All practice questions

Question 2081Question →

A solid right circular cone has a base radius of $r$ and a height of $h$ . A right circular cylinder has a base radius that is twice the base radius of the cone, and a height that is three times the height of the cone. What is the ratio of the volume of the cone to the volume of the cylinder?

\frac{1}{36}

\frac{1}{18}

\frac{1}{12}

\frac{1}{6}

Show answer & explanation

Answer: $\frac{1}{36}$

Answer

The ratio of the volume of the cone to the volume of the cylinder is

\frac{1}{36}

The volume of a right circular cone is given by

V_{\text{cone}} = \frac{1}{3}\pi r^2 h

. Since the cylinder has a radius that is twice that of the cone (

2r

) and a height that is three times that of the cone (

3h

), its volume is

V_{\text{cylinder}} = \pi (2r)^2(3h) = 12\pi r^2 h

. Dividing the cone's volume by the cylinder's volume gives

\frac{\frac{1}{3}\pi r^2 h}{12\pi r^2 h} = \frac{1}{36}

.

Step-by-Step Solution

1

Write the formula for the volume of the cone.

V_{\text{cone}} = \frac{1}{3}\pi r^2 h

This establishes the volume of the first solid in terms of

r

and

h

.

2

Write the formula for the volume of the cylinder using the scaled dimensions.

V_{\text{cylinder}} = \pi (2r)^2(3h) = 12\pi r^2 h

The cylinder has a radius of

2r

and a height of

3h

, and the volume formula is

\pi R^2 H

.

3

Calculate the ratio of the volume of the cone to the volume of the cylinder.

\frac{V_{\text{cone}}}{V_{\text{cylinder}}} = \frac{\frac{1}{3}\pi r^2 h}{12\pi r^2 h} = \frac{1}{36}

This compares the two volumes directly by dividing the cone's volume by the cylinder's volume.

Key Concept

Volume of cylinders and cones and dimensional scaling relationships.

Alternative Method

Choose convenient sample values for the variables, such as

r = 1

and

h = 3

. The volume of the cone is

V_{\text{cone}} = \frac{1}{3}\pi (1)^2 (3) = \pi

. The cylinder has radius

2(1) = 2

and height

3(3) = 9

, so its volume is

V_{\text{cylinder}} = \pi (2)^2 (9) = 36\pi

. The ratio of the cone's volume to the cylinder's volume is

\frac{\pi}{36\pi} = \frac{1}{36}

.

Estimated Time:1m 15s

Question 2082Question →

A solid metal sphere with radius $r$ is melted down and recast into a right circular cylinder with height $\frac{3}{4}r$ . If the total surface area of the cylinder is $k\pi r^2$ , what is the value of $k$ ?

\frac{17}{9}

\frac{7}{2}

\frac{34}{9}

\frac{50}{9}

Show answer & explanation

Answer: $\frac{50}{9}$

Answer

\frac{50}{9}

The volume of the sphere with radius

r

is

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

. When it is melted and recast into a right circular cylinder with radius

R

and height

h = \frac{3}{4}r

, the volume remains the same. The volume of the cylinder is

V = \pi R^2 h

. Equating the two volumes gives

\frac{4}{3}\pi r^3 = \pi R^2 \left(\frac{3}{4}r\right)

. Dividing both sides by

\pi r

yields

\frac{4}{3} r^2 = \frac{3}{4} R^2

, which simplifies to

R^2 = \frac{16}{9} r^2

. Taking the square root of both sides gives

R = \frac{4}{3}r

. The total surface area of the cylinder is given by the formula

S = 2\pi R^2 + 2\pi R h

. Substituting the values of

R

and

h

in terms of

r

yields

S = 2\pi \left(\frac{4}{3}r\right)^2 + 2\pi \left(\frac{4}{3}r\right)\left(\frac{3}{4}r\right) = 2\pi \left(\frac{16}{9}r^2\right) + 2\pi r^2 = \frac{32}{9}\pi r^2 + \frac{18}{9}\pi r^2 = \frac{50}{9}\pi r^2

. Since the surface area is

k\pi r^2

, the value of

k

is

\frac{50}{9}

.

Step-by-Step Solution

1

Express the volume of the sphere in terms of its radius

r

.

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

This is the standard formula for the volume of a sphere.

2

Equate the volume of the sphere to the volume of the cylinder to find the cylinder's radius

R

in terms of

r

.

\frac{4}{3}\pi r^3 = \pi R^2 \left(\frac{3}{4}r\right) \implies R^2 = \frac{16}{9}r^2 \implies R = \frac{4}{3}r

The volume of a cylinder is

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

. Since the sphere is melted and recast into the cylinder, their volumes must be equal. Solving for

R^2

and taking the square root gives the radius

R

.

3

Use the total surface area formula for the cylinder to express the surface area in terms of

r

.

A = 2\pi R^2 + 2\pi R h = 2\pi\left(\frac{16}{9}r^2\right) + 2\pi\left(\frac{4}{3}r\right)\left(\frac{3}{4}r\right) = \frac{32}{9}\pi r^2 + 2\pi r^2 = \frac{50}{9}\pi r^2

The total surface area of a right circular cylinder consists of the area of the two circular bases and the lateral surface area.

4

Compare the calculated surface area to the given expression

k\pi r^2

to find the value of

k

.

k = \frac{50}{9}

Equating

\frac{50}{9}\pi r^2

and

k\pi r^2

yields the constant value

k

.

Key Concept

Equating volumes of solids and calculating total surface area of a cylinder

Estimated Time:2m 30s

Question 2083Question →

For the function $f$ , selected values of $x$ and $f(x)$ are shown in the table below.

$x$	$f(x)$
$-1$	$4$
$1$	$2$
$3$	$-1$
$5$	$6$

The function $g$ is defined by $g(x) = a \cdot f(x - 2) + 5$ , where $a$ is a constant. If $g(5) = 3$ , what is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

2

Evaluating

g(5)

using the formula

g(x) = a \cdot f(x - 2) + 5

requires finding

f(5 - 2) = f(3)

. From the table,

f(3) = -1

. Substituting these values yields

3 = a(-1) + 5

, which simplifies to

-a = -2

, or

a = 2

.

Step-by-Step Solution

1

Express

g(5)

using the given definition of

g(x)

.

g(5) = a \cdot f(3) + 5

By substituting

x = 5

into the definition

g(x) = a \cdot f(x - 2) + 5

, we obtain

g(5) = a \cdot f(5 - 2) + 5 = a \cdot f(3) + 5

.

2

Find the value of

f(3)

from the table.

f(3) = -1

Looking at the row where

x = 3

in the table, the corresponding output

f(x)

is

-1

.

3

Substitute the known values into the equation for

g(5)

and solve for

a

.

a = 2

Substitute

g(5) = 3

and

f(3) = -1

into the equation to get

3 = a(-1) + 5

. Subtracting 5 from both sides gives

-2 = -a

, which simplifies to

a = 2

.

Key Concept

Evaluating a transformed function using a table of values.

Question 2084Question →

A wind turbine generates electricity at a constant rate of $1.5\text{ megawatt-hours (MWh)}$ per day. If $1\text{ megawatt-hour}$ is equivalent to $3.6\text{ gigajoules (GJ)}$ , how many gigajoules of energy does the wind turbine generate in $8\text{ hours}$ ?

16.2

1.8

43.2

0.14

Show answer & explanation

Answer: 1.8

Answer

1.8

The correct answer is 1.8. To find the total energy generated in gigajoules over 8 hours, the daily rate of

1.5\text{ MWh}

is first converted to an hourly rate by dividing by 24 hours, yielding

0.0625\text{ MWh}

per hour. Multiplying this rate by 8 hours results in

0.5\text{ MWh}

of energy generated. Finally, multiplying the energy in MWh by the conversion factor of

3.6\text{ GJ}

per MWh yields

1.8\text{ GJ}

.

Step-by-Step Solution

1

Convert the daily generation rate of

1.5\text{ MWh}

to an hourly generation rate by dividing by 24 hours.

0.0625\text{ MWh per hour}

Since there are 24 hours in a day, dividing the daily rate by 24 yields the rate of energy generated per hour.

2

Multiply the hourly generation rate by 8 hours to find the total energy generated in megawatt-hours.

0.5\text{ MWh}

Multiplying the rate per hour by the duration of 8 hours gives the cumulative energy generated over that period.

3

Convert the energy generated from megawatt-hours to gigajoules by multiplying by the conversion factor of

3.6\text{ GJ}

per MWh.

1.8\text{ GJ}

Multiplying the energy in MWh by

3.6\text{ GJ/MWh}

converts the unit to gigajoules.

Key Concept

Unit Conversions

Question 2085Question →

The graph of the quadratic function $f(x) = x^2 - 6x + c$ , where $c$ is a constant, has its vertex at $(h, k)$ in the $xy$ -plane. If the graph of $f$ is translated $3$ units to the right and $2$ units down, the vertex of the translated graph lies on the line $y = 2x$ . What is the value of $c$ ?

Show answer & explanation

Answer: 23

Answer

23

To find the constant

c

, we first determine the vertex of the function

f(x) = x^2 - 6x + c

. The x-coordinate of the vertex of a parabola in the form

y = ax^2 + bx + c

is given by

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

. For this function,

x = -\frac{-6}{2(1)} = 3

. Substituting this back into the function gives the y-coordinate of the vertex:

f(3) = (3)^2 - 6(3) + c = c - 9

. Thus, the original vertex is at

(3, c - 9)

. Translating the graph

3

units to the right increases the x-coordinate of the vertex by

3

, making it

3 + 3 = 6

. Translating the graph

2

units down decreases the y-coordinate of the vertex by

2

, making it

(c - 9) - 2 = c - 11

. The problem states that this new vertex

(6, c - 11)

lies on the line

y = 2x

. Substituting these coordinates into the linear equation gives

c - 11 = 2(6)

, which simplifies to

c - 11 = 12

. Adding

11

to both sides gives the value of

c

as

23

.

Step-by-Step Solution

1

Find the vertex

(h, k)

of the original quadratic function

f(x) = x^2 - 6x + c

.

The vertex is at

(3, c - 9)

.

The x-coordinate of the vertex of a quadratic function

y = ax^2 + bx + c

is given by

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

. For

f(x) = x^2 - 6x + c

, we have

h = -\frac{-6}{2(1)} = 3

. Substituting

x = 3

into the function gives the y-coordinate:

k = f(3) = 3^2 - 6(3) + c = c - 9

.

2

Determine the coordinates of the vertex after translating the graph

3

units to the right and

2

units down.

The new vertex is at

(6, c - 11)

.

A translation of

3

units to the right increases the x-coordinate of the vertex by

3

, so the new x-coordinate is

3 + 3 = 6

. A translation of

2

units down decreases the y-coordinate of the vertex by

2

, so the new y-coordinate is

(c - 9) - 2 = c - 11

.

3

Set up an equation using the line

y = 2x

and solve for

c

.

c = 23

Since the translated vertex

(6, c - 11)

lies on the line

y = 2x

, substituting

x = 6

and

y = c - 11

into the line's equation must satisfy it:

c - 11 = 2(6) \Rightarrow c - 11 = 12 \Rightarrow c = 23

.

Key Concept

Determining the vertex of a quadratic function and applying translations to its graph.

Estimated Time:1m 30s

Question 2086Question →

An administrator at a large university with 12,000 enrolled students wants to evaluate student satisfaction with campus services. The administrator designs two separate surveys:

* Survey A: A survey is sent to a random sample of 1,000 students selected from the registrar's list of all 12,000 enrolled students. In this sample, 60% of the students report being satisfied with campus services.
* Survey B: A survey is sent to a random sample of 250 students selected from the 3,000 students who live in on-campus dormitories. In this sample, 70% of the students report being satisfied with campus services.

Which of the following statements must be true?

I. The results of Survey B can be generalized to estimate the satisfaction of all 12,000 enrolled students at the university.
II. If the administrator wants to reduce the margin of error of Survey A to approximately half of its current value, the sample size of Survey A should be increased to 4,000 students.
III. Based on the results of Survey B, it is estimated that 2,100 students who live in on-campus dormitories are satisfied with campus services.

II only

II and III only

I and II only

I, II, and III

Show answer & explanation

Answer: II and III only

Answer

II and III only

The correct option is 'II and III only'. Statement II is true because the margin of error is inversely proportional to the square root of the sample size (

n

). To reduce the margin of error by half (a factor of

\frac{1}{2}

), the sample size must be multiplied by

2^2 = 4

. Thus, increasing the sample size of Survey A from 1,000 to 4,000 students will halve its margin of error. Statement III is true because Survey B selected a random sample of 250 students from the population of 3,000 students living in on-campus dormitories. Since the sample is representative of this population, the sample proportion of 70% can be generalized to the 3,000 dormitory residents, yielding an estimate of

0.70 \times 3,000 = 2,100

satisfied students. Statement I is false because the sample for Survey B was drawn exclusively from students living in on-campus dormitories, who may have different satisfaction levels than off-campus students; therefore, the results cannot be generalized to all 12,000 enrolled students.

Step-by-Step Solution

1

Evaluate Statement I by checking the target population of Survey B's sampling frame.

Statement I is false.

Survey B's sample was drawn specifically from the 3,000 students living in on-campus dormitories, not from the entire student body of 12,000. Therefore, the findings of Survey B can only be generalized to on-campus dormitory residents, not to the entire university.

2

Evaluate Statement II by analyzing the relationship between sample size and margin of error.

Statement II is true.

The margin of error for a sample proportion is inversely proportional to the square root of the sample size (

n

). To reduce the margin of error to

\frac{1}{2}

of its current value, the sample size must be increased by a factor of

2^2 = 4

. Since the original sample size of Survey A is 1,000, increasing it to

1,000 \times 4 = 4,000

will halve the margin of error.

3

Evaluate Statement III by calculating the estimated population value from the sample proportion.

Statement III is true.

Since Survey B used a random sample of the 3,000 dormitory residents, the sample proportion of 70% (

0.70

) can be used to estimate the number of satisfied dormitory residents in the population:

0.70 \times 3,000 = 2,100

students.

4

Combine the evaluations to select the correct option.

Only Statement II and Statement III must be true.

Since Statement I is false and both Statement II and Statement III are true, the correct choice is the option containing 'II and III only'.

Key Concept

Generalization limits of survey results based on sampling frames, and the mathematical relationship between sample size and margin of error.

Estimated Time:2m 30s

Question 2087Question →

To determine the atmospheric composition of distant exoplanets, astronomers use a technique known as transmission spectroscopy. As an exoplanet passes in front of its host star, some of the starlight passes through the planet’s atmosphere ______ gases in the atmosphere absorb specific wavelengths of light, leaving distinct dark lines in the star's spectrum.

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

; gases

, gases

gases

although gases

Show answer & explanation

Answer: ; gases

Answer

The choice '; gases' is correct because it uses a semicolon to properly join two independent clauses.

The correct option '; gases' is correct because a semicolon is an appropriate punctuation mark to link two independent clauses. The clause before the blank ('As an exoplanet... atmosphere') and the clause after the blank ('gases... spectrum') are both independent, meaning they can each stand alone as a complete sentence. A semicolon properly connects them without the need for a coordinating conjunction.

Step-by-Step Solution

1

Identify the clause boundaries in the sentence.

The text before the blank ('As an exoplanet passes in front of its host star, some of the starlight passes through the planet’s atmosphere') is a complete independent clause. The text after the blank ('gases in the atmosphere absorb specific wavelengths of light, leaving distinct dark lines in the star's spectrum') is also a complete independent clause.

Analyzing the grammatical structure of the clauses determines how they should be linked.

2

Evaluate the grammatical rules for linking two independent clauses.

Two independent clauses can be joined by a period, a semicolon, a colon, a dash, or a comma paired with a coordinating conjunction (such as 'and', 'but', or 'so').

Standard English conventions dictate which punctuation marks and conjunctions can successfully connect independent clauses.

3

Test the choices against the grammatical rules.

The option with a semicolon successfully links the clauses. The option with a comma creates a comma splice. The option with no punctuation creates a run-on. The option with 'although' creates a logical contrast error and a grammatical fragment.

Eliminating grammatically incorrect and logically contradictory options leaves the correct answer.

Key Concept

Clause Boundaries and Linking

Question 2088Question →

A student is reviewing notes on the mimic octopus:

* The mimic octopus (*Thaumoctopus mimicus*) is an Indo-Pacific species that deters predators by impersonating other marine animals.
* It can change its shape, color, and movement patterns to mimic creatures such as flatfish and sea snakes.
* Most other octopus species rely on background camouflage to blend in and hide from predators.
* Unlike the mimic octopus, most octopuses do not actively impersonate other animals.

The student wants to present a generalization about how the mimic octopus defends itself compared to other octopuses. Which choice most effectively uses information from the notes to accomplish this goal?

The mimic octopus can mimic flatfish and sea snakes by changing its shape, color, and movement patterns.

While most octopuses protect themselves by blending into their surroundings, the mimic octopus deters predators by actively impersonating other marine species.

Most octopus species are native to the Indo-Pacific and use background camouflage, whereas the mimic octopus relies on flatfish for protection.

Octopuses use a wide variety of advanced physical and behavioral techniques to survive in marine environments like the Indo-Pacific.

Show answer & explanation

Answer: While most octopuses protect themselves by blending into their surroundings, the mimic octopus deters predators by actively impersonating other marine species.

Answer

While most octopuses protect themselves by blending into their surroundings, the mimic octopus deters predators by actively impersonating other marine species.

The correct answer effectively satisfies the student's goal by generalizing the difference between the defense mechanism of the mimic octopus (active impersonation) and that of most other octopuses (background camouflage).

Step-by-Step Solution

1

Identify the student's specific rhetorical goal.

The student wants to present a generalization about how the mimic octopus defends itself compared to other octopuses.

Understanding the goal allows us to eliminate choices that do not compare the two subjects or that focus on irrelevant details.

2

Analyze the notes for the key contrast between the two subjects.

The notes state that the mimic octopus actively impersonates other animals to deter predators, whereas most other octopuses rely on background camouflage to blend in.

This contrast forms the basis of the required generalization.

3

Evaluate the choices to find the one that generalizes this contrast without introducing errors or focusing on narrow details.

The correct option successfully compares the camouflage strategy of most octopuses with the active impersonation strategy of the mimic octopus.

This directly satisfies the rhetorical goal while remaining factually accurate according to the notes.

Key Concept

Rhetorical Synthesis: Summarization and Generalization

Question 2089Question →

A community library is moving its collection of books to a new building. The number of books, $B$ , remaining to be packed $h$ hours after the packing begins is modeled by the linear equation $B = 12,500 - 450h$ , where $0 \leq h \leq 25$ . Which of the following is the best interpretation of $450$ in this context?

The number of books packed each hour

The initial number of books that need to be packed

The number of hours it takes to pack each book, which is

\frac{1}{450}

hour

The total number of hours it takes to pack all the books

Show answer & explanation

Answer: The number of books packed each hour

Answer

The number of books packed each hour

In the linear equation

B = 12,500 - 450h

, the coefficient of

h

is

-450

. This value represents the rate of change of the dependent variable,

B

(remaining books), with respect to the independent variable,

h

(hours). A slope of

-450

means that for every hour that passes, the number of remaining books decreases by

450

, which means

450

books are packed each hour.

Step-by-Step Solution

1

Identify the variable and constants in the given linear equation

B = 12,500 - 450h

.

B

represents the number of remaining books,

h

represents the number of hours spent packing,

12,500

is the constant term (

y

-intercept), and

-450

is the coefficient of

h

(slope).

Understanding the components of the linear model is essential to interpreting their real-world meanings.

2

Analyze the rate of change (slope) of the equation.

The slope is

-450

, which indicates that the number of remaining books decreases by

450

for every

1

hour that passes.

The coefficient of the independent variable in a linear equation in context represents the rate of change of the dependent variable with respect to the independent variable.

3

Translate the rate of change of the remaining books into the rate of packing.

A decrease of

450

remaining books per hour is equivalent to packing

450

books per hour.

Since books are being packed to be moved, a reduction in the remaining unpacked books corresponds directly to the number of books packed.

Key Concept

Interpreting the slope (rate of change) of a linear equation in a real-world context.

Estimated Time:1m 30s

Question 2090Question →

In the inequality $-3(2x - 5) + 8 \geq 2(kx - 3) - 1$ , $k$ is a constant. If the solution set for $x$ is $x \leq 2$ , what is the value of $k$ ?

Show answer & explanation

Answer: 4.5

Answer

4.5

To find the value of

k

, simplify both sides of the inequality:

-3(2x - 5) + 8 \geq 2(kx - 3) - 1

becomes

-6x + 23 \geq 2kx - 7

. Grouping the variable terms on the left and constants on the right gives

-(6 + 2k)x \geq -30

. Because the solution set is

x \leq 2

, dividing by the negative coefficient reverses the inequality sign to yield

x \leq \frac{30}{6 + 2k}

. Setting the boundary value

\frac{30}{6 + 2k}

equal to 2 gives the equation

30 = 2(6 + 2k) = 12 + 4k

. Subtracting 12 from both sides gives

18 = 4k

, which results in

k = 4.5

.

Step-by-Step Solution

1

Simplify both sides of the inequality

-6x + 23 \geq 2kx - 7

Distribute the constants on both sides:

-3(2x - 5) + 8 = -6x + 15 + 8 = -6x + 23

, and

2(kx - 3) - 1 = 2kx - 6 - 1 = 2kx - 7

.

2

Isolate the variable terms on the left side and constants on the right side

-(6 + 2k)x \geq -30

Subtract

2kx

and

23

from both sides to get

-6x - 2kx \geq -7 - 23

, then factor out

x

to get

-(6 + 2k)x \geq -30

.

3

Relate the inequality to the given solution boundary

x \leq \frac{30}{6 + 2k}

Since the solution is

x \leq 2

, dividing both sides by the negative coefficient

-(6 + 2k)

reverses the inequality sign, yielding

x \leq \frac{-30}{-(6 + 2k)} = \frac{30}{6 + 2k}

.

4

Solve for

k

using the boundary equation

k = 4.5

Set the boundary expression equal to 2:

\frac{30}{6 + 2k} = 2

. Multiply both sides by

6 + 2k

to get

30 = 12 + 4k

, which simplifies to

18 = 4k

, giving

k = 4.5

.

Key Concept

Solving linear inequalities in one variable involving variable coefficients and applying the inequality direction flip when dividing by a negative value.

Question 2091Question →

A water desalination facility uses two types of filtration systems, System X and System Y. System X processes saltwater at a constant rate of $15$ liters per minute, producing purified water and brine in a ratio of $4:1$ . System Y processes saltwater at a constant rate of $21$ liters per minute, producing purified water and brine in a ratio of $5:2$ . If both systems operate simultaneously, how many liters of brine will be produced in total during the time it takes the two systems combined to produce $324$ liters of purified water?

81

108

126

432

Show answer & explanation

Answer: 108

Answer

108

The correct value of 108 is found by calculating the rate of brine production for each system. System X produces brine at a rate of

15 \times \frac{1}{5} = 3

liters per minute, and System Y produces brine at

21 \times \frac{2}{7} = 6

liters per minute, giving a combined brine rate of

9

liters per minute. Meanwhile, the combined purified water rate is

(15 \times \frac{4}{5}) + (21 \times \frac{5}{7}) = 12 + 15 = 27

liters per minute. It takes the systems

\frac{324}{27} = 12

minutes to produce the purified water. During these 12 minutes, the total brine produced is

12 \times 9 = 108

liters.

Step-by-Step Solution

1

Find the purified water and brine production rates for System X.

Purified water rate =

12

liters per minute; Brine rate =

3

liters per minute.

Since the ratio of purified water to brine is

4:1

, there are

4 + 1 = 5

parts in total. System X processes

15

liters of saltwater per minute, so the purified water rate is

15 \times \frac{4}{5} = 12

liters per minute and the brine rate is

15 \times \frac{1}{5} = 3

liters per minute.

2

Find the purified water and brine production rates for System Y.

Purified water rate =

15

liters per minute; Brine rate =

6

liters per minute.

Since the ratio of purified water to brine is

5:2

, there are

5 + 2 = 7

parts in total. System Y processes

21

liters of saltwater per minute, so the purified water rate is

21 \times \frac{5}{7} = 15

liters per minute and the brine rate is

21 \times \frac{2}{7} = 6

liters per minute.

3

Calculate the combined purified water and brine rates.

Combined purified rate =

27

liters per minute; Combined brine rate =

9

liters per minute.

Adding the individual rates of System X and System Y gives the combined rates of

12 + 15 = 27

liters per minute for purified water and

3 + 6 = 9

liters per minute for brine.

4

Determine the time required to produce the target amount of purified water.

Time =

12

minutes.

Divide the target purified water volume of

324

liters by the combined purified water rate of

27

liters per minute to find the operating time:

\frac{324}{27} = 12

minutes.

5

Calculate the total amount of brine produced during this time.

Total brine =

108

liters.

Multiply the operating time of

12

minutes by the combined brine production rate of

9

liters per minute:

12 \times 9 = 108

liters.

Key Concept

Combining rates derived from part-to-whole ratios to solve multi-step rate problems.

Question 2092Question →

In the quadratic equation $x^2 - 6x - k = 0$ , $k$ is a positive constant. If the solutions to the equation are $x = 3 \pm \sqrt{17}$ , what is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

8

By applying the quadratic formula to

x^2 - 6x - k = 0

, we find the solutions are

x = 3 \pm \sqrt{9 + k}

. Equating the expression inside the radical to the given solutions

3 \pm \sqrt{17}

yields

9 + k = 17

. Solving for

k

gives 8.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation.

a = 1

,

b = -6

, and

c = -k

To apply the quadratic formula, we need to know the values of

a

,

b

, and

c

from the standard form

ax^2 + bx + c = 0

.

2

Substitute the coefficients into the quadratic formula.

x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(-k)}}{2(1)}

The quadratic formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

gives the solutions directly.

3

Simplify the expression inside the radical and divide by 2.

x = 3 \pm \sqrt{9 + k}

Simplifying the radical expression allows us to easily compare it to the given solutions.

4

Compare the simplified solution to the given solution form.

9 + k = 17

Since the solutions are given as

3 \pm \sqrt{17}

, the expression inside the radical must equal 17.

5

Solve for the constant

k

.

k = 8

Subtracting 9 from both sides of the equation yields the value of the constant.

Key Concept

Solving quadratic equations using the quadratic formula or by completing the square to find the relationship between coefficients and solutions.

Alternative Method

Instead of using the quadratic formula, the equation can be solved by completing the square. Rewrite the equation as

x^2 - 6x = k

. Adding 9 to both sides gives

x^2 - 6x + 9 = k + 9

, which can be factored as

(x - 3)^2 = k + 9

. Taking the square root of both sides yields

x = 3 \pm \sqrt{k + 9}

. Comparing this to the given solutions

3 \pm \sqrt{17}

, we get

k + 9 = 17

, so

k = 8

.

Estimated Time:1m 30s

Question 2093Question →

In right triangle $JKL$ , the measure of angle $K$ is $90^\circ$ . If $\cos(J) = 0.28$ , what is the value of $\sin(L)$ ?

Show answer & explanation

Answer: 0.28

Answer

The correct answer is 0.28 (or the equivalent fraction 7/25).

In a right triangle, the two acute angles are complementary, meaning they add up to

90^\circ

. The co-function identity states that the sine of an acute angle is equal to the cosine of its complement. Therefore, in right triangle

JKL

with right angle

K

,

\sin(L) = \cos(J)

. Given that

\cos(J) = 0.28

, the value of

\sin(L)

must also be

0.28

.

Step-by-Step Solution

1

Determine the relationship between angles

J

and

L

.

J + L = 90^\circ

Since the sum of angles in a triangle is

180^\circ

and angle

K

is

90^\circ

, the sum of the remaining two angles must be

180^\circ - 90^\circ = 90^\circ

.

2

Use the co-function trigonometric identity.

\sin(L) = \cos(J)

The sine of an acute angle in a right triangle is equal to the cosine of its complementary angle.

3

Substitute the given value of

\cos(J)

into the identity.

\sin(L) = 0.28

We are given that

\cos(J) = 0.28

.

Key Concept

Complementary angle trigonometric identity (co-function identity)

Question 2094Question →

A rectangular box has a square base with side length $x$ inches and a height of $5$ inches. The total surface area of the box is $192$ square inches. What is the value of $x$ ?

Show answer & explanation

Answer: 6

Answer

The value of

x

is

6

.

The total surface area is found by adding the areas of all six faces. The top and bottom faces are squares with side length

x

, so their combined area is

2x^2

. The four vertical sides are rectangles with dimensions

x

by

5

, so their combined area is

4(5x) = 20x

. Setting their sum equal to the total surface area gives

2x^2 + 20x = 192

. Dividing by

2

yields

x^2 + 10x - 96 = 0

, which factors as

(x + 16)(x - 6) = 0

. Since

x

must be positive,

x = 6

.

Step-by-Step Solution

1

Set up the equation for the total surface area of the rectangular box.

\text{Total Surface Area} = 2x^2 + 20x = 192

The total surface area of a rectangular box with a square base of side length

x

and height

h

consists of two square bases (top and bottom) of area

x^2

each, and four rectangular sides of area

xh

each. Substituting

h = 5

gives

2x^2 + 4(5x) = 2x^2 + 20x

.

2

Simplify and set the quadratic equation to zero.

x^2 + 10x - 96 = 0

Dividing all terms by

2

simplifies the equation to

x^2 + 10x = 96

. Subtracting

96

from both sides sets the quadratic equation to standard form

ax^2 + bx + c = 0

.

3

Factor the quadratic equation and solve for the positive value of

x

.

x = 6

Factoring the equation gives

(x + 16)(x - 6) = 0

, which yields solutions of

x = -16

and

x = 6

. Since the side length of a geometric solid must be positive, we reject the negative solution.

Key Concept

Calculating the surface area of a rectangular prism and solving the resulting quadratic relationship.

Estimated Time:1m 35s

Question 2095Question →

A marine biologist records the dive depths, in meters, of a seal on 6 consecutive dives. The recorded depths are 52, 68, 57, 65, 48, and $x$ . The median of these 6 dive depths is 58 meters. What is the value of $x$ ?

Show answer & explanation

Answer: 59

Answer

59

To find the median of a set of 6 numbers, the numbers must be listed in ascending order, and the median will be the average of the 3rd and 4th numbers. The five known numbers in ascending order are 48, 52, 57, 65, and 68. If the unknown value

x

is less than or equal to 57, the 3rd and 4th numbers in the sorted list would be at most 57, which would make the median at most 57. If

x

is greater than or equal to 65, the 3rd and 4th numbers in the sorted list would be 57 and 65, which would make the median

\frac{57+65}{2} = 61

. Since the median is given as 58,

x

must be between 57 and 65. Thus, when the 6 numbers are sorted, the 3rd number is 57 and the 4th number is

x

. Setting their average equal to 58 gives

\frac{57+x}{2} = 58

. Multiplying both sides by 2 yields

57 + x = 116

. Subtracting 57 from both sides gives

x = 59

.

Step-by-Step Solution

1

Order the five known depths from least to greatest.

48, 52, 57, 65, 68

To find or use the median of a dataset, the data points must first be arranged in ascending order.

2

Analyze the position of the median in a dataset of 6 values.

The median is the average of the 3rd and 4th values in the ordered list.

For an even number of data points, the median is the arithmetic mean of the two middle values.

3

Determine the constraints on

x

to achieve a median of 58.

The value of

x

must be between 57 and 65, making 57 the 3rd value and

x

the 4th value.

If

x \le 57

, the 3rd and 4th values are both 57 or less, yielding a median of at most 57. If

x \ge 65

, the 3rd and 4th values are 57 and 65, yielding a median of 61. Thus,

x

must be the 4th value.

4

Set up and solve the equation for the median.

\frac{57 + x}{2} = 58 \implies 57 + x = 116 \implies x = 59

Since the 3rd value is 57 and the 4th value is

x

, their average must equal the given median of 58.

Key Concept

Calculating and interpreting the median of a dataset with an even number of values, including handling unknown variables.

Question 2096Question →

A scientist surveyed $15$ different forest regions and recorded the number of endangered plant species in each region. The numbers of endangered species in $14$ of the regions are:

12, 15, 15, 16, 18, 20, 20, 22, 22, 22, 25, 28, 30, 34

The number of endangered species in the $15\text{th}$ region is $x$ , where $x$ is an integer greater than $25$ . If the median of the numbers of endangered species for all $15$ regions is equal to the mean, what is the value of $x$ ?

Show answer & explanation

Answer: 31

Answer

The correct answer is 31.

The correct answer is 31. The sum of the 14 known values is 299. Since the 15th value

x

is greater than 25, it must be placed at or after the 12th position when all 15 values are sorted. Thus, the 8th value (the median of the 15 values) remains the same as the 8th value of the known 14 sorted values, which is 22. Setting the mean equal to the median gives

\frac{299 + x}{15} = 22

, which simplifies to

299 + x = 330

, resulting in

x = 31

. Since

31 > 25

, this is the correct value.

Step-by-Step Solution

1

Find the sum of the 14 known values.

The sum of the 14 known values is 299.

This is needed to construct the expression for the mean of the 15 values.

2

Determine the median of the 15 values given the constraint

x > 25

.

The median is 22.

Since

x > 25

, it will be sorted after the first 11 values (the 11th value is 25). The median of 15 values is the 8th value, which remains 22.

3

Set the mean equal to the median and solve for

x

.

\frac{299 + x}{15} = 22 \implies 299 + x = 330 \implies x = 31

.

This yields the value of

x

that satisfies the condition that the mean equals the median.

Key Concept

Calculating and comparing the mean and median of a dataset, and analyzing how a new data point affects these measures.

Question 2097Question →

The function $f$ is defined by $f(x) = x^2 - 4x + 7$ . The function $g$ is defined by $g(x) = f(x - 3) + 2$ . If the minimum value of $f$ is $v$ , and the minimum value of $g$ occurs at $x = k$ , what is the value of $v + k$ ?

1

2

7

8

Show answer & explanation

Answer: 8

Answer

8

To find the minimum value of

f(x) = x^2 - 4x + 7

, we can complete the square to write it in vertex form:

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

. The vertex of the graph of

f

is

(2, 3)

, meaning the minimum value of

f

is

v = 3

, occurring at

x = 2

. The function

g

is defined as

g(x) = f(x - 3) + 2

, which represents a shift of the graph of

f

to the right by 3 units and up by 2 units. Since the minimum of

f

occurs at

x = 2

, the minimum of

g

occurs at

x = 2 + 3 = 5

, so

k = 5

. Thus, the value of

v + k

is

3 + 5 = 8

.

Step-by-Step Solution

1

Find the vertex form of the quadratic function

f(x)

to identify its minimum value

v

and the x-coordinate where it occurs.

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

, so the minimum value is

v = 3

, which occurs at

x = 2

.

Completing the square allows us to read the vertex

(h, k)

of the parabola directly, where

h

is the x-coordinate of the vertex and

k

is the minimum value.

2

Determine the x-coordinate

k

where the minimum value of

g(x)

occurs using function transformations.

k = 2 + 3 = 5

The function

g(x) = f(x - 3) + 2

shifts the graph of

f

to the right by 3 units. Therefore, the minimum point is translated from

x = 2

to

x = 2 + 3 = 5

.

3

Calculate the sum

v + k

.

3 + 5 = 8

Adding the minimum value of

f

(

v = 3

) to the x-coordinate of the minimum of

g

(

k = 5

) gives the final required value.

Key Concept

Identifying the vertex and minimum values of quadratic functions, and applying horizontal and vertical translations to their graphs.

Estimated Time:1m 30s

Question 2098Question →

While preparing a presentation on forest ecology, a student takes the following notes:
* *Pando* is a clonal colony in Utah consisting of over 47,000 genetically identical quaking aspen trees.
* The trees in the colony are connected by a shared underground root system that sends up new shoots.
* In a 2018 study, ecologist Dr. Paul Rogers found that young shoots are being consumed by mule deer and elk before reaching maturity.
* The study noted that in areas protected by fences that exclude these herbivores, young shoots are successfully growing into mature trees.

The student wants to support the claim that herbivore grazing is preventing the clonal colony from regenerating. Which choice best uses information from the notes to accomplish this goal?

In his 2018 study, ecologist Dr. Paul Rogers confirmed that the mature trees in the clonal colony are dying of natural causes rather than from herbivore grazing.

Because a decline in natural predators has caused mule deer and elk populations to increase, their heavy grazing on young shoots is preventing *Pando* from regenerating.

While young shoots are successfully growing into mature trees in areas protected by fences, those outside the fences are being consumed by mule deer and elk before they can reach maturity.

*Pando* is a clonal colony in Utah that consists of over 47,000 genetically identical quaking aspen trees connected by a shared underground root system.

Show answer & explanation

Answer: While young shoots are successfully growing into mature trees in areas protected by fences, those outside the fences are being consumed by mule deer and elk before they can reach maturity.

Answer

The option stating that young shoots grow into mature trees inside protected fences but are eaten by herbivores outside the fences.

The correct choice highlights that young shoots successfully grow into mature trees when protected by fences but are eaten by herbivores when unprotected. This directly supports the claim that herbivore grazing is preventing regeneration by drawing solely on the provided notes.

Step-by-Step Solution

1

Identify the student's rhetorical goal.

The goal is to support the claim that herbivore grazing is preventing the clonal colony from regenerating.

This sets the criteria for evaluating the choices based on the prompt's specific demand.

2

Evaluate the provided notes for evidence related to the claim.

The notes state that herbivores (mule deer and elk) consume young shoots, and that when fences exclude these herbivores, the shoots successfully grow into mature trees.

This identifies the facts necessary to build the supporting statement.

3

Analyze each option to see which one directly uses these facts to support the claim without adding outside assumptions.

The correct choice contrasts the fenced areas (where regeneration succeeds without herbivores) with the unfenced areas (where herbivores eat the shoots), showing that grazing indeed blocks regeneration.

This selects the option that fulfills the rhetorical goal using only the provided notes.

Key Concept

Rhetorical Synthesis: Supporting Claims

Question 2099Question →

In 1928, Scottish researcher Alexander Fleming accidentally discovered penicillin, which became the world's first effective antibiotic. Since Fleming's historic breakthrough, medical researchers ______ countless other antibiotics to treat bacterial infections, saving millions of lives and fundamentally transforming modern healthcare.

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

have developed

developed

had developed

will develop

Show answer & explanation

Answer: have developed

Answer

The present perfect tense 'have developed' is correct because the word 'since' indicates an action that began in the past and continues to be relevant in the present.

The correct answer is the phrase 'have developed' because the sentence establishes a time frame starting in the past and extending to the present with the prepositional phrase 'Since Fleming's historic breakthrough'. This requires the present perfect tense. Because the subject 'medical researchers' is plural, the plural auxiliary verb 'have' must be used.

Step-by-Step Solution

1

Identify the subject and temporal markers in the sentence.

The subject is 'medical researchers' (plural) and the time marker is 'Since Fleming's historic breakthrough'.

This establishes that the verb must be plural and in a tense that connects a past point to the present.

2

Select the verb form that matches the temporal marker.

The present perfect plural verb 'have developed' is the correct choice.

Present perfect tense is required to show action starting in the past and continuing up to or affecting the present.

Key Concept

Using the present perfect tense with the time marker 'since'

Estimated Time:45s

Question 2100Question →

The hoatzin (Opisthocomus hoazin), a tropical bird native to the Amazon and Orinoco basins, possesses an unusual digestive system. Unlike most birds, which rely on a muscular gizzard to grind seeds and insects, the hoatzin is a strict folivore, feeding almost exclusively on leaves. To digest this fibrous diet, the bird utilizes foregut fermentation, a process facilitated by a highly enlarged crop and lower esophagus. Specialized symbiotic bacteria in these organs break down cellulose before the food reaches the stomach. While this adaptation allows the hoatzin to extract nutrients from otherwise indigestible foliage, it has a significant trade-off: the heavy, filled crop restricts the development of the bird's breastbone and flight muscles, making the hoatzin a notoriously weak and clumsy flyer.

Based on the text, what is true of the hoatzin's digestive system?

It relies on a highly developed muscular gizzard to process seeds and insects.

It employs symbiotic bacteria to break down cellulose before food enters the stomach.

It prevents the bird from gaining enough nutrients from its diet of foliage.

It is a common adaptation shared by most other bird species in the Amazon and Orinoco basins.

Show answer & explanation

Answer: It employs symbiotic bacteria to break down cellulose before food enters the stomach.

Answer

The hoatzin's digestive system employs symbiotic bacteria to break down cellulose before food enters the stomach.

The correct answer is supported by the text, which explicitly states that specialized symbiotic bacteria in the hoatzin's enlarged crop and lower esophagus break down cellulose before the food reaches the stomach.

Step-by-Step Solution

1

Analyze the passage to locate details describing how the hoatzin's digestive system processes food.

The text states that the hoatzin is a folivore that uses foregut fermentation, where specialized symbiotic bacteria in the enlarged crop and lower esophagus break down cellulose before the food reaches the stomach.

This direct statement must match the correct option exactly without requiring speculative leaps.

2

Evaluate the choices against this explicit detail.

The option stating that the system employs symbiotic bacteria to break down cellulose before food enters the stomach directly mirrors the text's description.

To identify the correct choice, look for the option that restates explicit facts from the text.

Key Concept

Explicit Details

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

Answer

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Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer