All practice questions

Question 1841Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = x^2 - 4x - 5$ intersects the $x$ -axis at the points $(p, 0)$ and $(q, 0)$ and has vertex $(h, k)$ . What is the area of the triangle with vertices at $(p, 0)$ , $(q, 0)$ , and $(h, k)$ ?

Show answer & explanation

Answer: 27

Answer

The area of the triangle is 27.

To find the area of the triangle, we first determine the coordinates of its vertices. The base of the triangle lies on the

x

-axis, with endpoints at the

x

-intercepts of the function

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

. Solving

x^2 - 4x - 5 = 0

by factoring gives

(x - 5)(x + 1) = 0

, so the intercepts are at

x = -1

and

x = 5

. The distance between these two points is

5 - (-1) = 6

, which is the base of the triangle. The third vertex is the vertex of the parabola. The

x

-coordinate of the vertex is

h = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2

. Substituting

x = 2

into the function gives the

y

-coordinate:

k = f(2) = 2^2 - 4(2) - 5 = -9

. The height of the triangle is the vertical distance from the

x

-axis to the vertex, which is

|-9| = 9

. The area of the triangle is

\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 9 = 27

.

Step-by-Step Solution

1

Find the

x

-intercepts of the parabola.

The intercepts are

(-1, 0)

and

(5, 0)

.

Setting

f(x) = 0

gives

x^2 - 4x - 5 = 0

. Factoring the quadratic equation yields

(x - 5)(x + 1) = 0

, which gives

x = 5

and

x = -1

.

2

Calculate the base of the triangle.

The base length is

6

.

The base of the triangle is the segment along the

x

-axis between the two intercepts. The distance between

(-1, 0)

and

(5, 0)

is

5 - (-1) = 6

.

3

Find the vertex

(h, k)

of the parabola.

The vertex is at

(2, -9)

.

The

x

-coordinate of the vertex is the midpoint of the intercepts:

h = \frac{-1 + 5}{2} = 2

. The

y

-coordinate is

k = f(2) = 2^2 - 4(2) - 5 = 4 - 8 - 5 = -9

.

4

Calculate the area of the triangle.

The area is

27

.

The height of the triangle is the distance from the

x

-axis to the vertex, which is

|k| = |-9| = 9

. Using the formula for the area of a triangle,

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 9 = 27

.

Key Concept

Finding the

x

-intercepts and vertex of a quadratic function to solve geometric problems in the coordinate plane.

Question 1842Question →

The function $f$ is defined by $f(x) = (x - 3)(x^3 - kx^2 + 5x - 15)$ , where $k$ is a constant. In the $xy$ -plane, the graph of $y = f(x)$ is tangent to the $x$ -axis at the point $(3, 0)$ . What is the value of $k$ ?

Show answer & explanation

Answer: 3

Answer

3

For the graph of a polynomial function to be tangent to the

x

-axis at

(3, 0)

, the root

x = 3

must have an even multiplicity (at least 2). The function is defined as

f(x) = (x - 3)(x^3 - kx^2 + 5x - 15)

. Since there is already one factor of

(x - 3)

explicitly defined, the remaining cubic factor

g(x) = x^3 - kx^2 + 5x - 15

must also have a factor of

(x - 3)

to make the total multiplicity of the root

x = 3

at least 2. According to the Factor Theorem, if

(x - 3)

is a factor of

g(x)

, then

g(3) = 0

. Substituting

x = 3

into

g(x)

gives

3^3 - k(3)^2 + 5(3) - 15 = 0

, which simplifies to

27 - 9k = 0

. Solving this equation for

k

yields

k = 3

.

Step-by-Step Solution

1

Identify the relationship between graph tangency and factor multiplicity.

For the graph of a polynomial function to be tangent to the

x

-axis at a point

(c, 0)

, the factor

(x - c)

must have an even multiplicity of at least 2 in the polynomial's factored form.

An odd multiplicity root causes the graph to cross the

x

-axis, while an even multiplicity root causes the graph to touch the

x

-axis and turn around (tangency).

2

Apply the multiplicity requirement to the given function.

Since

f(x) = (x - 3)(x^3 - kx^2 + 5x - 15)

already contains one factor of

(x - 3)

, the cubic expression

g(x) = x^3 - kx^2 + 5x - 15

must also contain

(x - 3)

as a factor to ensure the total multiplicity of the root

x = 3

is at least 2.

This guarantees that

(x - 3)^2

is a factor of

f(x)

.

3

Apply the Factor Theorem to the cubic expression.

Since

(x - 3)

is a factor of

g(x)

, then

g(3) = 0

.

The Factor Theorem states that a polynomial

P(x)

has a factor

(x - c)

if and only if

P(c) = 0

.

4

Solve for the constant

k

by substituting

x = 3

into

g(x)

.

3^3 - k(3)^2 + 5(3) - 15 = 0 \implies 27 - 9k + 15 - 15 = 0 \implies 27 - 9k = 0 \implies 9k = 27 \implies k = 3

.

Arithmetic simplification yields the value of the constant.

Key Concept

The relationship between polynomial factors, root multiplicities, and the behavior of the graph at

x

-intercepts.

Question 1843Question →

A polynomial function $q$ with real coefficients satisfies the equation $q(x) + q(6-x) = 8$ for all real numbers $x$ . In the $xy$ -plane, the graph of $y = q(x)$ has an $x$ -intercept at $(5, 0)$ . What is the remainder when $q(x)$ is divided by $x-1$ ?

8

-8

0

2

Show answer & explanation

Answer: $8$

Answer

The correct answer is

8

, which is the remainder when the polynomial is divided by

x-1

.

To find the remainder when the polynomial

q(x)

is divided by

x-1

, we use the Remainder Theorem, which states that this remainder is equal to

q(1)

. Since the graph of

y = q(x)

has an

x

-intercept at

(5, 0)

, we know that

q(5) = 0

. Substituting

x = 5

into the given equation

q(x) + q(6-x) = 8

yields

q(5) + q(6-5) = 8

, which simplifies to

q(5) + q(1) = 8

. Substituting

q(5) = 0

into this equation gives

0 + q(1) = 8

, so

q(1) = 8

. Geometrically, the equation

\frac{q(x) + q(6-x)}{2} = 4

shows that the graph of

q

is symmetric under a point reflection about

(3, 4)

. Reflecting the point

(5, 0)

about

(3, 4)

gives the point

(1, 8)

, verifying that

q(1) = 8

.

Step-by-Step Solution

1

Translate the given graphical intercept into function notation.

q(5) = 0

An

x

-intercept at

(5, 0)

means that when the input

x

is

5

, the output of the function is

0

.

2

Apply the Remainder Theorem to the divisor

x-1

.

The remainder is equal to

q(1)

.

The Remainder Theorem states that when a polynomial

q(x)

is divided by

x-a

, the remainder is

q(a)

.

3

Substitute

x = 5

into the given functional equation

q(x) + q(6-x) = 8

.

q(5) + q(6-5) = 8 \implies q(5) + q(1) = 8

Substituting

x = 5

allows us to relate the known value

q(5)

directly to the target value

q(1)

.

4

Solve for

q(1)

using the value of

q(5)

.

0 + q(1) = 8 \implies q(1) = 8

Substituting

q(5) = 0

into the equation yields the value of

q(1)

, which is the remainder.

Key Concept

Remainder Theorem and Polynomial Graphs

Alternative Method

Use the point symmetry of the graph. The equation

q(x) + q(6-x) = 8

can be rewritten as

\frac{q(x) + q(6-x)}{2} = 4

. This indicates that the graph of

y = q(x)

has point symmetry about the point

(3, 4)

. Since the graph has an

x

-intercept at

(5, 0)

, point symmetry dictates that the reflected point across

(3, 4)

must also lie on the graph. The midpoint of

(5, 0)

and the symmetric point

(x_0, y_0)

is

(3, 4)

. Solving

\frac{5+x_0}{2} = 3

gives

x_0 = 1

, and

\frac{0+y_0}{2} = 4

gives

y_0 = 8

. Thus,

(1, 8)

lies on the graph, meaning

q(1) = 8

. By the Remainder Theorem, the remainder when

q(x)

is divided by

x-1

is

q(1) = 8

.

Estimated Time:1m 30s

Question 1844Question →

An equation is shown below.

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

What is the value of the real solution to the equation?

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

The correct answer is 2. Multiplying both sides by the least common denominator

(x - 1)(x + 2)

results in the quadratic equation

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

. Simplifying this equation yields

x^2 + 2 = 6

, which has solutions

x = 2

and

x = -2

. However, substituting

x = -2

into the original equation results in division by zero, making it an extraneous solution. Therefore,

2

is the only valid real solution.

Step-by-Step Solution

1

Multiply the entire equation by the least common denominator,

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

, to clear the denominators.

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

This simplifies the rational equation into a polynomial equation.

2

Expand the terms and simplify the equation.

x^2 + 2x - 2x + 2 = 6

, which simplifies to

x^2 + 2 = 6

.

Distributing the terms allows us to group like terms and solve for the variable.

3

Solve the quadratic equation for

x

.

x^2 = 4

, which gives

x = 2

or

x = -2

.

Subtracting 2 from both sides isolates the squared variable.

4

Check for extraneous solutions by substituting the potential solutions back into the original denominators.

For

x = -2

, the denominator

x + 2

becomes 0, which is undefined. For

x = 2

, all denominators are non-zero.

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

Key Concept

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

Question 1845Question →

A digital photography archive consists of RAW files and JPEG files. Initially, the RAW files account for $80\%$ of the total storage space used by the archive. To reduce the storage space, the photographer compresses $40\%$ of the RAW files into JPEGs, which reduces the storage space of those specific files by $62.5\%$ . If the remaining uncompressed RAW files now account for $x\%$ of the updated total storage space of the archive, what is the value of $x$ ?

Show answer & explanation

Answer: 60

Answer

The remaining uncompressed RAW files occupy 48% of the initial total storage space, and the updated total storage space is 80% of the initial storage space. The percentage of the updated total storage space occupied by the uncompressed RAW files is therefore 60%.

The remaining uncompressed RAW files occupy

48\%

of the initial total storage space, and the updated total storage space is

80\%

of the initial storage space. The percentage of the updated total storage space occupied by the uncompressed RAW files is therefore 60%.

Step-by-Step Solution

1

Define variables for initial storage space.

Initial RAW space is

0.80T

; initial JPEG space is

0.20T

.

Establishes the baseline values relative to the initial total storage space

T

.

2

Calculate the space occupied by the portion of RAW files to be compressed and the remaining uncompressed RAW files.

Compressed RAW space is

0.32T

; remaining uncompressed RAW space is

0.48T

.

Splits the RAW files into the group that changes size and the group that remains unchanged.

3

Determine the new space occupied by the compressed files after the

62.5\%

reduction.

Reduction is

0.20T

; new space of compressed files is

0.12T

.

Calculates the impact of the compression percent change on the target subpopulation.

4

Calculate the updated total storage space of the archive.

Updated total storage space is

0.80T

.

Summing all updated file spaces (

0.48T

uncompressed RAW +

0.20T

original JPEG +

0.12T

newly compressed files) provides the new base for the final percentage calculation.

5

Compute the final percentage of the updated total storage space occupied by the uncompressed RAW files.

x = 60

.

Dividing the remaining uncompressed RAW space (

0.48T

) by the new total space (

0.80T

) gives the updated percentage.

Key Concept

Multi-step percent change and relative base calculations.

Question 1846Question →

In the $xy$ -plane, the graph of the linear equation $y = mx + b$ , where $m$ and $b$ are constants, passes through the points $(-2, 5)$ and $(4, 7)$ . What is the value of $m + b$ ?

6

-2

4

\frac{14}{3}

Show answer & explanation

Answer: $6$

Answer

The value of

m + b

is

6

.

The correct answer is

6

. By finding the slope

m

using the change in

y

divided by the change in

x

, we get

m = \frac{1}{3}

. Using the slope-intercept form with the point

(4, 7)

allows us to solve for

b = \frac{17}{3}

. Adding the two values together yields

m + b = \frac{1}{3} + \frac{17}{3} = 6

.

Step-by-Step Solution

1

Calculate the slope

m

of the line using the two points

(-2, 5)

and

(4, 7)

.

m = \frac{7 - 5}{4 - (-2)} = \frac{2}{6} = \frac{1}{3}

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

m = \frac{y_2 - y_1}{x_2 - x_1}

.

2

Substitute the slope

m = \frac{1}{3}

and the coordinates of one point, such as

(4, 7)

, into the slope-intercept equation

y = mx + b

to solve for

b

.

7 = \frac{1}{3}(4) + b \Rightarrow 7 = \frac{4}{3} + b \Rightarrow b = \frac{17}{3}

Substituting a known point and the slope into the slope-intercept form allows us to isolate and solve for the constant

y

-intercept.

3

Add the calculated values of

m

and

b

to find

m + b

.

m + b = \frac{1}{3} + \frac{17}{3} = \frac{18}{3} = 6

The question asks for the sum of the slope

m

and the

y

-intercept

b

.

Key Concept

Linear Equations in Two Variables

Question 1847Question →

In the system of equations below, $x$ and $y$ are variables.

2x - y = 5

3x - 2y = -17

What is the value of $x$ ?

7

25

27

49

Show answer & explanation

Answer: 27

Answer

The value of

x

is 27.

Isolating

y

from the first equation gives

y = 2x - 5

. Substituting this expression into the second equation yields

3x - 2(2x - 5) = -17

. Distributing the

-2

gives

3x - 4x + 10 = -17

, which simplifies to

-x + 10 = -17

. Subtracting

10

from both sides results in

-x = -27

, meaning

x = 27

. Alternatively, multiplying the first equation by

2

gives

4x - 2y = 10

, and subtracting the second equation from it eliminates

y

directly to give

x = 10 - (-17) = 27

.

Step-by-Step Solution

1

Isolate the variable

y

in the first equation,

2x - y = 5

.

y = 2x - 5

This allows for substitution of the expression for

y

into the second equation.

2

Substitute the expression for

y

into the second equation,

3x - 2y = -17

.

3x - 2(2x - 5) = -17

To create a single-variable equation in terms of

x

.

3

Distribute the coefficient

-2

to the terms inside the parentheses and simplify the equation.

3x - 4x + 10 = -17

, which simplifies to

-x + 10 = -17

Applying the distributive property correctly, noting that

-2 \times -5 = 10

.

4

Subtract

10

from both sides of the equation to isolate the term with

x

.

-x = -27

To isolate the variable term on one side of the equation.

5

Multiply both sides of the equation by

-1

to solve for

x

.

x = 27

To find the positive value of the variable

x

.

Key Concept

Solving systems of linear equations using the substitution method.

Estimated Time:1m 30s

Question 1848Question →

In 1847, astronomer Maria Mitchell swept her telescope across the night sky and spotted a blurry streak that would become known as 'Miss Mitchell's Comet.' While many of Mitchell's contemporaries focused on documenting known celestial bodies, the _______ dedication to scanning the skies led to the historic discovery of a new comet and earned her international acclaim.

Which choice completes the text with the most logical and grammatically correct term?

astronomer's

astronomers

astronomers'

astronomer

Show answer & explanation

Answer: astronomer's

Answer

astronomer's

The correct answer is 'astronomer's'. The text refers to the dedication of one specific astronomer, Maria Mitchell. To show that the 'dedication' belongs to her, we must use the singular possessive form, which is created by adding an apostrophe and an 's' to the singular noun.

Step-by-Step Solution

1

Identify the relationship between the blank and the noun that follows it.

The blank precedes 'dedication', indicating that we need a possessive noun to show who the dedication belongs to.

Understanding the grammatical relationship helps determine if a possessive form is required.

2

Determine the number of the noun in context.

The sentence refers to Maria Mitchell, who is a single individual, so we need the singular form.

This establishes whether the noun should be singular or plural.

3

Combine the singular number and possessive case to choose the correct spelling.

The singular possessive form of 'astronomer' is 'astronomer's'.

This produces the grammatically correct word for the blank.

Key Concept

Singular possessive nouns require an apostrophe followed by an 's' to show ownership.

Question 1849Question →

For the quadratic function $f$ , the table below shows three points that lie on its graph in the $xy$ -plane, where $k$ is a constant.

$x$	$f(x)$
$2$	$0$
$6$	$0$
$1$	$-15$

If the vertex of the graph of $y = f(x)$ is $(4, k)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 12

Answer

The value of

k

is

12

.

The table indicates that the function has

x

-intercepts at

x = 2

and

x = 6

because

f(2) = 0

and

f(6) = 0

. Consequently, the quadratic function can be written in factored form as

f(x) = a(x - 2)(x - 6)

for some constant

a

. Using the point

(1, -15)

to find

a

, we substitute

x = 1

and

f(1) = -15

, which yields

-15 = a(1 - 2)(1 - 6)

, or

-15 = 5a

, so

a = -3

. Since the vertex of the graph is

(4, k)

, the value of

k

is the function value at the vertex's

x

-coordinate, which is

f(4)

. Evaluating the function gives

k = -3(4 - 2)(4 - 6) = -3(2)(-2) = 12

.

Step-by-Step Solution

1

Write the quadratic function in factored form using the given

x

-intercepts.

f(x) = a(x - 2)(x - 6)

The table shows that

f(2) = 0

and

f(6) = 0

, meaning the graph has

x

-intercepts at

x = 2

and

x = 6

.

2

Substitute the point

(1, -15)

into the factored equation to find the value of the constant

a

.

a = -3

Substituting

x = 1

and

f(x) = -15

gives

-15 = a(1 - 2)(1 - 6) \implies -15 = 5a \implies a = -3

.

3

Calculate the value of

k

by finding the function value at the vertex

x = 4

.

k = 12

Since the vertex is

(4, k)

, the value of

k

is

f(4)

. Substituting

x = 4

into

f(x) = -3(x - 2)(x - 6)

yields

k = -3(4 - 2)(4 - 6) = -3(2)(-2) = 12

.

Key Concept

Using intercepts and an additional point to determine the equation of a quadratic function, and evaluating it at the vertex.

Question 1850Question →

The function $f$ is defined by $f(x) = 2x - 3$ . If the function $g$ is defined by $g(x) = f(x + 4)$ , what is the value of $g(1)$ ?

-9

3

7

13

Show answer & explanation

Answer: 7

Answer

The correct answer is 7.

Since

g(x) = f(x + 4)

, evaluating

g(1)

requires finding

f(1 + 4)

, which is

f(5)

. Substituting

5

into the expression for

f(x)

gives

f(5) = 2(5) - 3 = 10 - 3 = 7

. Therefore, the correct answer is 7.

Step-by-Step Solution

1

Substitute

x = 1

into the definition of

g(x)

to express

g(1)

in terms of

f

.

g(1) = f(1 + 4) = f(5)

To find the value of

g(1)

, we must evaluate the input to the outer function first.

2

Substitute the input value

5

into the expression for

f(x)

.

f(5) = 2(5) - 3

Evaluating

f(5)

requires replacing

x

with

5

in the definition

f(x) = 2x - 3

.

3

Perform the operations to find the final value.

f(5) = 10 - 3 = 7

Simplifying the numerical expression gives the final value of

g(1)

.

Key Concept

Evaluating a transformed function at a given point using function notation.

Estimated Time:45s

Question 1851Question →

A specialized cooling system is used to lower the temperature of a chemical solution in a laboratory. The temperature of the solution, $T$ , in degrees Celsius, can be modeled by the linear equation $T = 85.4 - 1.25m$ , where $m$ is the number of minutes since the cooling process began. According to the model, how many minutes does it take for the temperature of the solution to decrease by $15$ degrees Celsius?

Show answer & explanation

Answer: 12

Answer

12 minutes

The correct answer is 12. In the equation

T = 85.4 - 1.25m

, the coefficient of

m

is

-1.25

, which indicates that the temperature decreases by

1.25

degrees Celsius for each minute that passes. To find how many minutes it takes for the temperature to decrease by

15

degrees Celsius, divide the total decrease by the rate of decrease:

\frac{15}{1.25} = 12

.

Step-by-Step Solution

1

Identify the rate of change from the linear equation.

The rate of decrease is

1.25

degrees Celsius per minute.

The slope of the linear equation

T = 85.4 - 1.25m

is

-1.25

, representing the change in temperature per minute.

2

Divide the target temperature change by the rate of change to find the time.

12 minutes

To find the number of minutes for a

15

-degree decrease at a rate of

1.25

degrees per minute, calculate

\frac{15}{1.25}

.

Key Concept

Interpreting the slope of a linear relationship in context

Question 1852Question →

A projectile is launched from the ground. Its height, in feet, $t$ seconds after launch is modeled by the function $h(t) = -16t^2 + v_0 t$ , where $v_0$ is the initial upward velocity in feet per second. If the projectile reaches its maximum height of $144$ feet, what is the value of $v_0$ ?

Show answer & explanation

Answer: 96

Answer

The correct answer is 96. The initial velocity of the projectile must be 96 feet per second to reach a maximum height of 144 feet.

The maximum height of a projectile modeled by a quadratic function is the

y

-value of its vertex. For

h(t) = -16t^2 + v_0 t

, the time

t

at the vertex is given by

t = -\frac{b}{2a} = -\frac{v_0}{2(-16)} = \frac{v_0}{32}

. Substituting this value of

t

back into the height equation yields the maximum height:

h\left(\frac{v_0}{32}\right) = -16\left(\frac{v_0}{32}\right)^2 + v_0\left(\frac{v_0}{32}\right) = -16\left(\frac{v_0^2}{1024}\right) + \frac{v_0^2}{32} = -\frac{v_0^2}{64} + \frac{v_0^2}{32} = \frac{v_0^2}{64}

. Given that the maximum height is

144

feet, we set

\frac{v_0^2}{64} = 144

, which simplifies to

v_0^2 = 144 \times 64

. Taking the square root of both sides gives

v_0 = 12 \times 8 = 96

.

Step-by-Step Solution

1

Find the time

t

at which the maximum height occurs using the vertex formula.

t = \frac{v_0}{32}

For a quadratic function of the form

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

, the vertex occurs at

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

. Here,

a = -16

and

b = v_0

, so the vertex is at

t = -\frac{v_0}{2(-16)} = \frac{v_0}{32}

.

2

Substitute the time

t = \frac{v_0}{32}

into the height function to express the maximum height in terms of

v_0

.

h\left(\frac{v_0}{32}\right) = \frac{v_0^2}{64}

Evaluating

h\left(\frac{v_0}{32}\right)

gives

-16\left(\frac{v_0}{32}\right)^2 + v_0\left(\frac{v_0}{32}\right) = -16\left(\frac{v_0^2}{1024}\right) + \frac{v_0^2}{32} = -\frac{v_0^2}{64} + \frac{v_0^2}{32} = \frac{v_0^2}{64}

.

3

Set the maximum height expression equal to the given maximum height of

144

feet and solve for

v_0

.

v_0 = 96

Setting

\frac{v_0^2}{64} = 144

yields

v_0^2 = 144 \times 64 = 9216

. Taking the positive square root (since

v_0

represents a positive initial upward velocity) gives

v_0 = \sqrt{144} \times \sqrt{64} = 12 \times 8 = 96

.

Key Concept

Determining the vertex coordinates of a quadratic function to find maximum value in context.

Question 1853Question →

If $27^{2x - 2} = \left(\frac{1}{3}\right)^{x - 8}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 2

Answer

The value of

x

is

2

.

To solve the equation

27^{2x - 2} = \left(\frac{1}{3}\right)^{x - 8}

, express both sides with the base

3

. Since

27 = 3^3

and

\frac{1}{3} = 3^{-1}

, the equation becomes

(3^3)^{2x - 2} = (3^{-1})^{x - 8}

. Applying the exponent rule

(a^m)^n = a^{mn}

yields

3^{6x - 6} = 3^{8 - x}

. Since the bases are now the same, set the exponents equal to each other:

6x - 6 = 8 - x

. Adding

x

to both sides gives

7x - 6 = 8

, and adding

6

to both sides gives

7x = 14

. Dividing by

7

results in

x = 2

.

Step-by-Step Solution

1

Rewrite

27

and

\frac{1}{3}

as powers of

3

.

27 = 3^3

and

\frac{1}{3} = 3^{-1}

This allows both sides of the equation to be expressed with a common base of

3

so their exponents can be equated.

2

Substitute these bases back into the equation and apply exponent rules.

(3^3)^{2x - 2} = (3^{-1})^{x - 8} \implies 3^{3(2x - 2)} = 3^{-(x - 8)} \implies 3^{6x - 6} = 3^{8 - x}

Using the power rule

(a^m)^n = a^{mn}

, multiply the outer exponent by the power of the base.

3

Equate the exponents and solve for

x

.

6x - 6 = 8 - x \implies 7x = 14 \implies x = 2

Since the bases are equal, their exponents must be equal for the equation to hold true.

Key Concept

Solving exponential equations by finding a common base and applying exponent rules.

Question 1854Question →

A bakery recipe requires $12$ ounces of butter to make one batch of bread. How many pounds of butter are needed to make $8$ batches of bread? (Given that $1\text{ pound} = 16\text{ ounces}$ )

Show answer & explanation

Answer: 6

Answer

The bakery needs

6

pounds of butter.

The total amount of butter needed for

8

batches is calculated by multiplying the butter required for one batch by the number of batches:

8 \times 12 = 96

ounces. Given that

1\text{ pound} = 16\text{ ounces}

, we convert ounces to pounds by dividing the total ounces by

16

. Therefore,

96 \div 16 = 6

pounds of butter are needed.

Step-by-Step Solution

1

Calculate the total ounces of butter needed.

96

ounces

Since each batch requires

12

ounces,

8

batches require

8 \times 12 = 96

ounces.

2

Convert the total ounces to pounds.

6

pounds

Divide the total number of ounces by the conversion factor (

16

ounces per pound):

\frac{96}{16} = 6

.

Key Concept

To convert a quantity from a smaller unit to a larger unit, multiply the initial quantity by the number of groups to find the total in the smaller unit, and then divide by the conversion factor that relates the two units.

Question 1855Question →

Deep-sea hydrothermal vents support complex communities in the complete absence of sunlight. Instead of relying on solar energy, the organisms in these extreme habitats depend on chemosynthetic bacteria that convert hydrogen sulfide into chemical energy ______ biological production forms the foundation of a food web that sustains unique species of tube worms, crabs, and shrimp.

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

, but this

this

; this

, this

Show answer & explanation

Answer: ; this

Answer

The semicolon followed by the pronoun 'this' ('; this') is the correct choice because it appropriately links the two independent clauses.

The semicolon successfully links two independent clauses ('Instead of relying on solar energy, the organisms in these extreme habitats depend on chemosynthetic bacteria that convert hydrogen sulfide into chemical energy' and 'this biological production forms the foundation of a food web that sustains unique species of tube worms, crabs, and shrimp') without the need for a coordinating conjunction.

Step-by-Step Solution

1

Analyze the structure of the two clauses surrounding the blank.

The first clause ('Instead of relying on solar energy, the organisms in these extreme habitats depend on chemosynthetic bacteria that convert hydrogen sulfide into chemical energy') and the second clause ('this biological production forms the foundation of a food web that sustains unique species of tube worms, crabs, and shrimp') are both independent clauses.

Identifying the clause types is necessary to determine the correct punctuation and connectors required.

2

Determine the relationship between the two clauses.

The second clause continues and elaborates on the idea in the first clause, demonstrating a continuous and direct relationship rather than a contrast.

The logical relationship helps eliminate coordinators or subordinators that introduce inappropriate transitions.

3

Select the correct punctuation to join two independent clauses without a coordinating conjunction.

A semicolon is the appropriate punctuation mark to join two closely related independent clauses. A comma alone creates a comma splice, no punctuation creates a run-on, and using the coordinating conjunction 'but' introduces an incorrect sense of contrast.

Applying grammatical rules ensures the sentence conforms to Standard English conventions.

Key Concept

Linking Independent Clauses

Question 1856Question →

A rideshare driver has a daily goal of earning at least $150. So far today, the driver has earned$ 45. The driver earns $12 per ride plus an average tip of$ 3 per ride. If the driver must pay a daily vehicle fee of $15, what is the minimum number of additional rides the driver must complete today to meet or exceed the daily earnings goal?

Show answer & explanation

Answer: 8

Answer

The minimum number of additional rides the driver must complete today is 8.

The driver earns

12 plus a

3 tip per ride, which is

15 per ride. Starting with

45 and subtracting the

15 fee leaves the driver with

30 before completing any new rides. To reach at least

150, the driver needs to earn at least

120 more. Dividing

120 by the

15 rate per ride gives a minimum of 8 rides.

Step-by-Step Solution

1

Define the variable

x

for the number of additional rides and write an inequality representing the total net earnings.

45 + 12x + 3x - 15 \geq 150

To represent the condition that the driver's total earnings, including initial earnings and new rides, minus the fee, must be at least $150.

2

Simplify the left side of the inequality by combining the constants and the

x

terms.

30 + 15x \geq 150

To group like terms and simplify the expression before solving.

3

Subtract 30 from both sides of the inequality to isolate the variable term.

15x \geq 120

To isolate the term with the variable on one side of the inequality.

4

Divide both sides of the inequality by 15 to solve for

x

.

x \geq 8

To find the minimum value of

x

that satisfies the inequality.

Key Concept

Solving linear inequalities in one variable to find a minimum threshold value in a real-world scenario.

Question 1857Question →

A medical study compared the effectiveness of two headache treatments, Treatment X and Treatment Y. The results of the study are shown in the table below.

Treatment	Improved	No Improvement	Total
Treatment X	40	10	50
Treatment Y	35	15	50
Total	75	25	100

If a participant who received Treatment X is selected at random, what is the probability that the participant's headache improved?

\frac{1}{5}

\frac{2}{5}

\frac{8}{15}

\frac{4}{5}

Show answer & explanation

Answer: $\frac{4}{5}$

Answer

The correct answer is

\frac{4}{5}

.

The correct answer is the fraction expressing the number of participants who received Treatment X and improved (40) out of the total number of participants who received Treatment X (50), which simplifies to

\frac{4}{5}

.

Step-by-Step Solution

1

Identify the total number of participants in the conditional group.

The total number of participants who received Treatment X is

50

.

The question specifies that a participant is selected from those who received Treatment X, which restricts the sample space to this row.

2

Identify the number of participants in that conditional group who experienced improvement.

There are

40

participants who received Treatment X and experienced improvement.

We need to find the number of favorable outcomes within the restricted sample space.

3

Calculate the conditional probability.

The probability is

\frac{40}{50} = \frac{4}{5}

.

Probability is the ratio of the number of favorable outcomes to the total number of possible outcomes in the restricted group.

Key Concept

Calculating conditional probability from a two-way table by identifying the restricted sample space.

Estimated Time:45s

Question 1858Question →

The quadratic function $f$ is defined by $f(x) = a(x - h)^2 + k$ , where $a$ , $h$ , and $k$ are constants. In the $xy$ -plane, the graph of $y = f(x)$ has a vertex at $(3, -4)$ and passes through the point $(5, 8)$ . If the function $g$ is defined by $g(x) = -2f(x - 1) + 5$ , what is the value of $g(2)$ ?

5

7

-11

13

Show answer & explanation

Answer: -11

Answer

-11

To find the value of

g(2)

, we first determine the equation of the quadratic function

f(x)

. Since the vertex is

(3, -4)

, the vertex form is

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

. Substituting the point

(5, 8)

yields

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

, which simplifies to

12 = 4a

, so

a = 3

. Therefore,

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

. We then substitute

x = 2

into the definition of

g(x)

, obtaining

g(2) = -2f(2 - 1) + 5 = -2f(1) + 5

. Evaluating

f(1)

gives

f(1) = 3(1 - 3)^2 - 4 = 8

. Substituting this back into the expression for

g(2)

gives

-2(8) + 5 = -11

. Thus, the option with value -11 is correct.

Step-by-Step Solution

1

Write the quadratic function

f(x)

in vertex form using the given vertex

(3, -4)

.

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

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex.

2

Substitute the point

(5, 8)

into the vertex form to solve for the constant

a

.

8 = a(5 - 3)^2 - 4 \implies 8 = 4a - 4 \implies 12 = 4a \implies a = 3

. Thus,

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

.

Since the graph of

f

passes through

(5, 8)

, these coordinates must satisfy the function's equation.

3

Substitute

x = 2

into the definition of

g(x)

to express

g(2)

in terms of

f

.

g(2) = -2f(2 - 1) + 5 = -2f(1) + 5

We need to evaluate the inner function transformation

f(x - 1)

at

x = 2

.

4

Evaluate

f(1)

using the formula determined in Step 2.

f(1) = 3(1 - 3)^2 - 4 = 3(-2)^2 - 4 = 3(4) - 4 = 8

To find

g(2)

, we must first compute the value of

f(1)

.

5

Substitute

f(1) = 8

back into the expression for

g(2)

and simplify.

g(2) = -2(8) + 5 = -16 + 5 = -11

This completes the evaluation of the multi-step transformation.

Key Concept

Function Notation and Transformations

Estimated Time:2m 0s

Question 1859Question →

A map has a scale where $2$ centimeters represents $15$ kilometers. If the distance between two cities on the map is $8$ centimeters, what is the actual distance between the two cities, in kilometers?

Show answer & explanation

Answer: 60

Answer

The actual distance between the two cities is

60

kilometers.

The correct answer is

60

. The map scale is

2

centimeters to

15

kilometers. The map distance between the cities is

8

centimeters, which is

4

times the scale distance of

2

centimeters (since

8 \div 2 = 4

). Therefore, the actual distance between the cities is

4

times the scale distance of

15

kilometers, which is

15 \times 4 = 60

kilometers.

Step-by-Step Solution

1

Set up a proportion using the map scale ratio of centimeters to kilometers.

\frac{2}{15} = \frac{8}{x}

This establishes that the ratio of map distance to actual distance remains constant.

2

Solve for the unknown actual distance,

x

, by cross-multiplying.

2x = 120

Multiplying the numerator of each fraction by the denominator of the other solves the proportion.

3

Divide by the coefficient of

x

to find the final actual distance.

x = 60

Dividing

120

by

2

isolates

x

and gives the actual distance in kilometers.

Key Concept

Setting up and solving proportions using scale factors.

Question 1860Question →

A manufacturer models the daily profit, $P(x)$ , in dollars, from producing and selling $x$ units of a product using the function $P(x) = -2x^2 + kx - 800$ , where $k$ is a constant. If the maximum daily profit is $1,000$ dollars, what is the number of units that must be sold to achieve this maximum profit?

10

30

60

120

Show answer & explanation

Answer: 30

Answer

The manufacturer must sell 30 units to achieve the maximum daily profit.

The correct answer is 30. The maximum value of a downward-opening quadratic function occurs at its vertex

(h, q)

. Given that the maximum value is 1000, we write the function in vertex form:

P(x) = -2(x - h)^2 + 1000

. Expanding this gives

P(x) = -2x^2 + 4hx - 2h^2 + 1000

. Comparing this to the given function

P(x) = -2x^2 + kx - 800

, the constant term must satisfy

-2h^2 + 1000 = -800

. Solving for

h

yields

2h^2 = 1800

, so

h^2 = 900

. Taking the positive square root because the number of units must be positive gives

h = 30

.

Step-by-Step Solution

1

Write the quadratic function in vertex form and identify the given parameters.

The vertex form of a quadratic function is

P(x) = a(x-h)^2 + q

, where

(h, q)

is the vertex. Here,

a = -2

and the maximum profit

q = 1000

, so the function is

P(x) = -2(x-h)^2 + 1000

.

Since the coefficient of the

x^2

term is negative (

a = -2

), the parabola opens downward, meaning the vertex represents the maximum value of the function.

2

Expand the vertex form equation to compare its coefficients with the standard form

P(x) = -2x^2 + kx - 800

.

P(x) = -2(x^2 - 2hx + h^2) + 1000 = -2x^2 + 4hx - (2h^2 - 1000)

Expanding the vertex form allows us to equate corresponding coefficients and constant terms between the two forms of the quadratic function.

3

Equate the constant terms from both expressions to solve for the vertex

x

-coordinate

h

.

-2h^2 + 1000 = -800 \Rightarrow -2h^2 = -1800 \Rightarrow h^2 = 900 \Rightarrow h = 30

(since the number of units must be positive).

By setting the constant terms equal, we can isolate and solve for

h

, which represents the number of units that maximizes the daily profit.

Key Concept

Quadratic functions in vertex form and their standard form equivalents.

Answer

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Answer

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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