radical equations by squaring both sides and checking for extraneous solutions.

Soru 426Soru →

A botanist conducted an experiment to study the germination rates of seeds in two different soil mixtures, Mixture A and Mixture B. The results of the experiment are partially shown in the table below, where $x$ is a positive constant.

Soil Mixture	Germinated	Did Not Germinate
Mixture A	$x$	$12$
Mixture B	$30$	$x + 6$

Of the seeds in the experiment that did not germinate, the probability that the seed was planted in Mixture A is $\frac{1}{3}$ . Given that a seed selected at random from the experiment germinated, what is the probability that it was planted in Mixture A?

Cevabı ve açıklamayı göster

Cevap: 0.375

Cevap

3/8 (or 0.375)

To find the probability that a germinated seed was planted in Mixture A, we must first solve for the variable

x

. The number of seeds in Mixture A that did not germinate is

12

, and the total number of seeds that did not germinate is

12 + (x + 6) = x + 18

. We are given that the probability a non-germinated seed was planted in Mixture A is

\frac{1}{3}

. Setting up the equation:

\frac{12}{x + 18} = \frac{1}{3}

yields

x + 18 = 36

, so

x = 18

. Substituting this value back into the table, the number of germinated seeds in Mixture A is

18

. The total number of germinated seeds is the sum of germinated seeds in Mixture A and Mixture B:

18 + 30 = 48

. Thus, the probability that a germinated seed was planted in Mixture A is

\frac{18}{48}

, which simplifies to

\frac{3}{8}

(or

0.375

).

Adım Adım Çözüm

1

Set up the conditional probability equation for seeds that did not germinate to find

x

.

x = 18

From the table, the number of seeds in Mixture A that did not germinate is

12

, and the number of seeds in Mixture B that did not germinate is

x + 6

. The total number of seeds that did not germinate is

12 + (x + 6) = x + 18

. The probability that a seed that did not germinate was from Mixture A is

\frac{12}{x + 18} = \frac{1}{3}

. Solving this equation gives

36 = x + 18

, which simplifies to

x = 18

.

2

Find the number of germinated seeds in Mixture A and the total number of germinated seeds.

Mixture A germinated =

18

, Total germinated =

48

Substituting

x = 18

, the number of germinated seeds in Mixture A is

18

. The number of germinated seeds in Mixture B is given as

30

. The total number of germinated seeds is

18 + 30 = 48

.

3

Calculate the conditional probability that a germinated seed was planted in Mixture A.

\frac{3}{8}

or

0.375

The probability is the number of germinated seeds in Mixture A divided by the total number of germinated seeds, which is

\frac{18}{48} = \frac{3}{8}

, or

0.375

as a decimal.

Anahtar Kavram

Conditional Probability in Two-Way Tables

Soru 427Soru →

A right circular cylinder has a height of $12$ and a base radius of $r$ . A sphere has a radius of $r$ . If the volume of the cylinder is equal to the volume of the sphere, what is the value of $r$ ?

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

9

The volume of a cylinder is given by

V = \pi r^2 h

and the volume of a sphere is given by

V = \frac{4}{3}\pi r^3

. Given that the cylinder's height is

12

, its volume is

12\pi r^2

. Setting the volumes equal yields

12\pi r^2 = \frac{4}{3}\pi r^3

. Since

r

is a non-zero radius, we can divide both sides by

\pi r^2

, resulting in

12 = \frac{4}{3}r

. Multiplying both sides by

\frac{3}{4}

gives

r = 9

.

Adım Adım Çözüm

1

State the standard volume formulas for a right circular cylinder and a sphere.

V_{\text{cylinder}} = \pi r^2 h

and

V_{\text{sphere}} = \frac{4}{3}\pi r^3

These formulas are needed to express the volumes of both solids in terms of

r

.

2

Equate the volume of the cylinder to the volume of the sphere and substitute the given height of

12

.

12\pi r^2 = \frac{4}{3}\pi r^3

The problem states that the volume of the cylinder is equal to the volume of the sphere.

3

Divide both sides of the equation by

\pi r^2

.

12 = \frac{4}{3}r

Simplifies the equation to a first-degree equation in terms of

r

.

4

Solve for

r

by multiplying both sides of the simplified equation by the reciprocal of

\frac{4}{3}

, which is

\frac{3}{4}

.

r = 9

Isolates the variable

r

to find the correct value.

Anahtar Kavram

Equating the volumes of geometric solids to solve for an unknown dimension.

Tahmini Süre:1m 30s

Soru 428Soru →

The graph of the function $f$ in the $xy$ -plane has a vertex at $(-2, 7)$ . The function $g$ is defined by $g(x) = f(-x + 3) + 12$ . If the vertex of the graph of $y = g(x)$ is the point $(a, b)$ , what is the value of $a + b$ ?

Cevabı ve açıklamayı göster

Cevap: 24

Cevap

24

The vertex of the parent function

f(x)

is at

(-2, 7)

, which means

f(-2) = 7

. The transformed function is

g(x) = f(-x + 3) + 12

. The vertex of

g(x)

occurs when the input to

f

, which is

-x + 3

, is equal to

-2

. Solving

-x + 3 = -2

gives

-x = -5

, or

x = 5

, so the x-coordinate of the vertex of

g(x)

is

a = 5

. To find the y-coordinate

b

, we evaluate

g(5) = f(-(5) + 3) + 12 = f(-2) + 12 = 7 + 12 = 19

. Therefore, the vertex of the graph of

y = g(x)

is

(5, 19)

, so

a = 5

and

b = 19

. The value of

a + b

is

5 + 19 = 24

.

Adım Adım Çözüm

1

Find the x-coordinate of the vertex of the transformed function

g(x)

by setting the inner expression equal to the x-coordinate of the original vertex.

a = 5

The vertex of

f(x)

is located at

x = -2

. For

g(x) = f(-x + 3) + 12

, the vertex occurs when the input to

f

,

-x + 3

, is equal to

-2

. Solving

-x + 3 = -2

yields

x = 5

.

2

Find the y-coordinate of the vertex of

g(x)

by evaluating

g(5)

.

b = 19

Substituting

x = 5

into the definition of

g(x)

gives

g(5) = f(-2) + 12

. Since the vertex of

f

is at

(-2, 7)

,

f(-2) = 7

. Thus,

g(5) = 7 + 12 = 19

.

3

Calculate the sum of the coordinates

a

and

b

.

24

The vertex of

g(x)

is

(5, 19)

, so

a = 5

and

b = 19

. The sum

a + b

is

5 + 19 = 24

.

Anahtar Kavram

Determining the coordinates of a transformed vertex using function notation.

Soru 429Soru →

A shipping container initially holds 450 packages. A crew unloads the container at a rate of 15 packages per hour, while an automated sorting machine unloads packages at a rate of $x$ packages per hour. The number of packages remaining in the container after 8 hours must be at most 130. The inequality $450 - 8(15 + x) \leq 130$ models this scenario. What is the minimum possible value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 25

Cevap

The minimum possible value of

x

is 25.

To find the minimum possible value of

x

, solve the inequality

450 - 8(15 + x) \leq 130

. First, distribute the

-8

to obtain

450 - 120 - 8x \leq 130

. Simplify the constant terms on the left side to get

330 - 8x \leq 130

. Subtract 330 from both sides, yielding

-8x \leq -200

. Finally, divide both sides by

-8

and reverse the inequality sign because of the division by a negative number, which results in

x \geq 25

. The minimum possible value of

x

is 25.

Adım Adım Çözüm

1

Distribute the factor of

-8

to both terms inside the parentheses.

450 - 120 - 8x \leq 130

To remove the parentheses and prepare to combine like terms.

2

Subtract 120 from 450 to simplify the constants on the left side.

330 - 8x \leq 130

To simplify the left side of the inequality before isolating the variable.

3

Subtract 330 from both sides of the inequality.

-8x \leq -200

To isolate the variable term on the left-hand side.

4

Divide both sides by

-8

and reverse the direction of the inequality symbol.

x \geq 25

Dividing by a negative value reverses the inequality sign. Since

x

must be greater than or equal to 25, the minimum possible value is 25.

Anahtar Kavram

Solving multi-step linear inequalities in one variable, including distributing coefficients and reversing the inequality sign when multiplying or dividing by a negative number.

Soru 430Soru →

In the $xy$ -plane, the graph of the quadratic function $f(x) = -x^2 + bx + c$ , where $b$ and $c$ are constants, has its vertex at $(4, 25)$ . If the positive $x$ -intercept of the graph of $f$ is $(d, 0)$ , what is the value of $d$ ?

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The value of

d

is

9

.

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex. Since the vertex is

(4, 25)

and the coefficient of

x^2

is

-1

, the function is

f(x) = -(x-4)^2 + 25

. Setting

f(x) = 0

to find the

x

-intercepts yields

-(x-4)^2 + 25 = 0

, which simplifies to

(x-4)^2 = 25

. Taking the square root of both sides gives

x - 4 = 5

or

x - 4 = -5

. Solving these equations gives

x = 9

or

x = -1

. The positive

x

-intercept is

(9, 0)

, so the value of

d

is

9

.

Adım Adım Çözüm

1

Write the function in vertex form.

f(x) = -(x-4)^2 + 25

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex. Here, the vertex is

(4, 25)

and the coefficient of

x^2

is

-1

, so

a = -1

,

h = 4

, and

k = 25

.

2

Set

f(x) = 0

to find the

x

-intercepts.

-(x-4)^2 + 25 = 0

The

x

-intercepts of a graph are the points where the function value is equal to 0.

3

Solve the equation for

x

.

x = 9

or

x = -1

Rearranging the equation gives

(x-4)^2 = 25

. Taking the square root of both sides gives

x-4 = 5

or

x-4 = -5

, which solves to

x = 9

or

x = -1

.

4

Identify the positive

x

-intercept coordinate

d

.

d = 9

The question asks for the positive

x

-intercept

(d, 0)

, which corresponds to

x = 9

.

Anahtar Kavram

Vertex form of a quadratic function and finding

x

-intercepts

Soru 431Soru →

A dataset consists of 10 positive integers: $4, 6, 8, 8, 10, 12, 12, 14, 16$ , and $x$ . If the median of the dataset is equal to the mean of the dataset, and $x > 16$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

20

The correct answer is 20. When the 10 positive integers are sorted in ascending order, the condition

x > 16

ensures that

x

is the largest value and occupies the final position:

4, 6, 8, 8, 10, 12, 12, 14, 16, x

. The median of this 10-value dataset is the average of the 5th and 6th values, which are 10 and 12. Thus, the median is

\frac{10 + 12}{2} = 11

. The mean of the dataset is the sum of the 10 values divided by 10. The sum of the 9 known values is

4 + 6 + 8 + 8 + 10 + 12 + 12 + 14 + 16 = 90

, so the mean is

\frac{90 + x}{10}

. Setting the mean equal to the median gives the equation

\frac{90 + x}{10} = 11

. Multiplying both sides by 10 gives

90 + x = 110

, and subtracting 90 yields

x = 20

. This value is consistent with the condition

x > 16

.

Adım Adım Çözüm

1

Sort the dataset including the variable

x

.

The sorted dataset of 10 integers is

4, 6, 8, 8, 10, 12, 12, 14, 16, x

.

Since

x > 16

, it must be the largest value in the dataset and will be positioned at the end of the sorted list.

2

Find the median of the sorted dataset.

The median is 11.

For an even number of values (10), the median is the average of the two middle values (the 5th and 6th values), which are 10 and 12:

(10 + 12) / 2 = 11

.

3

Find the sum of the known values and write an expression for the mean.

The mean is

\frac{90 + x}{10}

.

The sum of the 9 known integers is

4 + 6 + 8 + 8 + 10 + 12 + 12 + 14 + 16 = 90

. Adding

x

gives a total sum of

90 + x

, and dividing by the total count of 10 gives the mean.

4

Equate the mean to the median and solve for

x

.

x = 20

Setting the mean expression equal to the median yields

\frac{90 + x}{10} = 11

. Multiplying by 10 gives

90 + x = 110

, which solves to

x = 20

.

Anahtar Kavram

Calculating and equating the mean and median of a dataset containing a variable.

Soru 432Soru →

A solid sphere has a volume of $36\pi$ cubic centimeters. What is the radius, in centimeters, of the sphere?

Cevabı ve açıklamayı göster

Cevap: 3

Cevap

The radius of the sphere is 3 centimeters.

The volume of a sphere is given by the formula

V = \frac{4}{3}\pi r^3

. Substituting

36\pi

for

V

gives

36\pi = \frac{4}{3}\pi r^3

. Dividing both sides by

\pi

results in

36 = \frac{4}{3}r^3

. Multiplying both sides by

\frac{3}{4}

yields

27 = r^3

. Taking the cube root of both sides gives

r = 3

. Thus, the radius of the sphere is 3 centimeters.

Adım Adım Çözüm

1

State the sphere volume formula.

V = \frac{4}{3}\pi r^3

The volume of a sphere is calculated using this standard formula relating volume to radius.

2

Substitute the given volume value into the formula.

36\pi = \frac{4}{3}\pi r^3

The problem specifies the volume is

36\pi

cubic centimeters.

3

Solve for the radius cubed (

r^3

).

r^3 = 27

Dividing both sides of the equation by

\pi

yields

36 = \frac{4}{3}r^3

, and multiplying both sides by

\frac{3}{4}

isolates

r^3

to give 27.

4

Take the cube root of both sides to find the radius (

r

).

r = 3

The cube root of 27 is 3, since

3 \times 3 \times 3 = 27

.

Anahtar Kavram

Using the volume of a sphere formula to solve for an unknown radius.

Soru 433Soru →

At the beginning of 2015, City A and City B each had a daily water consumption of $1,200,000$ gallons. The daily water consumption of City A decreased linearly by $24,000$ gallons each year, and the daily water consumption of City B decreased exponentially by a constant percentage each year. At the beginning of 2025, the daily water consumption of City A was equal to the daily water consumption of City B. If these trends continue, what will be the daily water consumption of City B, in gallons, at the beginning of 2035?

Cevabı ve açıklamayı göster

Cevap: 768000

Cevap

768,000

To find the daily water consumption of City B at the beginning of 2035, we first determine the consumption of both cities at the beginning of 2025 (

t = 10

years after the beginning of 2015). For City A, which decreases linearly, the consumption is

1,200,000 - 10 \times 24,000 = 960,000

gallons. Since City B's consumption is equal to City A's at this time and decreases exponentially, we set up the equation

1,200,000 \cdot b^{10} = 960,000

, where

b

is the annual decay factor. This gives

b^{10} = 0.8

. The consumption of City B at the beginning of 2035 (

t = 20

) is given by

1,200,000 \cdot b^{20}

. Using exponent rules, we can rewrite

b^{20}

as

(b^{10})^2

. Substituting

b^{10} = 0.8

gives

1,200,000 \cdot (0.8)^2 = 1,200,000 \cdot 0.64 = 768,000

.

Adım Adım Çözüm

1

Calculate the consumption of City A at the beginning of 2025.

960,000

gallons

City A decreases linearly by a constant

24,000

gallons per year for

10

years starting from

1,200,000

gallons.

2

Set up the exponential model for City B at the beginning of 2025.

b^{10} = 0.8

City B's consumption decreases exponentially, so it is modeled by

W_B(t) = 1,200,000 \cdot b^t

. Since it equals City A's consumption at

t = 10

,

1,200,000 \cdot b^{10} = 960,000

.

3

Determine the consumption of City B at the beginning of 2035.

768,000

gallons

At the beginning of 2035 (

t = 20

), City B's consumption is

1,200,000 \cdot b^{20}

. Since

b^{20} = (b^{10})^2

, we substitute

0.8

for

b^{10}

to get

1,200,000 \cdot (0.8)^2 = 1,200,000 \cdot 0.64 = 768,000

.

Anahtar Kavram

Distinguishing between linear and exponential models and applying exponent properties to solve exponential growth/decay problems.

Alternatif Yöntem

Alternatively, you can solve for the annual decay factor

b

directly using a calculator. Since

b^{10} = 0.8

, we take the 10th root of both sides to get

b = 0.8^{0.1} \approx 0.97793

. The water consumption at

t = 20

is then calculated as

1,200,000 \cdot (0.97793)^{20} \approx 768,000

gallons. Recognizing the algebraic shortcut

(b^{10})^2 = b^{20}

avoids decimal approximations and is much faster.

Tahmini Süre:2m 30s

Soru 434Soru →

The population of a species of fish in a lake can be modeled by the function $P(t) = P_0 \cdot 2^{\frac{t}{d}}$ , where $P_0$ is the initial population when the population was first measured, $t$ represents the time in years since it was first measured, and $d$ is a constant representing the doubling time in years. If the population of the fish doubles every 6 years, and the population after 18 years is 3,200, what was the initial population of the fish?

Cevabı ve açıklamayı göster

Cevap: 400

Cevap

The initial population of the fish was 400.

By substituting the doubling time

d = 6

and the final population of 3,200 at

t = 18

into the exponential model

P(t) = P_0 \cdot 2^{\frac{t}{d}}

, we get

3,200 = P_0 \cdot 2^{\frac{18}{6}}

. Simplifying the exponent gives

3,200 = P_0 \cdot 2^3

, which simplifies further to

3,200 = 8P_0

. Dividing both sides by 8 yields

P_0 = 400

.

Adım Adım Çözüm

1

Identify the values for the known variables from the word problem.

d = 6

years and at

t = 18

years,

P(18) = 3,200

.

To substitute these values into the exponential growth function model.

2

Substitute the known values into the exponential function

P(t) = P_0 \cdot 2^{\frac{t}{d}}

.

3,200 = P_0 \cdot 2^{\frac{18}{6}}

To set up an equation to solve for the unknown parameter

P_0

.

3

Simplify the exponent and calculate the growth factor.

3,200 = P_0 \cdot 2^3 \Rightarrow 3,200 = 8P_0

Reducing the fractional exponent simplifies the equation.

4

Solve for the initial population

P_0

by dividing both sides of the equation by 8.

P_0 = 400

Isolating

P_0

gives the initial population of the fish.

Anahtar Kavram

Using an exponential function to model real-world growth and solving for the initial value.

Soru 435Soru →

A technology company surveyed a random sample of $128$ of its employees to determine their primary operating system and their job role. The table below summarizes the results of the survey.

Job Role	macOS	Windows	Total
Developer	$30$	$50$	$80$
Designer	$30$	$18$	$48$
Total	$60$	$68$	$128$

If an employee from the survey whose job role is developer is selected at random, what is the probability that the employee's primary operating system is Windows?

Cevabı ve açıklamayı göster

Cevap: 0.625

Cevap

5/8 (or 0.625)

The question asks for the probability that a randomly selected employee uses Windows, given that their job role is developer. This restricts the sample space to only the developers. From the table, there are

80

total developers, which serves as the denominator. Within this group of developers,

50

use Windows, which serves as the numerator. The probability is

\frac{50}{80}

, which simplifies to

\frac{5}{8}

or

0.625

. Both

5/8

and

.625

are correct formats that fit the standard SAT grid-in requirements.

Adım Adım Çözüm

1

Identify the subgroup defined by the condition.

The total number of developers is

80

.

The probability is conditional on the employee being a developer, which restricts the denominator of the probability to the total number of developers.

2

Identify the favorable outcomes within the subgroup.

There are

50

developers whose primary operating system is Windows.

This value represents the numerator for our conditional probability calculation.

3

Calculate the conditional probability.

The probability is

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

(or

0.625

).

Dividing the favorable outcomes by the total outcomes in the restricted sample space gives the correct conditional probability.

Anahtar Kavram

Conditional Probability in Two-Way Tables

Soru 436Soru →

An equation is shown below.

\frac{x+2}{x-1} - \frac{4}{x} = \frac{4}{x^2 - x}

What is the real solution to the equation above?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The only real solution to the equation is

2

.

To solve the rational equation, we multiply all terms by the common denominator

x(x-1)

, assuming

x \neq 0

and

x \neq 1

. This results in the equation

x(x+2) - 4(x-1) = 4

. Expanding and simplifying this gives

x^2 + 2x - 4x + 4 = 4

, which simplifies to

x^2 - 2x = 0

. Factoring the quadratic expression yields

x(x-2) = 0

, which gives the potential solutions

x = 0

and

x = 2

. However, substituting

x = 0

back into the original equation causes a division by zero. Therefore,

x = 0

is an extraneous solution, and the only valid real solution is

2

.

Adım Adım Çözüm

1

Find the common denominator for the terms in the rational equation.

The common denominator is

x(x-1) = x^2 - x

, which requires

x \neq 0

and

x \neq 1

.

Multiplying by the common denominator allows us to eliminate the fractions.

2

Multiply every term in the equation by the common denominator

x(x-1)

and simplify.

x(x+2) - 4(x-1) = 4

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

3

Expand and simplify the resulting equation to standard quadratic form.

x^2 + 2x - 4x + 4 = 4

, which simplifies to

x^2 - 2x = 0

.

Grouping like terms is necessary to solve the quadratic equation.

4

Factor the quadratic equation to determine the potential solutions.

x(x-2) = 0

, giving potential solutions of

x = 0

and

x = 2

.

Factoring allows us to find the roots of the quadratic expression.

5

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

x = 0

is extraneous because it leads to division by zero. Thus, the only real solution is

x = 2

.

Solutions that make any denominator in the original equation equal to zero must be excluded.

Anahtar Kavram

Solving rational equations and identifying extraneous solutions

Soru 437Soru →

An electric delivery van's remaining battery capacity, $B$ , in kilowatt-hours (kWh), after driving $d$ miles is modeled by a linear equation. After driving $45$ miles, the remaining battery capacity is $68$ kWh. After driving a total of $120$ miles, the remaining battery capacity is $38$ kWh. According to this model, how many miles can the van drive on a full charge before the battery capacity reaches $0$ kWh?

Cevabı ve açıklamayı göster

Cevap: 215

Cevap

215

To find the maximum distance the van can drive on a full charge, we model the relationship between the remaining battery capacity,

B

, and the distance driven,

d

, using the linear equation

B = md + B_0

. First, calculate the rate of consumption (slope,

m

) using the points

(45, 68)

and

(120, 38)

:

m = \frac{38 - 68}{120 - 45} = \frac{-30}{75} = -0.4

kWh per mile. Next, find the initial battery capacity (

B_0

) by substituting the values from one of the points:

68 = -0.4(45) + B_0

, which simplifies to

68 = -18 + B_0

, so

B_0 = 86

kWh. Finally, determine the distance

d

when the battery is fully depleted (

B = 0

):

0 = -0.4d + 86

, which yields

0.4d = 86

, and

d = 215

miles.

Adım Adım Çözüm

1

Calculate the rate of change of the battery capacity per mile driven.

The rate of change (slope) is

-0.4

kWh per mile.

The slope

m

represents the energy consumption rate and is calculated using the two data points

(45, 68)

and

(120, 38)

with the formula

m = \frac{38 - 68}{120 - 45} = \frac{-30}{75} = -0.4

.

2

Determine the initial battery capacity when the distance driven is 0 miles.

The initial battery capacity is

86

kWh.

Using the slope-intercept form

B = md + B_0

, substitute

m = -0.4

and the point

(45, 68)

to get

68 = -0.4(45) + B_0

. This simplifies to

68 = -18 + B_0

, so

B_0 = 86

.

3

Solve for the distance driven

d

when the battery capacity

B

is

0

kWh.

The distance is

215

miles.

Set

B = 0

in the linear model

B = -0.4d + 86

to get

0 = -0.4d + 86

. Solving for

d

gives

0.4d = 86

, which results in

d = \frac{86}{0.4} = 215

.

Anahtar Kavram

Interpreting and calculating slope, y-intercept, and x-intercept of a linear function in a real-world context.

Soru 438Soru →

For a third-degree polynomial $p(x)$ , the expression $x^2 - 4x + 4$ is a factor. When $p(x)$ is divided by $x - 3$ , the remainder is $5$ , and when $p(x)$ is divided by $x + 1$ , the remainder is $-27$ . What is the remainder when $p(x)$ is divided by $x - 1$ ?

Cevabı ve açıklamayı göster

Cevap: 1

Cevap

The remainder when the polynomial is divided by

x - 1

is

1

.

The correct answer is

1

. The polynomial is expressed as

p(x) = (x-2)^2(ax + b)

since it is a third-degree polynomial with a factor of

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

. Applying the Remainder Theorem, we evaluate the polynomial at

x = 3

and

x = -1

, giving the system of equations

3a + b = 5

and

-a + b = -3

. Solving this system yields

a = 2

and

b = -1

, meaning the polynomial is

p(x) = (x-2)^2(2x - 1)

. Finally, the remainder when

p(x)

is divided by

x - 1

is

p(1) = (1-2)^2(2(1) - 1) = 1

.

Adım Adım Çözüm

1

Express the third-degree polynomial in terms of its known quadratic factor.

p(x) = (x-2)^2(ax + b)

Since the polynomial is of degree 3 and has a quadratic factor of

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

, the remaining factor must be linear, of the form

ax+b

.

2

Apply the Remainder Theorem to set up the system of equations.

p(3) = 5

and

p(-1) = -27

The Remainder Theorem states that the remainder of a polynomial

p(x)

when divided by

x - c

is equal to

p(c)

.

3

Substitute the values of

x

into the polynomial expression.

3a + b = 5

and

9(-a + b) = -27

Evaluating

p(3) = (3-2)^2(3a + b) = 5

yields

3a + b = 5

, and evaluating

p(-1) = (-1-2)^2(-a + b) = -27

yields

9(-a + b) = -27

.

4

Simplify the second equation and solve the system of linear equations.

a = 2

and

b = -1

Dividing the second equation by

9

gives

-a + b = -3

. Subtracting this from the first equation (

3a + b = 5

) gives

4a = 8

, which means

a = 2

. Substituting

a = 2

into the first equation yields

b = -1

.

5

Calculate the remainder when

p(x)

is divided by

x - 1

.

p(1) = 1

According to the Remainder Theorem, the remainder of

p(x)

divided by

x - 1

is

p(1)

. Substituting

x = 1

into

p(x) = (x-2)^2(2x - 1)

yields

(1-2)^2(2(1) - 1) = 1

.

Anahtar Kavram

Using the Remainder Theorem and known factors of a polynomial to solve for unknown coefficients.

Soru 439Soru →

A forestry service worker randomly selected $80$ oak trees in a state park containing $2,000$ oak trees. The worker found that $12$ of the selected oak trees showed signs of a leaf disease. Based on this sample, what is the estimated number of oak trees in the entire state park that show signs of the leaf disease?

Cevabı ve açıklamayı göster

Cevap: 300

Cevap

300

Because the sample of

80

oak trees was selected at random, it can be assumed to be representative of the population of

2,000

oak trees in the state park. The proportion of diseased trees in the sample is

\frac{12}{80} = 0.15

. Multiplying this proportion by the total population of

2,000

oak trees yields the estimated total number of diseased trees:

0.15 \times 2,000 = 300

.

Adım Adım Çözüm

1

Calculate the sample proportion of diseased trees.

0.15

Dividing the

12

diseased trees by the sample size of

80

trees gives the proportion of diseased trees in the sample:

\frac{12}{80} = 0.15

.

2

Generalize the sample proportion to the entire population of oak trees in the park.

300

Multiplying the sample proportion of

0.15

by the total population of

2,000

oak trees yields the estimated total number of diseased trees:

0.15 \times 2,000 = 300

.

Anahtar Kavram

Generalizing results from a representative random sample to estimate a population parameter.

Soru 440Soru →

An electric vehicle consumes $0.35$ kilowatt-hours ( $\text{kWh}$ ) of electricity per mile traveled. The vehicle travels at a constant speed of $48$ miles per hour. What is the vehicle's rate of electricity consumption, in $\text{kWh}$ per minute?

Cevabı ve açıklamayı göster

Cevap: 0.28

Cevap

0.28

To find the rate of electricity consumption in kilowatt-hours (

\text{kWh}

) per minute, we must convert the given rate of

0.35\text{ kWh}

per mile into

\text{kWh}

per minute. First, convert the speed of

48\text{ miles per hour}

to miles per minute:

\frac{48\text{ miles}}{60\text{ minutes}} = 0.8\text{ miles per minute}

. Next, multiply the energy consumed per mile by the distance traveled per minute:

0.35\text{ kWh per mile} \times 0.8\text{ miles per minute} = 0.28\text{ kWh per minute}

. Therefore, the vehicle's rate of electricity consumption is

0.28\text{ kWh}

per minute.

Adım Adım Çözüm

1

Convert the vehicle's speed from miles per hour to miles per minute.

0.8

miles per minute

Since there are 60 minutes in 1 hour, dividing the speed in miles per hour by 60 gives the speed in miles per minute:

\frac{48\text{ miles}}{1\text{ hour}} \times \frac{1\text{ hour}}{60\text{ minutes}} = 0.8\text{ miles per minute}

.

2

Calculate the rate of electricity consumption in kWh per minute.

0.28

kWh per minute

Multiply the consumption rate per mile by the distance traveled per minute:

0.35\text{ kWh per mile} \times 0.8\text{ miles per minute} = 0.28\text{ kWh per minute}

.

Anahtar Kavram

Unit conversion involving rates and compound units.

Alternatif Yöntem

First, find the total electricity consumed in one hour of driving by multiplying the consumption rate per mile by the miles traveled in one hour:

0.35\text{ kWh per mile} \times 48\text{ miles} = 16.8\text{ kWh}

. Then, convert this hourly rate to a minute rate by dividing by 60 minutes:

\frac{16.8\text{ kWh}}{60\text{ minutes}} = 0.28\text{ kWh per minute}

.

Tahmini Süre:1m 15s

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Alternatif Yöntem

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