All practice questions

Question 1421Question →

While the international coalition of astronomers working on the new telescope array had expected to capture high-resolution images of the newly discovered galaxy cluster during the summer solstice, _______ efforts were ultimately thwarted by a series of severe atmospheric disturbances and unexpected software anomalies that disrupted the sensitive instruments.

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

their

its

they're

its'

Show answer & explanation

Answer: its

Answer

its

The singular possessive pronoun 'its' correctly agrees in number with the singular collective noun antecedent 'coalition' and correctly shows possession over 'efforts'.

Step-by-Step Solution

1

Identify the antecedent of the pronoun needed in the blank.

The pronoun modifies the noun 'efforts' and refers back to the subject 'the international coalition of astronomers working on the new telescope array'. The core noun of this subject is 'coalition'.

To ensure correct agreement, we must find the primary noun that the pronoun refers to, ignoring any intervening prepositional phrases.

2

Determine the grammatical number of the antecedent.

The noun 'coalition' is a singular collective noun, representing a single unified entity.

The pronoun must agree in number with its antecedent, meaning a singular pronoun is required.

3

Determine the required pronoun case and form.

The blank modifies the noun 'efforts' ('_______ efforts'), indicating ownership or possession. Therefore, a possessive pronoun is required. The singular possessive pronoun is 'its'.

Using the correct case and number ensures the pronoun conforms to the grammatical rules of the sentence.

Key Concept

Pronoun-Antecedent Agreement (Collective Nouns)

Question 1422Question →

For all $x > 0$ , which of the following expressions is equivalent to $\frac{x^2 - 8}{x^{4/3} + 2x^{2/3} + 4} + \frac{x - 4x^{1/3}}{x^{2/3} + 2x^{1/3}}$ ?

x^{2/3} + x^{1/3} + 4

x^{4/3} + x^{2/3} - 4

x^{2/3} + x^{1/3} - 4

x^{2/3} + x^{1/3} - 6

Show answer & explanation

Answer: $x^{2/3} + x^{1/3} - 4$

Answer

The expression

x^{2/3} + x^{1/3} - 4

The correct answer shows the sum of the simplified terms, which is

x^{2/3} + x^{1/3} - 4

. First, the numerator of the first term,

x^2 - 8

, can be written as a difference of cubes:

(x^{2/3})^3 - 2^3 = (x^{2/3} - 2)(x^{4/3} + 2x^{2/3} + 4)

. Dividing by the denominator leaves

x^{2/3} - 2

. Second, the second term can be factored by extracting

x^{1/3}

from both the numerator and denominator, leaving

\frac{x^{2/3} - 4}{x^{1/3} + 2}

. Factoring the numerator as a difference of squares,

(x^{1/3} - 2)(x^{1/3} + 2)

, and dividing by the denominator leaves

x^{1/3} - 2

. Adding the two simplified parts,

(x^{2/3} - 2) + (x^{1/3} - 2)

, gives

x^{2/3} + x^{1/3} - 4

.

Step-by-Step Solution

1

Simplify the first term,

\frac{x^2 - 8}{x^{4/3} + 2x^{2/3} + 4}

x^{2/3} - 2

Rewrite

x^2 - 8

as

(x^{2/3})^3 - 2^3

and expand using the difference of cubes identity:

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

, where

a = x^{2/3}

and

b = 2

. The term

x^{4/3} + 2x^{2/3} + 4

in the numerator and denominator cancels out.

2

Simplify the second term,

\frac{x - 4x^{1/3}}{x^{2/3} + 2x^{1/3}}

x^{1/3} - 2

Factor out

x^{1/3}

from both the numerator and denominator to get

\frac{x^{2/3} - 4}{x^{1/3} + 2}

. Then rewrite

x^{2/3} - 4

as

(x^{1/3})^2 - 2^2

and expand using the difference of squares identity:

a^2 - b^2 = (a - b)(a + b)

, where

a = x^{1/3}

and

b = 2

. The term

x^{1/3} + 2

in the numerator and denominator cancels out.

3

Sum the two simplified terms

x^{2/3} + x^{1/3} - 4

Add the two simplified expressions:

(x^{2/3} - 2) + (x^{1/3} - 2) = x^{2/3} + x^{1/3} - 4

.

Key Concept

Simplification of rational expressions involving fractional exponents, difference of cubes, and difference of squares.

Estimated Time:2m 0s

Question 1423Question →

Glacial surfaces are not entirely lifeless zones; instead, they host vibrant micro-ecosystems known as cryoconite holes. These water-filled depressions form when dark, windblown dust absorbs solar radiation and melts the surrounding ice. Researchers have discovered that these holes act as crucial carbon sinks ______ the diverse microbial communities residing within them actively photosynthesize and accumulate organic matter.

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

, the

, but the

; the

the

Show answer & explanation

Answer: ; the

Answer

The choice that uses a semicolon correctly links the two independent clauses without introducing a comma splice, run-on, or illogical transition.

The correct choice appropriately uses a semicolon to separate two independent clauses. The clause before the blank ('Researchers have discovered that these holes act as crucial carbon sinks') and the clause after the blank ('the diverse microbial communities residing within them actively photosynthesize and accumulate organic matter') are both independent. A semicolon is a standard grammatical way to link two closely related independent clauses.

Step-by-Step Solution

1

Analyze the clause structure before the blank.

The clause 'Researchers have discovered that these holes act as crucial carbon sinks' is independent.

Identifying that this clause can stand alone as a complete sentence helps determine the required punctuation.

2

Analyze the clause structure after the blank.

The clause 'the diverse microbial communities residing within them actively photosynthesize and accumulate organic matter' is also independent.

Because both clauses are independent, they cannot be joined with just a comma or no punctuation.

3

Determine the logical relationship between the two clauses to select the correct punctuation or conjunction.

The second clause explains how or why the holes act as carbon sinks. A semicolon is a grammatically correct way to join these two independent clauses.

A semicolon links the independent clauses cleanly without creating a run-on sentence, comma splice, or introducing an incorrect coordinating conjunction like 'but'.

Key Concept

Clause Boundaries and Linking

Question 1424Question →

For all $x > 0$ , the expression $\left( \frac{x^{4/3} + 4x^{2/3} + 16}{x^{2/3} + 2x^{1/3} + 4} + 2x^{1/3} \right)^3 - x^2 - 12x^{4/3} - 48x^{2/3}$ is equivalent to a constant $C$ . What is the value of $C$ ?

Show answer & explanation

Answer: 64

Answer

The constant value is 64.

The expression inside the parentheses simplifies to

x^{2/3} + 4

after factoring the numerator as

(x^{2/3} + 2x^{1/3} + 4)(x^{2/3} - 2x^{1/3} + 4)

and canceling the common factor in the denominator. Cubing

x^{2/3} + 4

yields

x^2 + 12x^{4/3} + 48x^{2/3} + 64

. Subtracting the remaining terms

x^2 + 12x^{4/3} + 48x^{2/3}

from this expansion results in the constant value 64.

Step-by-Step Solution

1

Substitute

u = x^{1/3}

into the expression to simplify the fractional exponents.

The terms become

x^{1/3} = u

,

x^{2/3} = u^2

,

x^{4/3} = u^4

, and

x^2 = u^6

. The expression inside the parentheses is rewritten as

\frac{u^4 + 4u^2 + 16}{u^2 + 2u + 4} + 2u

.

Using a temporary variable

u

simplifies the algebraic factoring and manipulation of terms with fractional exponents.

2

Factor the numerator

u^4 + 4u^2 + 16

by completing the square.

u^4 + 4u^2 + 16 = (u^2 + 4)^2 - 4u^2 = (u^2 + 2u + 4)(u^2 - 2u + 4)

.

Expressing the quartic polynomial as a difference of squares allows it to be factored into two quadratic polynomials.

3

Simplify the rational expression and add

2u

.

\frac{(u^2 + 2u + 4)(u^2 - 2u + 4)}{u^2 + 2u + 4} + 2u = (u^2 - 2u + 4) + 2u = u^2 + 4

.

Canceling the common factor

u^2 + 2u + 4

in the numerator and denominator simplifies the expression inside the parentheses to

u^2 + 4

.

4

Substitute

u = x^{1/3}

back into

u^2 + 4

and cube the expression.

(x^{2/3} + 4)^3 = (x^{2/3})^3 + 3(x^{2/3})^2(4) + 3(x^{2/3})(16) + 64 = x^2 + 12x^{4/3} + 48x^{2/3} + 64

.

Applying the binomial expansion formula

(A + B)^3 = A^3 + 3A^2B + 3AB^2 + B^3

expands the cubed expression.

5

Subtract the remaining terms from the expanded expression.

(x^2 + 12x^{4/3} + 48x^{2/3} + 64) - x^2 - 12x^{4/3} - 48x^{2/3} = 64

.

Subtracting the variable terms cancels them out entirely, leaving the constant value 64.

Key Concept

Factoring quartic polynomials using the difference of squares and simplifying rational expressions with fractional exponents.

Question 1425Question →

A landscaping company is planting $x$ maple trees and $y$ pine trees in a park. The number of trees of each type must satisfy the system of inequalities below:

\begin{aligned} 30x + 40y &\le 360 \\ x + y &\ge 10 \\ x &\ge 4 \end{aligned}

What is the maximum number of pine trees the company can plant?

Show answer & explanation

Answer: 6

Answer

The maximum number of pine trees the company can plant is 6.

To find the maximum number of pine trees,

y

, we look at the boundary constraints. The constraint

x \ge 4

states that at least

4

maple trees must be planted. Since planting fewer maple trees leaves more of the budget for pine trees, we minimize

x

by setting

x = 4

. Substituting this value into the budget inequality

30x + 40y \le 360

gives

120 + 40y \le 360

. Solving for

y

yields

40y \le 240

, which simplifies to

y \le 6

. We then verify that the solution

(4, 6)

satisfies the total tree constraint

x + y \ge 10

, which it does since

4 + 6 = 10

. Thus, the maximum number of pine trees is 6.

Step-by-Step Solution

1

Substitute the minimum possible value of

x

into the first inequality.

Since

x \ge 4

, the smallest possible value for

x

is

4

. Substituting

x = 4

into

30x + 40y \le 360

yields:

30(4) + 40y \le 360

120 + 40y \le 360

To maximize the value of

y

under the resource constraint, we must minimize the value of

x

.

2

Solve the inequality for

y

.

40y \le 240

y \le 6

This establishes that the maximum possible value for

y

based on the budget constraint is

6

.

3

Verify that

(4, 6)

satisfies all inequalities in the system.

Checking the second inequality:

x + y \ge 10 \implies 4 + 6 = 10 \ge 10

, which is true. The third inequality

x \ge 4 \implies 4 \ge 4

is also true.

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

Key Concept

To find the maximum value of a variable in a system of inequalities with constraints, analyze the boundary lines and the intersection points of the feasible region.

Question 1426Question →

If $3^{x - 2} = 81$ , what is the value of $x$ ?

2

4

6

29

Show answer & explanation

Answer: 6

Answer

6

To solve the equation

3^{x - 2} = 81

, we first express the number 81 as a power of 3, which is

3^4

. This gives us

3^{x - 2} = 3^4

. Since the bases are equal, we can set their exponents equal to each other, resulting in the equation

x - 2 = 4

. Solving for

x

by adding 2 to both sides gives the correct value of 6.

Step-by-Step Solution

1

Express both sides of the equation with a common base of 3.

3^{x - 2} = 3^4

Since 81 is equal to

3 \times 3 \times 3 \times 3

, it can be written as

3^4

.

2

Set the exponents equal to each other because the bases are now the same.

x - 2 = 4

If

b^y = b^z

for a positive base

b \neq 1

, then

y = z

.

3

Solve the linear equation for

x

by adding 2 to both sides of the equation.

x = 6

Adding 2 to both sides isolates the variable

x

.

Key Concept

Solving exponential equations by expressing both sides with a common base and equating their exponents.

Question 1427Question →

In the $xy$ -plane, the graph of a line $l$ passes through the points $(0, 1)$ and $(3, 5)$ . If another point on line $l$ has coordinates $(t, 9)$ , what is the value of $t$ ?

Show answer & explanation

Answer: 6

Answer

The value of

t

is

6

.

The slope of line

l

is

m = \frac{5 - 1}{3 - 0} = \frac{4}{3}

. Using the y-intercept

(0, 1)

, the equation of the line is

y = \frac{4}{3}x + 1

. Setting

y = 9

gives

9 = \frac{4}{3}t + 1

. Subtracting 1 from both sides gives

8 = \frac{4}{3}t

. Multiplying both sides by

\frac{3}{4}

yields

t = 6

.

Step-by-Step Solution

1

Calculate the slope of line

l

using the slope formula

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

with the points

(0, 1)

and

(3, 5)

.

m = \frac{5 - 1}{3 - 0} = \frac{4}{3}

The slope of a line represents its constant rate of change and is needed to determine the line's equation.

2

Write the equation of the line in slope-intercept form,

y = mx + b

, using the slope

m = \frac{4}{3}

and the y-intercept

b = 1

(from the point

(0, 1)

).

y = \frac{4}{3}x + 1

The slope-intercept equation defines the relationship between the

x

- and

y

-coordinates of any point on the line.

3

Substitute the point

(t, 9)

into the line's equation and solve for

t

.

9 = \frac{4}{3}t + 1 \implies 8 = \frac{4}{3}t \implies t = 6

Since the point lies on the line, its coordinates must satisfy the line's equation.

Key Concept

Determining the equation of a linear function from a graph or points and evaluating it for a given value.

Question 1428Question →

A circle in the $xy$ -plane is defined by the equation $(x - 2)^2 + (y + 1)^2 = 10$ . The line $y = 3x + k$ , where $k$ is a constant, is tangent to the circle. If $k < 0$ , what is the value of $k$ ?

-3

-5

-17

3

Show answer & explanation

Answer: -17

Answer

The correct value of

k

is

-17

.

Substituting the line equation

y = 3x + k

into the circle equation yields

(x - 2)^2 + (3x + k + 1)^2 = 10

. Expanding and writing this in standard form gives

10x^2 + (6k + 2)x + (k^2 + 2k - 5) = 0

. For the line to be tangent to the circle, the quadratic equation must have exactly one real solution, meaning its discriminant must be

0

. Setting the discriminant

\Delta = (6k+2)^2 - 4(10)(k^2 + 2k - 5)

to

0

and simplifying gives

-4k^2 - 56k + 204 = 0

. Dividing by

-4

yields

k^2 + 14k - 51 = 0

, which factors as

(k+17)(k-3)=0

. Since

k < 0

, the value of

k

must be

-17

.

Step-by-Step Solution

1

Substitute the linear equation

y = 3x + k

into the circle equation.

(x - 2)^2 + (3x + k + 1)^2 = 10

To find the coordinates where the line and the circle intersect.

2

Expand both squared terms and simplify the equation to standard quadratic form

Ax^2 + Bx + C = 0

.

10x^2 + (6k + 2)x + (k^2 + 2k - 5) = 0

To write the system as a single quadratic equation in terms of

x

.

3

Set the discriminant of the quadratic equation to zero.

(6k + 2)^2 - 4(10)(k^2 + 2k - 5) = 0

which simplifies to

-4k^2 - 56k + 204 = 0

Since the line is tangent to the circle, there must be exactly one intersection point, which means the quadratic equation must have exactly one real solution.

4

Divide the simplified equation by

-4

and solve for

k

.

k^2 + 14k - 51 = 0 \implies (k + 17)(k - 3) = 0 \implies k = -17 \text{ or } k = 3

To find the values of

k

that make the line tangent to the circle.

5

Apply the given constraint

k < 0

.

k = -17

The problem specifies that

k

must be a negative value.

Key Concept

Solving nonlinear systems of equations involving circles and lines by substitution and using the discriminant to determine tangency.

Estimated Time:2m 30s

Question 1429Question →

A deep space communications satellite transmits a telemetry data file to a ground station on Earth. The remaining size of the file to be received at the ground station, $S$ , in megabytes (MB), can be modeled by the equation $S = 850 - 4.5(t - 18)$ , where $t$ is the number of seconds since the satellite initiated its transmission sequence, and $t \ge 18$ . Which of the following is the best interpretation of the number $18$ in this context?

The rate, in megabytes per second, at which the data is received by the ground station.

The size, in megabytes, of the telemetry file when the ground station begins receiving it.

The number of seconds after the transmission sequence is initiated before the ground station begins receiving the file.

The total number of seconds needed to fully transmit the telemetry file.

Show answer & explanation

Answer: The number of seconds after the transmission sequence is initiated before the ground station begins receiving the file.

Answer

The number of seconds after the transmission sequence is initiated before the ground station begins receiving the file.

The model

S = 850 - 4.5(t - 18)

is valid for

t \ge 18

. Substituting

t = 18

into the equation yields

S = 850

megabytes, which is the total size of the file before any data has been received. As

t

increases beyond

18

, the remaining file size decreases at a rate of

4.5

megabytes per second. Therefore, the first

18

seconds after the transmission sequence is initiated represent the time delay before the ground station starts receiving the file.

Step-by-Step Solution

1

Analyze the structure of the linear equation

S = 850 - 4.5(t - 18)

in context.

The variable

S

represents the remaining file size in megabytes, and

t

represents the time in seconds since the sequence was initiated. The model is defined only for

t \ge 18

.

Understanding the variables and constraints is the first step to interpreting the components of the equation.

2

Evaluate the equation at the boundary value

t = 18

.

When

t = 18

,

S = 850 - 4.5(18 - 18) = 850

megabytes.

This determines the starting state of the data reception modeled by the equation.

3

Analyze how

S

changes as

t

increases beyond

18

.

For every second

t

increases beyond

18

,

S

decreases by

4.5

megabytes.

This confirms that the transmission starts at

t = 18

seconds and proceeds at a rate of

4.5

megabytes per second, meaning the first

18

seconds represent the delay before data reception starts.

Key Concept

Interpreting Linear Relationships in Context

Question 1430Question →

For the system of equations shown below,

(x, y)

is the unique solution:

\begin{aligned} \frac{3}{4}(2x - y) &= \frac{2}{3}(x + 2y) \\ \frac{1}{2}(x - y) + \frac{5}{6}(2x + y) &= 11.5 \end{aligned}

What is the value of

x + y

?

Show answer & explanation

Answer: 7

Answer

The value of

x + y

is 7.

To find the value of

x + y

, we first simplify the first equation by multiplying both sides by the least common multiple of

3

and

4

, which is

12

. This yields

9(2x - y) = 8(x + 2y)

, which simplifies to

18x - 9y = 8x + 16y

, and further simplifies to

10x = 25y

, or

x = 2.5y

. Next, we substitute

x = 2.5y

into the second equation:

\frac{1}{2}(2.5y - y) + \frac{5}{6}(2(2.5y) + y) = 11.5

. Simplifying the terms gives

0.75y + 5y = 11.5

, or

5.75y = 11.5

, which yields

y = 2

. Substituting

y = 2

back into

x = 2.5y

gives

x = 5

. Thus, the value of

x + y

is

5 + 2 = 7

.

Step-by-Step Solution

1

Clear the fractions in the first equation by multiplying by the least common multiple of the denominators.

10x = 25y

, which simplifies to

x = 2.5y

.

To express one variable in terms of the other for substitution.

2

Substitute the expression for

x

into the second equation and solve for

y

.

y = 2

To find the numerical value of one of the variables.

3

Substitute the value of

y

back into the simplified first equation to find

x

.

x = 5

To find the numerical value of the remaining variable.

4

Add the values of

x

and

y

to find

x + y

.

7

To answer the specific question asked.

Key Concept

Solving systems of linear equations using algebraic simplification and substitution

Question 1431Question →

For all $x > 2$ , which of the following is equivalent to the expression $\frac{3x^2 - 5x - 2}{x^2 - 4} - \frac{x - 1}{x + 2}$ ?

\frac{2x}{x + 2}

\frac{2x + 2}{x + 2}

\frac{2x + 2}{x - 2}

\frac{x + 1}{x + 2}

Show answer & explanation

Answer: $\frac{2x + 2}{x + 2}$

Answer

The expression is equivalent to

\frac{2x + 2}{x + 2}

.

The correct answer is obtained by first factoring the first term:

\frac{3x^2 - 5x - 2}{x^2 - 4} = \frac{(3x + 1)(x - 2)}{(x - 2)(x + 2)}

. Canceling the common factor of

x - 2

yields

\frac{3x + 1}{x + 2}

. Subtracting the second term gives

\frac{3x + 1 - (x - 1)}{x + 2} = \frac{3x + 1 - x + 1}{x + 2} = \frac{2x + 2}{x + 2}

.

Step-by-Step Solution

1

Factor the numerator and the denominator of the first term of the expression.

The numerator factors as

3x^2 - 5x - 2 = (3x + 1)(x - 2)

, and the denominator factors as

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

.

This allows common factors in the numerator and denominator to be identified and canceled.

2

Simplify the first term by canceling the common factor

(x - 2)

for

x > 2

.

The first term simplifies to

\frac{3x + 1}{x + 2}

.

For

x > 2

,

x - 2 \neq 0

, so we can divide both numerator and denominator by

x - 2

to simplify the fraction.

3

Subtract the second term from the simplified first term.

The expression becomes

\frac{3x + 1}{x + 2} - \frac{x - 1}{x + 2} = \frac{(3x + 1) - (x - 1)}{x + 2}

.

Since both fractions have the same denominator,

x + 2

, their numerators can be subtracted directly.

4

Distribute the negative sign in the numerator and combine like terms.

\frac{3x + 1 - x + 1}{x + 2} = \frac{2x + 2}{x + 2}

.

Distributing the subtraction to both terms in the parenthesis

(x - 1)

gives

-x + 1

. Combining

3x - x

yields

2x

, and

1 + 1

yields

2

.

Key Concept

Simplifying rational expressions by factoring and performing algebraic operations with common denominators.

Estimated Time:1m 30s

Question 1432Question →

A landscaping service uses a water tank to irrigate lawns. The volume of water, $V$ , in gallons, remaining in the tank after $n$ lawns have been irrigated is modeled by the equation $V = 850 - 25n$ . According to the model, by how many gallons does the volume of water in the tank decrease for each lawn that is irrigated?

Show answer & explanation

Answer: 25

Answer

The correct answer is 25, which represents the decrease in the volume of water in the tank, in gallons, for each lawn irrigated.

In the linear equation

V = 850 - 25n

, the coefficient of

n

is

-25

. This coefficient represents the rate of change of the volume of water with respect to the number of lawns irrigated. The negative sign shows that the volume is decreasing, and the magnitude, 25, indicates that the volume decreases by 25 gallons for each lawn irrigated.

Step-by-Step Solution

1

Identify the coefficient of the variable

n

in the equation

V = 850 - 25n

.

The coefficient of

n

is

-25

.

The coefficient of the independent variable in a linear equation represents the rate of change of the dependent variable.

2

Interpret the coefficient in terms of the real-world context.

The coefficient

-25

means the volume of water decreases by 25 gallons for each lawn irrigated.

The negative sign indicates a decrease, and the magnitude represents the amount of change per unit.

Key Concept

Interpreting the slope of a linear relationship in context.

Question 1433Question →

In 1948, physicist Richard Feynman introduced his namesake diagrams, which visually represented the complex mathematical expressions governing subatomic particle behavior. Initially met with skepticism by some traditionalists who favored algebraic formulas, these intuitive drawings ultimately revolutionized theoretical physics ______ they allowed researchers to bypass tedious calculations and directly visualize quantum interactions.

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

physics, they

physics; allowing

physics; they

physics they

Show answer & explanation

Answer: physics; they

Answer

physics; they

The correct answer is the option that ends with a semicolon followed by 'they'. The transition from the first part of the sentence to the second part requires a punctuation mark that can link two independent clauses. A semicolon is the correct choice because both the statement before the blank and the statement after the blank function as grammatically complete, independent sentences.

Step-by-Step Solution

1

Analyze the clauses surrounding the blank space.

The clause before the blank ('Initially met with skepticism by some traditionalists who favored algebraic formulas, these intuitive drawings ultimately revolutionized theoretical physics') is independent. The clause after the blank ('they allowed researchers to bypass tedious calculations and directly visualize quantum interactions') is also independent.

Determining the grammatical status of the clauses is necessary to choose the correct punctuation mark.

2

Select the punctuation that correctly links two independent clauses.

A semicolon is appropriate to link two closely related independent clauses without a coordinating conjunction.

Standard English conventions require a semicolon, colon, dash, or period with a coordinating conjunction to connect two independent clauses.

Key Concept

Semicolons are used to link two grammatically independent clauses that are closely related in thought.

Question 1434Question →

An environmental cleanup crew is removing a contaminant from a soil site. The remaining mass of the contaminant, $C$ , in kilograms, after $d$ days of treatment is modeled by the equation $C = 400 - p d$ , where $p$ is the daily removal rate, in kilograms per day, under the original protocol. Under a new treatment protocol, the daily removal rate is increased by $25\%$ , and the treatment time required to completely remove the contaminant is reduced by $8$ days. What was the daily removal rate, in kilograms per day, under the original protocol?

Show answer & explanation

Answer: 10

Answer

The daily removal rate under the original protocol was 10 kilograms per day.

Under the original protocol, complete removal of the

400

kg contaminant occurs when

C = 0

, giving a duration of

d = \frac{400}{p}

days. Under the new protocol, the removal rate increases by

25\%

to

1.25p

, and the duration is reduced by

8

days to

\frac{400}{p} - 8

. Because the total mass removed must still equal

400

kg, we write the equation

(1.25p)\left(\frac{400}{p} - 8\right) = 400

. Distributing

1.25p

yields

500 - 10p = 400

. Solving for

p

gives

10p = 100

, which simplifies to

p = 10

kilograms per day.

Step-by-Step Solution

1

Set

C = 0

in the original equation to represent complete removal.

0 = 400 - p d \implies d = \frac{400}{p}

Complete removal of the contaminant means that the remaining mass

C

is

0

kilograms.

2

Express the new daily removal rate and the new treatment duration using the given percentage increase and day reduction.

p_{\text{new}} = 1.25p

and

d_{\text{new}} = d - 8 = \frac{400}{p} - 8

The new protocol increases the daily removal rate by

25\%

and reduces the total treatment time by

8

days.

3

Set up the equation for complete removal under the new protocol using the new rate and duration.

400 - (1.25p) \left(\frac{400}{p} - 8\right) = 0

The total initial contaminant mass of

400

kilograms must be completely removed by the new daily rate over the new duration.

4

Solve the equation for

p

.

1.25p \left(\frac{400}{p} - 8\right) = 400 \implies 500 - 10p = 400 \implies 10p = 100 \implies p = 10

Distribute

1.25p

into the parentheses to eliminate the fraction, then isolate the variable

p

.

Key Concept

Interpreting the rate (slope) and intercepts of a linear relationship in context, and modeling variations of those parameters.

Question 1435Question →

In a certain video game, players earn points for completing quests and defeating bosses. Completing a quest earns $q$ points, and defeating a boss earns $b$ points. Leo completed $5$ quests and defeated $3$ bosses, earning a total of $250$ points. Maya completed $7$ quests and defeated $2$ bosses, earning a total of $240$ points. What is the value of $q$ ?

Show answer & explanation

Answer: 20

Answer

The value of

q

is

20

.

To find the value of

q

, we translate the given scenarios into a system of two linear equations:

5q + 3b = 250

and

7q + 2b = 240

. Multiplying the first equation by

2

gives

10q + 6b = 500

. Multiplying the second equation by

3

gives

21q + 6b = 720

. Subtracting the first equation from the second yields

(21q - 10q) + (6b - 6b) = 720 - 500

, which simplifies to

11q = 220

. Dividing both sides by

11

results in

q = 20

.

Step-by-Step Solution

1

Set up the system of linear equations from the given information.

5q + 3b = 250

and

7q + 2b = 240

To represent the points earned by Leo and Maya mathematically.

2

Multiply the first equation by

2

and the second equation by

3

.

10q + 6b = 500

and

21q + 6b = 720

To make the coefficients of

b

equal so they can be eliminated.

3

Subtract the first modified equation from the second modified equation.

11q = 220

To eliminate

b

and solve for

q

directly.

4

Divide both sides of the equation by

11

.

q = 20

To find the number of points earned per completed quest.

Key Concept

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

Question 1436Question →

In the quadratic equation $x^2 - kx + 9 = 0$ , $k$ is a positive constant. If the equation has exactly one real solution, what is the value of $k$ ?

Show answer & explanation

Answer: 6

Answer

6

For a quadratic equation

ax^2 + bx + c = 0

to have exactly one real solution, its discriminant must equal zero (

b^2 - 4ac = 0

). In the equation

x^2 - kx + 9 = 0

, the coefficients are

a = 1

,

b = -k

, and

c = 9

. Setting the discriminant to zero gives

(-k)^2 - 4(1)(9) = 0

, which simplifies to

k^2 - 36 = 0

. Solving this equation yields

k^2 = 36

, so

k = 6

or

k = -6

. Since

k

is a positive constant, we reject the negative root, leaving

k = 6

.

Step-by-Step Solution

1

Identify the condition for a quadratic equation to have exactly one real solution.

The discriminant of the quadratic equation must be equal to 0, which is represented by the formula

b^2 - 4ac = 0

.

The discriminant determines the number of real solutions of a quadratic equation. If the discriminant is 0, there is exactly one real solution.

2

Identify the coefficients

a

,

b

, and

c

of the given equation

x^2 - kx + 9 = 0

.

a = 1

,

b = -k

, and

c = 9

.

These coefficients are required to compute the value of the discriminant.

3

Set the discriminant equal to 0 and simplify the equation.

(-k)^2 - 4(1)(9) = 0 \Rightarrow k^2 - 36 = 0

.

Substituting the coefficients into the discriminant formula sets up the algebraic relationship to find

k

.

4

Solve for the positive constant

k

.

k^2 = 36 \Rightarrow k = 6

or

k = -6

. Since

k

must be positive,

k = 6

.

Solving the equation gives two possible values, but the negative solution is discarded because the problem specifies

k

is a positive constant.

Key Concept

Discriminant of a quadratic equation

Alternative Method

Alternatively, a quadratic equation has exactly one real solution if it can be written as a perfect square trinomial in the form

(x - d)^2 = 0

, which expands to

x^2 - 2dx + d^2 = 0

. Comparing this with

x^2 - kx + 9 = 0

, we get

d^2 = 9

and

2d = k

. Since

d^2 = 9

,

d

can be

3

or

-3

. Given that

k

is positive and

k = 2d

,

d

must also be positive, meaning

d = 3

. Substituting this back gives

k = 2(3) = 6

.

Estimated Time:45s

Question 1437Question →

Based on the rules of Standard English conventions, enter the punctuation mark that should be placed in the blank to grammatically connect the clauses in the passage.

Fill in the blanks below

In his research on temperate forest canopies, ecologist Liam O'Connor documented a fascinating behavior among certain tree species. Some trees in the dense study plots exhibited a high degree of crown shyness, maintaining a distinct boundary of air between their uppermost branchesothers grew in close proximity, their leaves overlapping to form a continuous canopy.

Show answer & explanation

Answer

A semicolon (;)

A semicolon is used to link two independent clauses that are closely related in thought but not joined by a coordinating conjunction. In the passage, the clause before the blank ('Some trees in the dense study plots exhibited a high degree of crown shyness, maintaining a distinct boundary of air between their uppermost branches') and the clause after the blank ('others grew in close proximity, their leaves overlapping to form a continuous canopy') are both independent clauses. Therefore, a semicolon is the correct grammatical choice.

Step-by-Step Solution

1

Identify the grammatical structure of the clauses surrounding the blank.

Both the clause before the blank ('Some trees in the dense study plots... uppermost branches') and the clause after the blank ('others grew in close proximity... continuous canopy') are independent clauses.

Determining if the clauses are independent or dependent is necessary to apply the correct punctuation rules.

2

Analyze the relationship between the two independent clauses.

The two clauses present contrasting but closely related observations about different trees' behaviors.

A colon is used when the second clause explains or illustrates the first, whereas a semicolon is appropriate for linking closely related but contrasting independent clauses without a coordinating conjunction.

3

Select the correct punctuation mark.

A semicolon is the standard way to connect these two independent clauses.

Using a semicolon avoids a comma splice and maintains the grammatical integrity of the sentence.

Key Concept

Using semicolons to connect independent clauses

Question 1438Question →

Although the board of directors, which consists of highly experienced representatives from five international technology subsidiaries, meets quarterly in Geneva to review the company's annual financial performance, _______ must ultimately seek shareholder approval before initiating any major corporate acquisitions or significant structural changes.

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

they

it

them

their

Show answer & explanation

Answer: it

Answer

The correct option is the pronoun 'it' because it is singular and in the subjective case, properly agreeing with the singular collective noun 'board' and functioning as the subject of the verb phrase.

The pronoun 'it' is singular and in the subjective case. It correctly agrees with the singular collective noun 'board' (the head of the noun phrase 'the board of directors') and functions as the subject of the clause.

Step-by-Step Solution

1

Identify the antecedent of the pronoun that needs to fill the blank.

The antecedent is the noun phrase 'the board of directors'. The head noun of this phrase is 'board', which is a singular collective noun.

Determining the correct head noun is necessary to establish pronoun-antecedent agreement in number.

2

Determine the grammatical case required for the pronoun in the blank.

The blank functions as the subject of the verb phrase 'must ultimately seek'. Therefore, a subjective case pronoun is required.

Grammatical case must match the pronoun's syntactic role in the clause.

3

Evaluate the choices to find a singular subjective pronoun.

The pronoun 'it' is singular and subjective, whereas the other options are plural or incorrect in case.

Selecting the grammatically correct option satisfies both agreement and case rules.

Key Concept

Pronoun-Antecedent Agreement and Case

Estimated Time:1m 0s

Question 1439Question →

In the quadratic equation $3x^2 - kx + 12 = 0$ , $k$ is a constant. The equation has two distinct real solutions, and the difference between these two solutions is less than $2$ . Which of the following describes all possible values of $k$ ?

-6\sqrt{5} < k < -12

or

12 < k < 6\sqrt{5}

-6\sqrt{5} < k < 6\sqrt{5}

-12\sqrt{2} < k < -12

or

12 < k < 12\sqrt{2}

k < -12

or

k > 12

Show answer & explanation

Answer: $-6\sqrt{5} < k < -12$ or $12 < k < 6\sqrt{5}$

Answer

The possible values of

k

are

-6\sqrt{5} < k < -12

or

12 < k < 6\sqrt{5}

The correct answer is found by combining two constraints. First, for the quadratic equation to have two distinct real solutions, the discriminant must be strictly positive:

(-k)^2 - 4(3)(12) > 0 \Rightarrow k^2 > 144

, which means

|k| > 12

. Second, the difference between the roots of

ax^2 + bx + c = 0

is

\frac{\sqrt{b^2 - 4ac}}{|a|}

. Here, the difference is

\frac{\sqrt{k^2 - 144}}{3}

. Setting this difference to be less than 2 gives

\frac{\sqrt{k^2 - 144}}{3} < 2 \Rightarrow \sqrt{k^2 - 144} < 6 \Rightarrow k^2 - 144 < 36 \Rightarrow k^2 < 180

, which means

|k| < 6\sqrt{5}

. Combining these inequalities yields

12 < |k| < 6\sqrt{5}

, which translates to

-6\sqrt{5} < k < -12

or

12 < k < 6\sqrt{5}

.

Step-by-Step Solution

1

Find the condition for the quadratic equation to have two distinct real solutions.

k^2 > 144

, which means

k < -12

or

k > 12

.

For the equation

3x^2 - kx + 12 = 0

to have two distinct real solutions, its discriminant

\Delta = b^2 - 4ac

must be strictly positive. Here,

\Delta = (-k)^2 - 4(3)(12) = k^2 - 144 > 0

.

2

Express the difference between the two solutions using the quadratic formula.

|x_1 - x_2| = \frac{\sqrt{k^2 - 144}}{3}

.

The solutions to the quadratic equation are given by

x = \frac{k \pm \sqrt{k^2 - 144}}{6}

. The difference between these solutions is

x_1 - x_2 = \frac{(k + \sqrt{k^2 - 144}) - (k - \sqrt{k^2 - 144})}{6} = \frac{2\sqrt{k^2 - 144}}{6} = \frac{\sqrt{k^2 - 144}}{3}

.

3

Apply the condition that the difference between the solutions is less than 2.

k^2 < 180

, which means

-6\sqrt{5} < k < 6\sqrt{5}

.

We set the difference expression to be less than 2:

\frac{\sqrt{k^2 - 144}}{3} < 2 \Rightarrow \sqrt{k^2 - 144} < 6

. Squaring both sides gives

k^2 - 144 < 36 \Rightarrow k^2 < 180

. Since

\sqrt{180} = 6\sqrt{5}

, this yields

-6\sqrt{5} < k < 6\sqrt{5}

.

4

Combine the inequalities from Step 1 and Step 3 to find the final overlapping range for

k

.

-6\sqrt{5} < k < -12

or

12 < k < 6\sqrt{5}

.

We must satisfy both

k^2 > 144

(for real solutions) and

k^2 < 180

(for the difference to be less than 2). This gives

144 < k^2 < 180

, which corresponds to the union of intervals

-6\sqrt{5} < k < -12

and

12 < k < 6\sqrt{5}

.

Key Concept

Using the discriminant and quadratic formula to analyze properties of roots under inequality constraints.

Question 1440Question →

A laboratory technician is cooling a liquid sample. The initial temperature of the sample is $80^\circ\text{C}$ . The technician uses a cooling program that decreases the temperature at a constant rate of $1.5^\circ\text{C}$ per minute. After $t$ minutes, the technician increases the cooling rate by $0.75^\circ\text{C}$ per minute and runs the cooling program for another $12$ minutes. If the final temperature of the sample is $44^\circ\text{C}$ , what is the value of $t$ ?

8

6

18

42

Show answer & explanation

Answer: 6

Answer

6

The correct answer is

6

. The initial temperature of the sample is

80^\circ\text{C}

. For the first

t

minutes, the temperature decreases by

1.5t^\circ\text{C}

. For the next

12

minutes, the rate is

1.5 + 0.75 = 2.25^\circ\text{C}

per minute, resulting in a temperature decrease of

2.25 \times 12 = 27^\circ\text{C}

. The final temperature equation is

80 - 1.5t - 27 = 44

, which simplifies to

53 - 1.5t = 44

. Solving for

t

gives

1.5t = 9

, or

t = 6

.

Step-by-Step Solution

1

Determine the cooling rate for both phases of the program.

The initial rate is

1.5^\circ\text{C}

per minute. The increased rate is

1.5 + 0.75 = 2.25^\circ\text{C}

per minute.

The rate in the second phase is increased by

0.75^\circ\text{C}

per minute from the initial rate.

2

Set up the linear equation representing the temperature change over time.

80 - 1.5t - 2.25(12) = 44

The final temperature is the initial temperature minus the temperature drops during each phase.

3

Solve the equation to isolate the variable

t

.

80 - 1.5t - 27 = 44 \implies 53 - 1.5t = 44 \implies -1.5t = -9 \implies t = 6

Perform basic algebraic operations to find the value of

t

.

Key Concept

Setting up and solving a linear equation in one variable from a real-world scenario.

Answer

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