All practice questions

Question 1441Question →

Each of the pioneering female photojournalists who documented the rapid industrialization of the American Midwest in the early twentieth century sought to capture the human cost of labor in ______ work. By emphasizing the grueling conditions of factory worker families, these photographers challenged the era's idealized narratives of technological progress.

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

her

their

she

they

Show answer & explanation

Answer: her

Answer

her

The singular possessive pronoun 'her' is correct because the antecedent of the pronoun is 'Each', which is a singular indefinite pronoun. Because the photojournalists are specified as female, the singular feminine possessive pronoun 'her' is appropriate to show possession of the noun 'work'.

Step-by-Step Solution

1

Identify the antecedent of the pronoun needed in the blank.

The antecedent is the indefinite pronoun 'Each'.

The pronoun must agree in number with its antecedent.

2

Determine the grammatical number of the antecedent.

'Each' is a singular indefinite pronoun.

Although 'photojournalists' is plural, it is the object of the prepositional phrase 'of the pioneering female photojournalists' and does not function as the antecedent.

3

Select the correct pronoun case and number to fit the context.

The blank modifies the noun 'work', so a possessive pronoun is needed. Since the photojournalists are female, the singular possessive pronoun 'her' is the correct choice.

A singular possessive pronoun is required to show possession and maintain agreement with 'Each'.

Key Concept

Pronoun-antecedent agreement in number requires that a pronoun match its antecedent (singular or plural), regardless of any intervening prepositional phrases or relative clauses.

Estimated Time:1m 0s

Question 1442Question →

A circle in the $xy$ -plane is defined by the equation $(x - h)^2 + (y - k)^2 = 16$ , where $h$ and $k$ are constants. The center $(h, k)$ of the circle lies on the line $y = x$ . A second line, which passes through the origin and has a slope of $-\frac{3}{4}$ , is tangent to the circle at exactly one point $(x, y)$ . If this point of tangency lies in a quadrant where $x > 0$ and $y < 0$ , what is the value of $h$ ?

20

-\frac{20}{7}

\frac{20}{7}

\frac{80}{7}

Show answer & explanation

Answer: $\frac{20}{7}$

Answer

\frac{20}{7}

The correct option is

\frac{20}{7}

. By setting the center of the circle to

(h, h)

and the equation of the line to

3x + 4y = 0

, we find that the distance from the center to the line is

\frac{|7h|}{5}

. Since the line is tangent to the circle, this distance must equal the radius, which is

4

. This gives

h = \pm\frac{20}{7}

. Finding the point of tangency shows that

x = \frac{4}{25}h

and

y = -\frac{3}{25}h

. For the point of tangency to lie in Quadrant IV (

x > 0

and

y < 0

),

h

must be positive, which yields

h = \frac{20}{7}

.

Step-by-Step Solution

1

Express the center of the circle and the equation of the tangent line in terms of the given parameters.

Since the center

(h, k)

lies on the line

y = x

, we have

k = h

. The circle has radius

R = \sqrt{16} = 4

and is centered at

(h, h)

. The tangent line passes through the origin with slope

-\frac{3}{4}

, so its equation is

y = -\frac{3}{4}x

, which simplifies to

3x + 4y = 0

.

Setting up the algebraic expressions for both geometric entities is necessary to relate them using coordinate geometry formulas.

2

Apply the tangency condition using the point-to-line distance formula.

The distance from the center

(h, h)

to the line

3x + 4y = 0

must equal the radius

4

. Thus:

\frac{|3h + 4h|}{\sqrt{3^2 + 4^2}} = 4 \implies \frac{|7h|}{5} = 4 \implies |7h| = 20

.

A line is tangent to a circle if and only if the perpendicular distance from the center of the circle to the line equals the radius.

3

Solve for the possible values of

h

.

h = \pm\frac{20}{7}

.

Solving the absolute value equation yields two symmetric possibilities for the x-coordinate of the circle's center.

4

Determine the relationship between the center

h

and the coordinates of the point of tangency

(x, y)

to apply the quadrant constraint.

The radius connecting the center

(h, h)

to the point of tangency

(x, y)

is perpendicular to the tangent line. Since the tangent line has a slope of

-\frac{3}{4}

, the perpendicular radius line has a slope of

\frac{4}{3}

. Its equation is:

y - h = \frac{4}{3}(x - h) \implies y = \frac{4}{3}x - \frac{1}{3}h

.

The intersection of the perpendicular radius line and the tangent line will locate the exact point of tangency.

5

Solve the system of equations for the point of tangency

(x, y)

in terms of

h

.

Equating the tangent line and the perpendicular line:

-\frac{3}{4}x = \frac{4}{3}x - \frac{1}{3}h \implies -\frac{9}{12}x = \frac{16}{12}x - \frac{4}{12}h \implies -\frac{25}{12}x = -\frac{4}{12}h \implies x = \frac{4}{25}h

. Substituting back:

y = -\frac{3}{4}\left(\frac{4}{25}h\right) = -\frac{3}{25}h

.

This yields the coordinates of the tangency point as a function of the parameter

h

.

6

Apply the quadrant constraint (

x > 0

and

y < 0

) to choose the correct sign of

h

.

We require

x = \frac{4}{25}h > 0

and

y = -\frac{3}{25}h < 0

. Both inequalities are satisfied if and only if

h > 0

. Therefore,

h = \frac{20}{7}

.

This filters out the extraneous geometric solution that lies in Quadrant II.

Key Concept

Solving systems of nonlinear equations representing circles and lines by utilizing geometric relations, distance formulas, and quadrant constraints.

Estimated Time:3m 0s

Question 1443Question →

If $(2x - 3)(x + 4) = 0$ and $x > 0$ , what is the value of $x$ ?

-4

\frac{3}{2}

-\frac{3}{2}

4

Show answer & explanation

Answer: $\frac{3}{2}$

Answer

The value of

x

is

\frac{3}{2}

.

To find the solutions to the equation

(2x - 3)(x + 4) = 0

, we set each factor equal to zero. Setting

2x - 3 = 0

gives

x = \frac{3}{2}

, and setting

x + 4 = 0

gives

x = -4

. The problem states that

x > 0

, meaning the value of

x

must be positive. Therefore, the only valid solution is

\frac{3}{2}

.

Step-by-Step Solution

1

Set each factor of the quadratic equation equal to zero using the zero product property.

2x - 3 = 0

or

x + 4 = 0

If the product of two factors is zero, then at least one of the individual factors must equal zero.

2

Solve each linear equation to find the possible values of

x

.

x = \frac{3}{2}

or

x = -4

Isolating

x

in each equation gives the roots of the quadratic equation.

3

Apply the given constraint

x > 0

to identify the correct solution.

x = \frac{3}{2}

is the only valid solution because

\frac{3}{2} > 0

and

-4 \ngtr 0

.

The problem restricts the solution to values of

x

that are strictly greater than zero, so the negative root must be discarded.

Key Concept

Solving factored quadratic equations with inequality constraints

Question 1444Question →

In sociology, 'third places' refer to social surroundings separate from the two primary environments of home and work. These spaces, such as coffee shops, libraries, and parks, are essential for civil society and civic engagement ______ they foster a sense of place and facilitate regular, voluntary gatherings. Without these informal meeting grounds, communities often experience a decline in social cohesion and mutual trust.

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

, they

they

; they

, but they

Show answer & explanation

Answer: ; they

Answer

The choice that uses a semicolon to link the two independent clauses ('; they') is correct.

The correct choice uses a semicolon to separate two independent clauses ('These spaces... are essential for civil society and civic engagement' and 'they foster a sense of place...'). A semicolon is a standard grammatical way to link two closely related independent clauses without a coordinating conjunction.

Step-by-Step Solution

1

Analyze the structure of the two clauses surrounding the blank.

The first clause ('These spaces, such as coffee shops, libraries, and parks, are essential for civil society and civic engagement') and the second clause ('they foster a sense of place and facilitate regular, voluntary gatherings') are both independent clauses because they can each stand alone as complete sentences.

Identifying clause types determines how they must be punctuated and joined.

2

Determine the appropriate punctuation or conjunction to connect two independent clauses.

Two independent clauses must be separated by a period, a semicolon, a colon (if the second explains the first), or a comma followed by a coordinating conjunction.

Linking independent clauses without proper punctuation results in run-on sentences or comma splices.

3

Evaluate the logical relationship between the two clauses.

The second clause explains why third places are essential for civil society. This is a continuous/explanatory relationship, not a contrasting one.

This relationship rules out contrasting coordinators like 'but' and supports a direct link like a semicolon.

Key Concept

Clause Boundaries and Linking

Question 1445Question →

A community food bank is preparing two types of relief packages: standard boxes and family boxes. Let $x$ represent the number of standard boxes and $y$ represent the number of family boxes. The food bank needs to prepare at least 120 boxes in total. Each standard box contains 3 cans of soup and 2 packages of grain. Each family box contains 6 cans of soup and 5 packages of grain. The food bank has a maximum of 600 cans of soup and a maximum of 450 packages of grain available. Which of the following systems of inequalities represents this situation?

\begin{aligned} x + y &\le 120 \\ 3x + 6y &\ge 600 \\ 2x + 5y &\ge 450 \end{aligned}

\begin{aligned} x + y &\ge 120 \\ 6x + 3y &\le 600 \\ 5x + 2y &\le 450 \end{aligned}

\begin{aligned} x + y &\ge 120 \\ 3x + 6y &\le 600 \\ 2x + 5y &\le 450 \end{aligned}

\begin{aligned} x + y &\ge 120 \\ 3x + 2y &\le 600 \\ 6x + 5y &\le 450 \end{aligned}

Show answer & explanation

Answer: $\begin{aligned} x + y &\ge 120 \\ 3x + 6y &\le 600 \\ 2x + 5y &\le 450 \end{aligned}$

Answer

The system of inequalities with

x + y \ge 120

,

3x + 6y \le 600

, and

2x + 5y \le 450

.

The correct system of inequalities translates the constraint of preparing at least 120 boxes as

x + y \ge 120

. The maximum limit of 600 cans of soup translates as

3x + 6y \le 600

, since each standard box contains 3 cans and each family box contains 6. The maximum limit of 450 packages of grain translates as

2x + 5y \le 450

, since each standard box contains 2 packages and each family box contains 5.

Step-by-Step Solution

1

Translate the total package constraint.

x + y \ge 120

The phrase 'at least 120 boxes in total' indicates that the sum of

x

and

y

must be greater than or equal to 120.

2

Translate the soup constraint.

3x + 6y \le 600

Since each standard box (

x

) has 3 cans of soup and each family box (

y

) has 6 cans of soup, the total soup used is

3x + 6y

. The food bank has 'a maximum of 600 cans,' so this total must be less than or equal to 600.

3

Translate the grain constraint.

2x + 5y \le 450

Since each standard box (

x

) has 2 packages of grain and each family box (

y

) has 5 packages of grain, the total grain used is

2x + 5y

. The food bank has 'a maximum of 450 packages,' so this total must be less than or equal to 450.

Key Concept

Systems of Linear Inequalities in Two Variables

Estimated Time:1m 30s

Question 1446Question →

The graphs of the equations $y - 2x = 3$ and $y = x^2$ intersect at two points in the $xy$ -plane. If $(x, y)$ represents an intersection point with a positive $x$ -coordinate, what is the value of $y$ ?

1

3

6

9

Show answer & explanation

Answer: 9

Answer

9

Substituting

y = x^2

into the first equation yields

x^2 - 2x = 3

. Setting the quadratic equation to zero gives

x^2 - 2x - 3 = 0

, which factors as

(x - 3)(x + 1) = 0

. This gives the solutions

x = 3

and

x = -1

. Because the question specifies a positive

x

-coordinate,

x

must be 3. Substituting

x = 3

back into

y = x^2

yields

y = 9

. Thus, the correct answer is 9.

Step-by-Step Solution

1

Substitute

y = x^2

into the equation

y - 2x = 3

.

x^2 - 2x = 3

This eliminates the variable

y

and creates a single quadratic equation in terms of

x

to find the

x

-coordinates of the intersection points.

2

Rewrite the quadratic equation in standard form.

x^2 - 2x - 3 = 0

Subtracting 3 from both sides sets the quadratic equation to 0, which is necessary for factoring.

3

Factor the quadratic equation to find its solutions.

(x - 3)(x + 1) = 0

, which gives

x = 3

or

x = -1

.

Factoring allows us to find the roots, which represent the

x

-coordinates of the intersection points.

4

Apply the constraint that the

x

-coordinate must be positive.

x = 3

The question specifies that

x > 0

, so we discard

x = -1

.

5

Substitute

x = 3

back into

y = x^2

to find the corresponding value of

y

.

y = 3^2 = 9

This determines the

y

-coordinate of the intersection point with the positive

x

-coordinate.

Key Concept

Solving a system of a linear equation and a quadratic equation by substitution.

Question 1447Question →

A population of bacteria doubles every 3 hours. If the initial population of the bacteria is 500, what is the population of the bacteria after 9 hours?

Show answer & explanation

Answer: 4000

Answer

The population of the bacteria after 9 hours is 4,000.

The final population is calculated using the formula

P(t) = P_0 \times 2^{t/d}

, where

P_0

is the initial population of 500,

d

is the doubling period of 3 hours, and

t

is the total time of 9 hours. Evaluating this gives

P(9) = 500 \times 2^{9/3} = 500 \times 2^3 = 500 \times 8 = 4000

.

Step-by-Step Solution

1

Identify the initial population (

P_0

), doubling time (

d

), and total time (

t

).

P_0 = 500

,

d = 3

, and

t = 9

.

These parameters are required to set up the exponential growth model.

2

Calculate the number of doubling periods.

The number of doubling periods is

\frac{9}{3} = 3

.

The population doubles once for every 3-hour interval.

3

Calculate the final population using the exponential growth formula.

500 \times 2^3 = 500 \times 8 = 4000

.

Applying the 3 doubling cycles to the initial population of 500 yields the final population.

Key Concept

Exponential Growth Model

Estimated Time:1m 0s

Question 1448Question →

In the system of equations below, $k$ is a positive constant.

x - y = k

x^2 - 3xy + y^2 = 5

If the system has exactly one real solution $(x, y)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 2

Answer

The value of

k

is 2.

Substituting

y = x - k

into the second equation yields

-x^2 + kx + k^2 - 5 = 0

, which can be rewritten in standard form as

x^2 - kx + (5 - k^2) = 0

. For this quadratic equation to have exactly one real solution, its discriminant must equal 0:

b^2 - 4ac = (-k)^2 - 4(1)(5 - k^2) = 5k^2 - 20 = 0

. Solving

5k^2 - 20 = 0

gives

k^2 = 4

, and since

k

must be positive,

k = 2

.

Step-by-Step Solution

1

Rearrange the first equation to express

y

in terms of

x

.

y = x - k

This allows for substitution into the second equation to eliminate

y

.

2

Substitute

y = x - k

into the second equation and expand.

x^2 - 3x(x - k) + (x - k)^2 = 5 \implies -x^2 + kx + k^2 - 5 = 0

To create a single quadratic equation in terms of

x

.

3

Multiply by

-1

to write the quadratic equation in standard form

ax^2 + bx + c = 0

.

x^2 - kx + (5 - k^2) = 0

Standard form makes it easier to identify the coefficients

a = 1

,

b = -k

, and

c = 5 - k^2

.

4

Set the discriminant

b^2 - 4ac

equal to 0.

(-k)^2 - 4(1)(5 - k^2) = 0

A quadratic equation has exactly one real solution if and only if its discriminant is zero.

5

Simplify the discriminant equation and solve for

k

.

5k^2 - 20 = 0 \implies k^2 = 4 \implies k = 2

(since

k

must be positive)

To find the positive constant

k

that satisfies the condition.

Key Concept

Determining the number of solutions to a nonlinear system by substituting and setting the discriminant of the resulting quadratic equation to zero.

Alternative Method

Alternatively, one can rewrite the second equation by grouping:

x^2 - 3xy + y^2 = (x - y)^2 - xy = 5

. Since

x - y = k

, we have

k^2 - xy = 5

, so

xy = k^2 - 5

. We now have a system of

x - y = k

and

xy = k^2 - 5

. Substituting

y = x - k

gives

x(x - k) = k^2 - 5

, leading to

x^2 - kx + (5 - k^2) = 0

, which can be solved using the discriminant as shown in the primary method.

Estimated Time:2m 30s

Question 1449Question →

For all $x > 0$ and $y > 0$ , the expression $\frac{(x^2y^3)^a}{(xy^2)^2}$ is equivalent to $x^6y^8$ , where $a$ is a constant. What is the value of $a$ ?

Show answer & explanation

Answer: 4

Answer

4

Applying the rules of exponents, the expression

\frac{(x^2y^3)^a}{(xy^2)^2}

simplifies to

\frac{x^{2a}y^{3a}}{x^2y^4} = x^{2a-2}y^{3a-4}

. Setting this equal to the equivalent expression

x^6y^8

gives the system of equations

2a - 2 = 6

and

3a - 4 = 8

. Solving either equation yields

a = 4

.

Step-by-Step Solution

1

Apply the power rule of exponents to the numerator and denominator.

Numerator:

x^{2a}y^{3a}

, Denominator:

x^2y^4

To expand the parentheses by multiplying the outer exponent with the inner exponents.

2

Apply the quotient rule of exponents to divide the numerator by the denominator.

x^{2a-2}y^{3a-4}

To simplify the rational expression by subtracting the exponents in the denominator from the exponents in the numerator.

3

Equate the simplified exponent of

x

to the exponent of

x

in the given equivalent expression.

2a - 2 = 6

Equivalent expressions must have identical exponents for corresponding variable bases.

4

Solve the linear equation for

a

.

a = 4

To find the constant value that satisfies the equation.

5

Verify the value of

a

using the exponents of

y

.

3(4) - 4 = 8

, which is true.

To ensure consistency across both variable exponents.

Key Concept

Equivalent algebraic expressions involving exponent rules

Question 1450Question →

A parabola and a line intersect at exactly one point in the $xy$ -plane. The parabola is defined by the equation $y = -x^2 + 6x - 2$ and the line is defined by the equation $y = 2x + k$ , where $k$ is a constant. What is the value of $k$ ?

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

To find the value of

k

where the parabola and the line intersect at exactly one point, we equate the two equations:

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

. Rearranging this into standard quadratic form yields

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

. For a quadratic equation to have exactly one real solution, its discriminant,

b^2 - 4ac

, must be equal to zero. Substituting

a = 1

,

b = -4

, and

c = k + 2

into the discriminant formula gives

(-4)^2 - 4(1)(k + 2) = 0

, which simplifies to

16 - 4k - 8 = 0

, or

8 - 4k = 0

. Solving for

k

gives

k = 2

.

Step-by-Step Solution

1

Equate the equations of the parabola and the line to set up an equation for the x-coordinates of their intersection points.

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

At the points of intersection, the y-values of both equations are equal.

2

Rearrange the equation into the standard quadratic form

ax^2 + bx + c = 0

.

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

This allows us to identify the coefficients

a = 1

,

b = -4

, and

c = k + 2

to apply the quadratic discriminant.

3

Set the discriminant

b^2 - 4ac

equal to zero.

(-4)^2 - 4(1)(k + 2) = 0

A quadratic system has exactly one real solution (tangency) if and only if the discriminant of the resulting quadratic equation is zero.

4

Solve the linear equation for

k

.

k = 2

Simplifying the expression yields

16 - 4k - 8 = 0

, which simplifies to

8 - 4k = 0

, giving

k = 2

.

Key Concept

Nonlinear Systems of Equations

Question 1451Question →

In the $xy$ -plane, the system of equations below has infinitely many solutions:

\begin{aligned} \frac{1}{2}(ax - by) &= 7 \\ 3x - 2y &= c \end{aligned}

where $a$ , $b$ , and $c$ are constants. If the line representing the first equation passes through the point $(4, 1)$ , what is the value of $c$ ?

5

8

10

20

Show answer & explanation

Answer: 10

Answer

10

The correct answer is 10. For a system of linear equations in two variables to have infinitely many solutions, the two equations must describe the exact same line, meaning their coefficients and constant terms are proportional. Multiplying the first equation by

2

yields

ax - by = 14

. Comparing this with

3x - 2y = c

shows that

\frac{a}{3} = \frac{b}{2} = \frac{14}{c}

, which simplifies to

b = \frac{2}{3}a

and

c = \frac{42}{a}

. Since the first line passes through

(4, 1)

, we substitute

x = 4

and

y = 1

to get

4a - b = 14

. Substituting

b = \frac{2}{3}a

gives

4a - \frac{2}{3}a = 14

, which simplifies to

\frac{10}{3}a = 14

, or

a = 4.2

. Finally, solving for

c

gives

c = \frac{42}{4.2} = 10

.

Step-by-Step Solution

1

Clear the fraction in the first equation by multiplying both sides by

2

.

ax - by = 14

This puts the first equation into standard form, making it easier to compare with the second equation.

2

Set up the proportionality of the coefficients for the two equations to represent the same line.

\frac{a}{3} = \frac{-b}{-2} = \frac{14}{c} \implies b = \frac{2}{3}a

and

c = \frac{42}{a}

For a system of two linear equations to have infinitely many solutions, the equations must be equivalent, meaning their corresponding coefficients and constants must be proportional.

3

Substitute the given point

(4, 1)

into the first equation.

a(4) - b(1) = 14 \implies 4a - b = 14

Since the line passes through the point

(4, 1)

, the coordinates must satisfy the equation of the line.

4

Substitute

b = \frac{2}{3}a

into

4a - b = 14

and solve for

a

.

4a - \frac{2}{3}a = 14 \implies \frac{10}{3}a = 14 \implies a = 4.2

This reduces the equation to a single variable, allowing us to find the value of the parameter

a

.

5

Substitute

a = 4.2

into the expression for

c

to find its value.

c = \frac{42}{4.2} = 10

This uses the coefficient proportionality relation from step 2 to determine the constant term of the second equation.

Key Concept

Systems of linear equations with infinitely many solutions require the equations to represent the same line, meaning their coefficients and constant terms are proportional.

Question 1452Question →

An industrial oven is used in a bakery. The temperature of the oven chamber, $C$ , in degrees Fahrenheit ( $^\circ\text{F}$ ), $t$ minutes after the heating element is turned on is modeled by the linear equation:

C = 18.5t + 72

After a system upgrade, the starting temperature of the oven is $8^\circ\text{F}$ warmer, and the rate at which the oven heats up is $20\%$ faster. During a test of the upgraded oven, the heating element is turned on for $15$ minutes, after which the oven is turned off and cools down at a constant rate of $12^\circ\text{F}$ per minute. If the cooling process is also linear, which of the following functions models the temperature of the upgraded oven, $U$ , in degrees Fahrenheit, $m$ minutes after it is turned off?

U(m) = -12m + 357.5

U(m) = -12m + 483.9

U(m) = -12m + 413

U(m) = 12m + 413

Show answer & explanation

Answer: $U(m) = -12m + 413$

Answer

The function

U(m) = -12m + 413

models the temperature of the upgraded oven,

U

, in degrees Fahrenheit,

m

minutes after it is turned off.

To find the temperature model during the cooling phase, we must first determine the state of the oven when the cooling begins. The upgraded oven has a starting temperature of

72 + 8 = 80^\circ\text{F}

and a heating rate of

18.5 \times 1.20 = 22.2^\circ\text{F}

per minute. After

15

minutes of heating, the temperature reaches

22.2 \times 15 + 80 = 413^\circ\text{F}

. When the oven is turned off at

m = 0

minutes, its temperature is

413^\circ\text{F}

, which serves as the y-intercept of the cooling function. Since the temperature decreases at a constant rate of

12^\circ\text{F}

per minute, the rate of change (slope) is

-12

. Therefore, the linear model is the function showing a rate of change of

-12

and a starting value of

413

.

Step-by-Step Solution

1

Determine the upgraded starting temperature and heating rate of the oven.

The upgraded starting temperature is

80^\circ\text{F}

and the upgraded heating rate is

22.2^\circ\text{F}

per minute.

The original starting temperature of

72^\circ\text{F}

is increased by

8^\circ\text{F}

to get

72 + 8 = 80^\circ\text{F}

. The original heating rate (slope) of

18.5^\circ\text{F}

per minute is increased by

20\%

, which is calculated as

18.5 \times 1.20 = 22.2^\circ\text{F}

per minute.

2

Calculate the temperature of the upgraded oven at the moment it is turned off.

The temperature is

413^\circ\text{F}

at

t = 15

minutes.

The heating phase is modeled by the linear relationship

H(t) = 22.2t + 80

. Substituting

t = 15

yields

H(15) = 22.2(15) + 80 = 333 + 80 = 413^\circ\text{F}

.

3

Construct the linear function for the cooling phase.

U(m) = -12m + 413

When the oven is turned off (

m = 0

), its temperature is

413^\circ\text{F}

. Since it cools down at a constant rate of

12^\circ\text{F}

per minute, the slope of the cooling function is

-12

. Thus, the linear model is

U(m) = -12m + 413

.

Key Concept

Interpreting and modifying parameters of linear models in multi-stage contextual scenarios.

Alternative Method

Instead of writing the heating function explicitly, you can calculate the total temperature increase directly: the temperature rises by

18.5 \times 1.20 = 22.2^\circ\text{F}

per minute for

15

minutes, which is a total increase of

22.2 \times 15 = 333^\circ\text{F}

. Adding this increase to the upgraded starting temperature of

72 + 8 = 80^\circ\text{F}

gives the peak temperature of

80 + 333 = 413^\circ\text{F}

. Since the cooling phase is linear with a slope of

-12

and a y-intercept of

413

, the function is immediately determined.

Estimated Time:3m 0s

Question 1453Question →

To rent an electric scooter, a rider pays a flat unlocking fee plus a fee for each minute of the ride. The total cost $C$ , in dollars, for a ride of $m$ minutes is given by the equation $C = 0.15m + 1.20$ . Which of the following is the best interpretation of the number $1.20$ in this context?

The cost, in dollars, for each minute of the scooter ride

The flat unlocking fee, in dollars, to rent the scooter

The total cost, in dollars, of a scooter ride that lasts 1 minute

The cost, in dollars, for a ride that lasts 15 minutes

Show answer & explanation

Answer: The flat unlocking fee, in dollars, to rent the scooter

Answer

The flat unlocking fee, in dollars, to rent the scooter

In the linear model

C = 0.15m + 1.20

, the constant term

1.20

represents the value of

C

when

m = 0

. Since

m

represents the number of minutes,

m = 0

corresponds to the beginning of the rental. Therefore,

1.20

represents the initial flat unlocking fee, in dollars, to rent the scooter.

Step-by-Step Solution

1

Identify the component of the linear equation

C = 0.15m + 1.20

that corresponds to the number

1.20

.

The number

1.20

is the constant term (y-intercept) of the equation.

A linear equation in slope-intercept form is

y = mx + b

, where

b

is the y-intercept (the value of

y

when

x = 0

).

2

Determine the physical meaning of

m = 0

in the given context.

m = 0

represents a ride that lasts

0

minutes, meaning no time has elapsed yet.

Evaluating the relationship at the initial state helps identify the physical meaning of the y-intercept.

3

Substitute

m = 0

into the equation to find the corresponding cost.

C = 0.15(0) + 1.20 = 1.20

dollars.

This shows that the initial cost, or flat unlocking fee before starting the ride, is

1.20

dollars.

Key Concept

Interpreting the y-intercept of a linear relationship in context

Question 1454Question →

During a chemical reaction, the temperature $T$ , in degrees Celsius, of a solution $s$ seconds after the reaction begins is modeled by the equation $T = -0.04(s - 150) + 92$ , where $150 \le s \le 900$ . According to the model, how many minutes does it take for the temperature of the solution to decrease by $12$ degrees Celsius?

Show answer & explanation

Answer: 5

Answer

5

To find the number of minutes it takes for the temperature to decrease by

12^\circ\text{C}

, we first determine the rate of temperature change from the linear model. The equation is given in the form

T = m(s - s_0) + T_0

, where the slope

m = -0.04

represents the rate of change of temperature in degrees Celsius per second. Thus, the temperature decreases at a rate of

0.04^\circ\text{C}

per second. To achieve a total decrease of

12^\circ\text{C}

, the time in seconds required is

\frac{12}{0.04} = 300

seconds. Converting

300

seconds to minutes gives

\frac{300}{60} = 5

minutes.

Step-by-Step Solution

1

Identify the rate of change from the linear equation.

The rate of temperature decrease is

0.04^\circ\text{C}

per second.

The slope of the linear equation

T = -0.04(s - 150) + 92

is

-0.04

, which represents a change of

-0.04^\circ\text{C}

for every

1

second increase in time.

2

Calculate the time in seconds for a decrease of

12^\circ\text{C}

.

300

seconds

Divide the target temperature change of

-12^\circ\text{C}

by the rate of change of

-0.04^\circ\text{C}

per second:

\frac{-12}{-0.04} = 300

seconds.

3

Convert the time from seconds to minutes.

5

minutes

Since there are

60

seconds in

1

minute, divide

300

seconds by

60

:

\frac{300}{60} = 5

minutes.

Key Concept

Interpreting Linear Relationships in Context

Question 1455Question →

A nutritionist is designing a meal plan containing $x$ grams of protein and $y$ grams of carbohydrates. The meal plan must satisfy the following system of inequalities:

\begin{aligned} y &\ge 1.5x + 15 \\ 3x + 2y &\le 120 \end{aligned}

What is the maximum possible number of grams of protein, $x$ , that can be included in the meal plan?

Show answer & explanation

Answer: 15

Answer

The maximum possible number of grams of protein that can be included is 15.

To find the maximum possible value of

x

, we determine the region defined by the system of inequalities. The system restricts the values to the region above the line

y = 1.5x + 15

and below the line

3x + 2y = 120

. Since the first inequality limits

y

from below and the second limits

y

from above, the feasible region narrows as

x

increases, terminating at the intersection of the two boundary lines. Substituting

y = 1.5x + 15

into

3x + 2y = 120

gives

3x + 2(1.5x + 15) = 120

. Simplifying this yields

3x + 3x + 30 = 120

, which simplifies further to

6x = 90

, giving

x = 15

. Thus, the maximum value of

x

is 15.

Step-by-Step Solution

1

Identify the boundary lines of the system of inequalities.

The boundary lines are

y = 1.5x + 15

and

3x + 2y = 120

.

The maximum value of

x

under these linear constraints occurs at the intersection of the boundary lines of the feasible region.

2

Substitute the expression for

y

from the first boundary equation into the second equation.

3x + 2(1.5x + 15) = 120

This allows us to solve for

x

by eliminating

y

.

3

Simplify the equation and solve for

x

.

3x + 3x + 30 = 120 \implies 6x + 30 = 120 \implies 6x = 90 \implies x = 15

.

Solving the linear equation gives the

x

-coordinate of the intersection point.

4

Verify that this point lies in the feasible region and represents the maximum possible value of

x

.

At

x=15

,

y=37.5

. Since

y \ge 1.5x + 15

restricts the region above the line and

3x + 2y \le 120

restricts it below the line, the region lies to the left of the intersection point

(15, 37.5)

. Thus, the maximum value of

x

is 15.

Confirming the geometry of the feasible region ensures the intersection point is indeed the maximum value.

Key Concept

Solving systems of linear inequalities to find the boundaries and extreme values of a feasible region.

Question 1456Question →

y = x^2 - 5x + 8

y = 2x + 2

The system of equations above has two real solutions, $(x_1, y_1)$ and $(x_2, y_2)$ . If $y_1 > y_2$ , what is the value of $x_1 - x_2$ ?

-5

5

7

10

Show answer & explanation

Answer: 5

Answer

The value of the difference between the two x-coordinates is 5.

To solve the system, substitute

y = 2x + 2

into

y = x^2 - 5x + 8

to obtain

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

. Subtracting

2x + 2

from both sides gives the quadratic equation

x^2 - 7x + 6 = 0

. Factoring this equation yields

(x - 6)(x - 1) = 0

, which gives the x-coordinates

x = 6

and

x = 1

. Substituting these back into

y = 2x + 2

gives the corresponding y-coordinates:

y = 14

when

x = 6

, and

y = 4

when

x = 1

. Thus, the two solutions are

(6, 14)

and

(1, 4)

. Because

y_1 > y_2

, we must have

(x_1, y_1) = (6, 14)

and

(x_2, y_2) = (1, 4)

. The value of

x_1 - x_2

is therefore

6 - 1 = 5

.

Step-by-Step Solution

1

Substitute the expression for

y

from the second equation into the first equation.

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

This substitution reduces the system of two equations to a single quadratic equation in terms of

x

.

2

Rearrange the quadratic equation into standard form by subtracting

2x

and

2

from both sides.

x^2 - 7x + 6 = 0

Putting the equation in standard form is necessary to factor it and find its roots.

3

Factor the quadratic equation to find the two possible values of

x

.

(x - 6)(x - 1) = 0

, which gives

x = 6

or

x = 1

.

Factoring allows us to find the x-coordinates of the points where the two graphs intersect.

4

Substitute the x-values back into the linear equation

y = 2x + 2

to find their corresponding y-values.

For

x = 6

,

y = 2(6) + 2 = 14

. For

x = 1

,

y = 2(1) + 2 = 4

. The two solutions are

(6, 14)

and

(1, 4)

.

Finding the y-values helps identify which coordinate pair corresponds to

(x_1, y_1)

and which to

(x_2, y_2)

using the given condition.

5

Apply the condition

y_1 > y_2

to assign the variables and calculate

x_1 - x_2

.

Since

14 > 4

, the solution with the larger y-value is

(x_1, y_1) = (6, 14)

and the other is

(x_2, y_2) = (1, 4)

. Thus,

x_1 - x_2 = 6 - 1 = 5

.

This calculation yields the final requested value.

Key Concept

Solving a system of a linear equation and a quadratic equation by substitution.

Question 1457Question →

Unlike other cephalopods that rely purely on camouflage, the mimic octopus (*Thaumoctopus mimicus*) employs a dynamic predator-deterrence strategy ______ by contorting its body and adjusting its swimming style, it can impersonate venomous species such as lionfish and sea snakes.

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

strategy, by

strategy; which, by

strategy: by

strategy by

Show answer & explanation

Answer: strategy: by

Answer

The correct punctuation is a colon followed by the preposition 'by' to introduce an explanation of the predator-deterrence strategy.

The correct option is the one that uses a colon to link the independent clauses. The clause preceding the blank is grammatically complete, and the clause following the blank explains the specific nature of the octopus's strategy. A colon is standard punctuation for introducing an explanation or illustration in this context.

Step-by-Step Solution

1

Identify the grammatical structure of the clauses before and after the blank.

The clause before the blank is independent, and the clause after the blank is also independent.

Determining clause independence helps identify which punctuation marks can be used to join them.

2

Evaluate the logical relationship between the two independent clauses.

The second clause explains and elaborates on the predator-deterrence strategy mentioned in the first clause.

A colon is the appropriate punctuation mark to introduce an explanation or elaboration that follows an independent clause.

Key Concept

Using colons to link independent clauses when the second clause explains or illustrates the first.

Question 1458Question →

If $x(x - 9) = -20$ , which of the following is a possible value of $x$ ?

-5

-4

5

9

Show answer & explanation

Answer: 5

Answer

5

To solve the equation

x(x - 9) = -20

, first distribute the

x

to get

x^2 - 9x = -20

. Next, rewrite the equation in standard form by adding

20

to both sides, which gives

x^2 - 9x + 20 = 0

. Factoring the quadratic expression yields

(x - 4)(x - 5) = 0

. Applying the zero product property gives two possible solutions:

x = 4

and

x = 5

. Therefore, the value

5

is a possible value of

x

.

Step-by-Step Solution

1

Distribute

x

on the left side of the equation.

x^2 - 9x = -20

Expanding the product allows the quadratic equation to be rewritten in standard form.

2

Add

20

to both sides to set the equation to standard form

ax^2 + bx + c = 0

.

x^2 - 9x + 20 = 0

Setting the quadratic equation to equal zero is necessary to solve by factoring or using the quadratic formula.

3

Factor the quadratic trinomial by finding two integers that multiply to

20

and add to

-9

.

(x - 4)(x - 5) = 0

The numbers

-4

and

-5

satisfy these conditions, allowing the expression to be factored.

4

Set each factor equal to zero to find the possible values of

x

.

x = 4

or

x = 5

According to the zero product property, if the product of two factors is zero, then at least one of the factors must be zero.

Key Concept

Solving quadratic equations by factoring

Question 1459Question →

Consider the system of inequalities below:

\begin{aligned} 3x + y &> 5 \\ x - 2y &> 4 \end{aligned}

Which of the following coordinate pairs $(x, y)$ is a solution to the system?

(3, 1)

(2, -1)

(4, -2)

(3, 2)

Show answer & explanation

Answer: $(4, -2)$

Answer

(4, -2)

The coordinate pair

(4, -2)

is the correct answer because substituting these values into both inequalities of the system produces true statements:

3(4) + (-2) = 10 > 5

and

4 - 2(-2) = 8 > 4

.

Step-by-Step Solution

1

Substitute the coordinates of the candidate point into the first inequality,

3x + y > 5

.

For

(4, -2)

, we get

3(4) + (-2) = 12 - 2 = 10

. Since

10 > 5

, the first inequality is satisfied.

A coordinate pair must satisfy both inequalities in the system to be a solution.

2

Substitute the coordinates of the candidate point into the second inequality,

x - 2y > 4

.

For

(4, -2)

, we get

4 - 2(-2) = 4 + 4 = 8

. Since

8 > 4

, the second inequality is also satisfied.

Since both inequalities are true for

(4, -2)

, it is a valid solution to the system.

Key Concept

A coordinate pair

(x, y)

is a solution to a system of linear inequalities if and only if it makes all inequalities in the system true when substituted.

Estimated Time:1m 30s

Question 1460Question →

A chemist needs to mix a $10\%$ acid solution with a $30\%$ acid solution to create a $100$ -milliliter mixture. If the final mixture must be $18\%$ acid, how many milliliters of the $10\%$ acid solution should the chemist use?

40

60

70

80

Show answer & explanation

Answer: 60

Answer

60 milliliters

The correct answer is 60 milliliters. Defining

x

as the volume of the

10\%

solution and

y

as the volume of the

30\%

solution gives the system of equations

x + y = 100

and

0.10x + 0.30y = 18

. Substituting

y = 100 - x

into the second equation yields

0.10x + 30 - 0.30x = 18

. Simplifying this equation gives

-0.20x = -12

, which simplifies to

x = 60

.

Step-by-Step Solution

1

Define variables for the volume of each solution and set up the system of equations representing the total volume and the total amount of pure acid.

Let

x

be the number of milliliters of the

10\%

acid solution, and let

y

be the number of milliliters of the

30\%

acid solution. The system is:

\begin{aligned} x + y &= 100 \\ 0.10x + 0.30y &= 18 \end{aligned}

To represent the physical relationships between the two solutions mathematically.

2

Solve the first equation for

y

in terms of

x

and substitute this expression into the second equation.

y = 100 - x

0.10x + 0.30(100 - x) = 18

To reduce the system to a single linear equation in terms of

x

.

3

Distribute the coefficients, combine like terms, and solve for

x

.

0.10x + 30 - 0.30x = 18

-0.20x + 30 = 18

-0.20x = -12

x = 60

To isolate

x

and determine the volume of the

10\%

acid solution.

Key Concept

Solving systems of linear equations in two variables using substitution or elimination.

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer