All practice questions

Question 2521Question →

A security camera is mounted on a vertical wall at point $C$ , exactly $15$ feet above the flat ground. The camera is programmed to monitor two objects, $A$ and $B$ , on the ground. The line of sight from the camera to object $A$ makes a $30^\circ$ angle with the wall, and the line of sight from the camera to object $B$ makes a $45^\circ$ angle with the wall. On the ground, the path from the base of the wall directly below the camera to object $A$ is perpendicular to the path from the base of the wall to object $B$ . What is the straight-line distance, in feet, between object $A$ and object $B$ ?

5\sqrt{6}

4\sqrt{15}

10\sqrt{3}

15 + 5\sqrt{3}

30

Show answer & explanation

Answer: $10\sqrt{3}$

Answer

The distance between object

A

and object

B

is

10\sqrt{3}

feet.

The horizontal distances from the base of the wall to objects

A

and

B

are calculated using the trigonometric ratios of the

30^\circ

-

60^\circ

-

90^\circ

and

45^\circ

-

45^\circ

-

90^\circ

triangles formed by the vertical wall. This gives legs of

5\sqrt{3}

feet and

15

feet. Applying the Pythagorean theorem to the right triangle on the ground yields a hypotenuse of

10\sqrt{3}

feet.

Step-by-Step Solution

1

Find the horizontal distance from the base of the wall to object

A

.

Let

O

be the base of the wall directly below the camera

C

, so

OC = 15

feet. The line of sight

CA

makes a

30^\circ

angle with the wall, so

\triangle OCA

is a

30^\circ

-

60^\circ

-

90^\circ

right triangle with leg

OA = 15 \tan(30^\circ) = \frac{15}{\sqrt{3}} = 5\sqrt{3}

feet.

We need to determine the length of one of the perpendicular legs on the ground.

2

Find the horizontal distance from the base of the wall to object

B

.

The line of sight

CB

makes a

45^\circ

angle with the wall, so

\triangle OCB

is a

45^\circ

-

45^\circ

-

90^\circ

right triangle with leg

OB = 15 \tan(45^\circ) = 15

feet.

We need to determine the length of the other perpendicular leg on the ground.

3

Use the Pythagorean theorem to calculate the straight-line distance

AB

on the ground.

Since the paths

OA

and

OB

are perpendicular,

\triangle AOB

is a right triangle with legs

OA = 5\sqrt{3}

and

OB = 15

. The hypotenuse

AB

is

\sqrt{(5\sqrt{3})^2 + 15^2} = \sqrt{75 + 225} = \sqrt{300} = 10\sqrt{3}

feet.

The straight-line distance between the two objects corresponds to the hypotenuse of the right triangle formed by their ground distances.

Key Concept

Applying special right triangle ratios and the Pythagorean theorem to solve multi-step problems in three-dimensional contexts.

Question 2522Question →

In the standard $(x, y)$ coordinate plane, a line segment has endpoints at $M(1, -3)$ and $N(5, 5)$ . If a second line is perpendicular to segment $MN$ at its midpoint, what is the $y$ -intercept of this second line?

Show answer & explanation

Answer: 2.5

Answer

The y-intercept of the second line is 2.5.

The slope of segment

MN

is

2

, meaning the perpendicular line must have a slope of

-0.5

. Since this perpendicular line bisects the segment, it must pass through the midpoint

(3, 1)

. Substituting these values into the slope-intercept form

y = mx + b

yields

1 = -0.5(3) + b

, which simplifies to

b = 2.5

.

Step-by-Step Solution

1

Calculate the coordinates of the midpoint of segment

MN

.

The midpoint is

(3, 1)

.

The perpendicular bisector must pass through the midpoint of the bisected segment.

2

Calculate the slope of segment

MN

.

The slope of

MN

is

2

.

The slope is needed to find the perpendicular slope.

3

Calculate the slope of the perpendicular line.

The perpendicular slope is

-0.5

.

Perpendicular lines have slopes that are negative reciprocals of each other.

4

Find the

y

-intercept (

b

) using the slope

-0.5

and the point

(3, 1)

.

The

y

-intercept is

2.5

.

Substituting the coordinates of the midpoint and the perpendicular slope into the equation

y = mx + b

gives

1 = -0.5(3) + b

, which simplifies to

b = 2.5

.

Key Concept

Finding the equation and y-intercept of a perpendicular bisector using midpoint and negative reciprocal slope.

Question 2523Question →

In the standard $(x, y)$ coordinate plane, the midpoint of a line segment with endpoints $A(r, -2)$ and $B(10, s)$ is $M(6, 3)$ . What is the distance between the point $A$ and the origin $(0, 0)$ ?

2\sqrt{2}

3\sqrt{5}

2\sqrt{17}

4

2\sqrt{41}

Show answer & explanation

Answer: $2\sqrt{2}$

Answer

The distance between the point

A

and the origin is

2\sqrt{2}

.

To find the coordinates of point

A(r, -2)

, we set up the midpoint formula with the coordinates of

B(10, s)

and the midpoint

M(6, 3)

. For the

x

-coordinate,

\frac{r + 10}{2} = 6 \implies r + 10 = 12 \implies r = 2

. For the

y

-coordinate,

\frac{-2 + s}{2} = 3 \implies -2 + s = 6 \implies s = 8

. Thus, point

A

is

(2, -2)

. The distance from

A(2, -2)

to the origin

(0, 0)

is

\sqrt{(2 - 0)^2 + (-2 - 0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}

.

Step-by-Step Solution

1

Use the midpoint formula to set up equations for the coordinates of the midpoint

M(6, 3)

given the endpoints

A(r, -2)

and

B(10, s)

.

The equations are

\frac{r + 10}{2} = 6

and

\frac{-2 + s}{2} = 3

.

The midpoint coordinates are the averages of the corresponding coordinates of the endpoints.

2

Solve these equations for

r

and

s

.

r + 10 = 12 \implies r = 2

, and

-2 + s = 6 \implies s = 8

.

This determines the coordinates of point

A

as

(2, -2)

and point

B

as

(10, 8)

.

3

Calculate the distance between point

A(2, -2)

and the origin

(0, 0)

using the distance formula.

d = \sqrt{(2 - 0)^2 + (-2 - 0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}

.

The distance formula is used to find the length of the segment connecting the point to the origin.

Key Concept

Finding the endpoint of a line segment using the midpoint formula, and then finding the distance between a point and the origin using the distance formula.

Alternative Method

Instead of solving for the

y

-coordinate

s

of point

B

, we only need the

x

-coordinate of

A

, which is

r

, to find the distance between

A(r, -2)

and the origin. Since

A

's

y

-coordinate is already given as

-2

, we only set up the equation for the

x

-coordinate of the midpoint:

\frac{r+10}{2} = 6

, which gives

r = 2

. Thus,

A

is

(2, -2)

, and the distance is

\sqrt{2^2 + (-2)^2} = 2\sqrt{2}

. This saves time by not calculating

s

.

Estimated Time:1m 30s

Question 2524Question →

A geometry student claims that every rectangle is also a parallelogram. Is this claim true or false?

True

False

Show answer & explanation

Answer: True

Answer

True

The claim is true because a rectangle must have four right angles, which mathematically forces opposite sides to be parallel, thereby meeting the definition of a parallelogram.

Step-by-Step Solution

1

Define a rectangle based on its angle properties.

A rectangle is a quadrilateral with four right angles.

This is the primary defining characteristic of a rectangle.

2

Define a parallelogram based on its side properties.

A parallelogram is a quadrilateral with two pairs of parallel opposite sides.

This is the primary defining characteristic of a parallelogram.

3

Analyze the relationship between the two definitions.

In a quadrilateral with four

90^\circ

angles, consecutive angles sum to

180^\circ

. By the consecutive interior angles converse, the opposite sides are parallel. Since both pairs of opposite sides are parallel, the quadrilateral must be a parallelogram. Thus, every rectangle is a parallelogram.

To determine whether the student's statement is correct.

Key Concept

Hierarchical classification of quadrilaterals

Estimated Time:30s

Question 2525Question →

A circle in the standard $(x, y)$ coordinate plane is described by the equation $x^2 + y^2 = 68$ . A line is described by the equation $y = x - 6$ . The line intersects the circle at two points. What is the distance between these two points of intersection?

8\sqrt{2}

2\sqrt{17}

10\sqrt{2}

20

6\sqrt{2}

Show answer & explanation

Answer: $10\sqrt{2}$

Answer

The distance between the two points of intersection is

10\sqrt{2}

.

To find the points of intersection, substitute

y = x - 6

into

x^2 + y^2 = 68

, which yields

x^2 + (x - 6)^2 = 68

. Expanding the binomial correctly results in

x^2 + x^2 - 12x + 36 = 68

. Combining like terms and writing the equation in standard form gives

2x^2 - 12x - 32 = 0

. Dividing the entire equation by 2 gives

x^2 - 6x - 16 = 0

. Factoring this quadratic equation yields

(x - 8)(x + 2) = 0

, giving solutions

x = 8

and

x = -2

. Substituting these values back into the linear equation gives the points

(8, 2)

and

(-2, -8)

. Finally, using the distance formula, the distance between the two points is

\sqrt{(8 - (-2))^2 + (2 - (-8))^2} = \sqrt{10^2 + 10^2} = \sqrt{200} = 10\sqrt{2}

.

Step-by-Step Solution

1

Substitute the linear equation into the circle equation.

x^2 + (x - 6)^2 = 68

To eliminate

y

and solve for the

x

-coordinates of the intersection points.

2

Expand the binomial and collect like terms.

2x^2 - 12x - 32 = 0

Expanding

(x - 6)^2

yields

x^2 - 12x + 36

, and combining it with

x^2

and subtracting

68

from both sides puts the equation in quadratic form.

3

Divide the quadratic equation by 2 and solve by factoring.

(x - 8)(x + 2) = 0

, so

x = 8

or

x = -2

Simplifying the equation to

x^2 - 6x - 16 = 0

makes it easy to find the roots of the quadratic equation.

4

Find the corresponding

y

-coordinates by substituting the

x

-values back into the linear equation.

For

x = 8

,

y = 2

, yielding point

(8, 2)

. For

x = -2

,

y = -8

, yielding point

(-2, -8)

.

To determine the full coordinates of the two intersection points.

5

Apply the distance formula to find the distance between

(8, 2)

and

(-2, -8)

.

d = \sqrt{(8 - (-2))^2 + (2 - (-8))^2} = \sqrt{10^2 + 10^2} = \sqrt{200} = 10\sqrt{2}

To calculate the straight-line distance between the two coordinate points.

Key Concept

Solving systems of linear and non-linear equations by substitution and finding the distance between intersection points.

Estimated Time:1m 30s

Question 2526Question →

On a coordinate grid, a line segment connects the points $A(1, 3)$ and $B(9, 15)$ . The midpoint of this segment is $M$ , and the midpoint of the segment connecting $A$ and $M$ is $C$ . What is the $y$ -coordinate of point $C$ ?

Show answer & explanation

Answer: 6

Answer

The

y

-coordinate of point

C

is 6.

Applying the midpoint formula to the endpoints

A(1, 3)

and

B(9, 15)

yields

M(5, 9)

. Applying the midpoint formula again to

A(1, 3)

and

M(5, 9)

yields

C(3, 6)

. The

y

-coordinate of point

C

is therefore 6.

Step-by-Step Solution

1

Calculate the coordinates of the midpoint

M

of segment

AB

.

M = (5, 9)

The midpoint formula is defined as

\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

.

2

Calculate the coordinates of the midpoint

C

of segment

AM

.

C = (3, 6)

Apply the midpoint formula using the coordinates of

A(1, 3)

and the newly found midpoint

M(5, 9)

.

3

Identify the

y

-coordinate of point

C

.

6

For the point

C(3, 6)

, the

x

-coordinate is 3 and the

y

-coordinate is 6.

Key Concept

Midpoint Formula

Estimated Time:1m 15s

Question 2527Question →

The following paragraphs are from a natural science essay about deep-sea ecosystems:

[Paragraph 1] In 1977, researchers aboard the research submersible Alvin made a discovery that transformed biology. Deep on the ocean floor near the Galápagos Rift, they found hydrothermal vents spewing superheated, mineral-rich water. Surrounding these vents were thriving communities of giant tube worms, clams, and crabs, existing in complete darkness. This finding shattered the long-held scientific consensus that all biological communities on Earth require sunlight as their primary source of energy.

[Paragraph 2] At the heart of this dark ecosystem are specialized bacteria that have bypassed the need for solar radiation entirely. Rather than relying on photosynthesis, these microbes perform chemosynthesis. They metabolize hydrogen sulfide—a compound toxic to most land-based organisms—flowing from the vents and convert it into organic matter. This process forms the base of the food web, nourishing the larger animals that live clustered around the vent openings.

[Paragraph 3] The realization that life can flourish in such extreme, sunless environments has profound implications for astrobiology. Scientists now look to the icy moons of the outer solar system, such as Jupiter's Europa and Saturn's Enceladus, with renewed optimism. If liquid oceans exist beneath their frozen crusts, heated by tidal forces, hydrothermal vent communities could theoretically survive there, completely isolated from any star's light.

Match each paragraph from the passage to its primary paragraph-level main idea.

Click a left item, then click its matching right item

Show answer & explanation

Answer

Paragraph 1 matches the introduction of a finding that challenged assumptions about solar energy; Paragraph 2 matches the explanation of how organisms generate energy without sunlight; Paragraph 3 matches the implications for life on icy moons in the outer solar system.

The correct pairings align each paragraph with its specific focus: Paragraph 1 introduces the discovery challenging solar dependency, Paragraph 2 explains the chemical process of chemosynthesis, and Paragraph 3 outlines the implications for life on outer solar system moons.

Step-by-Step Solution

1

Analyze Paragraph 1 to identify its main focus.

Paragraph 1 describes the discovery of hydrothermal vents in 1977 and how it shattered the consensus that all life requires sunlight.

Understanding the paragraph's primary purpose helps match it to the correct statement.

2

Analyze Paragraph 2 to identify its main focus.

Paragraph 2 explains chemosynthesis, the chemical process bacteria use to generate food from hydrogen sulfide without sunlight.

This establishes the biological mechanism described in the second section.

3

Analyze Paragraph 3 to identify its main focus.

Paragraph 3 discusses astrobiology, icy moons like Europa and Enceladus, and the potential for life in extraterrestrial oceans.

This connects the local discovery to broader celestial search efforts.

Key Concept

Determining Paragraph-Level Main Ideas

Question 2528Question →

A square metal sheet has a side length of $3x^3$ inches. A square cutout with a side length of $(x^3 - 2)$ inches is removed from the center of the sheet. Which of the following polynomial expressions represents the area, in square inches, of the remaining metal sheet?

8x^6 - 4x^3 + 4

8x^5 + 4x^3 - 4

8x^6 + 4

8x^6 + 4x^3 - 4

2x^6 + 4x^3 - 4

Show answer & explanation

Answer: $8x^6 + 4x^3 - 4$

Answer

The expression

8x^6 + 4x^3 - 4

represents the remaining area of the metal sheet.

To find the remaining area, subtract the area of the square cutout from the total area of the metal sheet. The total area is the square of the side length

3x^3

, which is

(3x^3)^2 = 9x^6

. The area of the cutout is the square of

(x^3 - 2)

, which is

(x^3 - 2)^2 = x^6 - 4x^3 + 4

. Subtracting the cutout area from the total area yields

9x^6 - (x^6 - 4x^3 + 4) = 9x^6 - x^6 + 4x^3 - 4 = 8x^6 + 4x^3 - 4

. This matches the correct expression.

Step-by-Step Solution

1

Calculate the total area of the square metal sheet.

Area_{total} = (3x^3)^2 = 9x^6

The area of a square is the square of its side length. According to the power of a product rule,

(ab)^n = a^n b^n

, and the power of a power rule,

(x^a)^b = x^{ab}

.

2

Calculate the area of the square cutout.

Area_{cutout} = (x^3 - 2)^2 = x^6 - 4x^3 + 4

Squaring the binomial yields

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

, which simplifies to

x^6 - 4x^3 + 4

.

3

Subtract the area of the cutout from the total area of the metal sheet.

Area_{remaining} = 9x^6 - (x^6 - 4x^3 + 4) = 8x^6 + 4x^3 - 4

Distributing the negative sign through the parentheses yields

9x^6 - x^6 + 4x^3 - 4

. Combining the like terms

9x^6

and

-x^6

gives the final simplified expression.

Key Concept

Polynomial subtraction and binomial expansion

Estimated Time:1m 30s

Question 2529Question →

A toy rocket is launched upward from a platform. Its height $h$ , in meters, above the ground $t$ seconds after launch is modeled by the function $h(t) = -4.9t^2 + 7.35t + 12.25$ . According to this model, how many seconds after launch does the rocket strike the ground?

Show answer & explanation

Answer: 2.5

Answer

The rocket strikes the ground

2.5

seconds after launch.

The correct answer of

2.5

is determined by setting the height

h(t)

to

0

and solving the resulting quadratic equation using the quadratic formula. Since time must be non-negative in this physical context, the negative solution of

-1

is discarded, leaving

2.5

seconds as the time when the rocket strikes the ground.

Step-by-Step Solution

1

Set the height function

h(t)

to

0

.

-4.9t^2 + 7.35t + 12.25 = 0

The rocket strikes the ground when its height above the ground is

0

meters.

2

Apply the quadratic formula

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

.

t = \frac{-7.35 \pm \sqrt{(7.35)^2 - 4(-4.9)(12.25)}}{2(-4.9)}

This formula provides the solutions to any quadratic equation of the form

at^2 + bt + c = 0

.

3

Calculate the discriminant and its square root.

b^2 - 4ac = 294.1225

and

\sqrt{294.1225} = 17.15

Evaluating the term under the radical simplifies the quadratic formula expression.

4

Evaluate the two possible values for

t

.

t = -1

or

t = 2.5

Solving the simplified expression gives the two mathematical roots of the quadratic equation.

5

Choose the physically valid solution.

t = 2.5

Time must be positive in this scenario, so the negative solution

t = -1

is discarded.

Key Concept

Solving a quadratic equation with decimal coefficients using the quadratic formula in a real-world motion context.

Question 2530Question →

In $\triangle ABC$ , the lengths of the sides are $AB = 8$ , $BC = 11$ , and $AC = 14$ . Which of the following inequalities correctly compares the measures of the interior angles of $\triangle ABC$ ?

m\angle A < m\angle B < m\angle C

m\angle B < m\angle A < m\angle C

m\angle A < m\angle C < m\angle B

m\angle C < m\angle A < m\angle B

m\angle B < m\angle C < m\angle A

Show answer & explanation

Answer: $m\angle C < m\angle A < m\angle B$

Answer

The correct inequality is

m\angle C < m\angle A < m\angle B

.

In any triangle, the order of the measures of the interior angles matches the order of the lengths of their opposite sides. Since the side lengths are ordered

AB < BC < AC

(

8 < 11 < 14

), their opposite angles must be ordered in the same way. The angle opposite side

AB

is

\angle C

, the angle opposite side

BC

is

\angle A

, and the angle opposite side

AC

is

\angle B

. Therefore, the correct relationship is

m\angle C < m\angle A < m\angle B

.

Step-by-Step Solution

1

Identify the theorem relating triangle side lengths to their opposite angles.

The Side-Angle Relationship Theorem states that in any triangle, the order of the measures of the angles is the same as the order of the lengths of the sides opposite to those angles.

This establishes the rule needed to compare the angle measures based on the given side lengths.

2

Determine the opposite angle for each side length in

\triangle ABC

.

The angle opposite side

AB

is

\angle C

. The angle opposite side

BC

is

\angle A

. The angle opposite side

AC

is

\angle B

.

This maps each side length to the correct angle it controls.

3

Order the side lengths and apply the mapping to order the angle measures.

Since

8 < 11 < 14

, we write the side inequality as

AB < BC < AC

. Substituting the corresponding opposite angles gives

m\angle C < m\angle A < m\angle B

.

This yields the final correct inequality comparing the angle measures.

Key Concept

Triangle Side-Angle Relationship Theorem

Alternative Method

Another way to solve this is to sketch the triangle to scale. By drawing the longest side

AC = 14

as the horizontal base, the shortest side

AB = 8

on the left, and the side

BC = 11

on the right, it becomes visually apparent that the angle opposite the longest side (

\angle B

at the top vertex) is the largest angle, and the angle opposite the shortest side (

\angle C

at the bottom-right vertex) is the smallest angle.

Estimated Time:1m 0s

Question 2531Question →

In right triangle $ABC$ , the measure of $\angle B$ is $90^\circ$ , the measure of $\angle A$ is $60^\circ$ , and the length of $AC$ is $16$ units. Point $D$ lies on side $BC$ such that the measure of $\angle ADB$ is $45^\circ$ . What is the length of segment $CD$ , rounded to the nearest tenth?

Show answer & explanation

Answer: 5.9

Answer

The length of segment CD is approximately 5.9 units.

By recognizing that triangle ABC is a 30-60-90 right triangle, the shorter leg AB is found to be 8 (half of the hypotenuse 16), and the longer leg BC is 8*sqrt(3). Because triangle ABD is a 45-45-90 right triangle, the leg BD is equal to the leg AB, which is 8. Subtracting BD from BC yields CD = 8*sqrt(3) - 8, which is approximately 5.9.

Step-by-Step Solution

1

Determine the type of triangle ABC

Triangle ABC is a 30-60-90 special right triangle.

The triangle has a right angle (90 degrees) at B and an angle of 60 degrees at A, which leaves 30 degrees for angle C.

2

Calculate the lengths of sides AB and BC

AB = 8 units and BC = 8*sqrt(3) units.

In a 30-60-90 triangle with hypotenuse AC = 16, the side opposite 30 degrees (AB) is half the hypotenuse, and the side opposite 60 degrees (BC) is the shorter leg multiplied by sqrt(3).

3

Determine the type of triangle ABD

Triangle ABD is a 45-45-90 special right triangle.

Since D lies on BC, angle ABD is a right angle (90 degrees). Given that angle ADB is 45 degrees, the remaining angle BAD must also be 45 degrees.

4

Calculate the length of side BD

BD = 8 units.

In a 45-45-90 right triangle, the two legs opposite the 45-degree angles are equal in length, so BD = AB.

5

Calculate the length of segment CD and round to the nearest tenth

CD ≈ 5.9 units.

Since D lies on side BC, CD = BC - BD = 8*sqrt(3) - 8 ≈ 8(1.732) - 8 = 13.856 - 8 = 5.856, which rounds to 5.9.

Key Concept

Applying properties of 30-60-90 and 45-45-90 special right triangles to find segment lengths within nested figures.

Question 2532Question →

Preparing a standard solution of a specific molarity requires a series of precise laboratory steps to ensure accuracy. First, the technician accurately weighs the solid solute on an analytical balance. The weighed solute is then transferred into a clean beaker, where a small volume of distilled water is added. The mixture is stirred gently with a glass rod until the solid is completely dissolved. Following dissolution, the liquid is carefully poured through a funnel into a volumetric flask of the desired volume. To guarantee that no solute is left behind, the beaker and glass rod must be rinsed multiple times with distilled water, and these washings must be added directly to the flask. Next, the technician adds distilled water to the flask until the level approaches the graduation mark on the neck. Using a Pasteur pipette, water is added dropwise until the bottom of the meniscus is exactly aligned with the mark. Finally, the flask is stoppered and inverted several times to ensure the solution is thoroughly mixed and homogeneous. Skipping the rinsing step, or failing to add the rinsings to the flask, will result in an inaccurate, lower concentration than intended.

According to the passage, what is the direct consequence of failing to rinse the beaker and add the washings to the volumetric flask?

The solid solute will not dissolve completely when water is added.

The final concentration of the standard solution will be lower than intended.

The bottom of the meniscus will fail to align with the graduation mark.

The solution will remain unmixed and non-homogeneous after inversion.

Show answer & explanation

Answer: The final concentration of the standard solution will be lower than intended.

Answer

The final concentration of the standard solution will be lower than intended.

The correct answer is correct because the final sentence of the passage states that skipping the rinsing step, or failing to add the rinsings to the flask, will result in a lower concentration than intended. Rinsing ensures all solute is transferred into the volumetric flask.

Step-by-Step Solution

1

Locate the portion of the passage describing the rinsing step and its purpose.

The text states that the beaker and glass rod must be rinsed to guarantee no solute is left behind, and the washings must be added to the flask.

This establishes the procedural purpose of the rinsing step.

2

Locate the sentence addressing the consequence of skipping this step.

The final sentence explicitly states that skipping the rinsing step, or failing to add the rinsings, will result in a lower concentration than intended.

This directly answers the question about the consequence of omitting the step.

3

Evaluate the choices to find the one matching the identified consequence.

The option stating the final concentration will be lower than intended matches the text directly.

This ensures the correct option is selected based on literal comprehension of the sequence.

Key Concept

Identifying the consequences of omitting a specific step in a described laboratory procedure.

Question 2533Question →

What is the maximum value of $x$ that satisfies the inequality $\frac{2 - 5x}{3} \ge 2x + 8$ ?

Show answer & explanation

Answer: -2

Answer

-2

Solving the inequality by isolating

x

leads to

x \le -2

, meaning that any value of

x

less than or equal to

-2

satisfies the inequality. Therefore, the maximum possible value is

-2

.

Step-by-Step Solution

1

Multiply both sides of the inequality by 3 to clear the denominator.

2 - 5x \ge 6x + 24

Multiplying by a positive number eliminates the fraction without changing the inequality direction.

2

Subtract

6x

from both sides of the inequality.

2 - 11x \ge 24

This gathers the variable terms on the left side of the inequality.

3

Subtract 2 from both sides of the inequality.

-11x \ge 22

This isolates the variable term on the left side.

4

Divide both sides by

-11

and reverse the inequality sign.

x \le -2

Dividing by a negative number requires reversing the direction of the inequality sign. The resulting inequality defines the upper bound for

x

.

Key Concept

Solving linear inequalities and applying the sign-flip rule when multiplying or dividing by a negative number.

Question 2534Question →

In the standard $(x, y)$ coordinate plane, a vertex $V$ of a square is translated $3$ units to the left and $4$ units up, and then reflected across the $x$ -axis. If the coordinates of the final image of $V$ are $(2, -5)$ , what are the coordinates of the original vertex $V$ ?

(-1, 9)

(5, 9)

(1, -9)

(5, 1)

(5, -9)

Show answer & explanation

Answer: $(5, 1)$

Answer

The coordinates of the original vertex V are (5, 1)

The correct answer is the coordinates

(5, 1)

. Starting with the original vertex

V(x, y)

, a horizontal translation of

3

units left results in

x - 3

, and a vertical translation of

4

units up results in

y + 4

, yielding the intermediate point

(x - 3, y + 4)

. Reflecting this point across the

x

-axis negates the entire y-coordinate expression, giving

(x - 3, -y - 4)

. Equating this to the final coordinates

(2, -5)

results in

x - 3 = 2

, which solves to

x = 5

, and

-y - 4 = -5

, which simplifies to

-y = -1

, or

y = 1

.

Step-by-Step Solution

1

Represent the transformations algebraically starting from the original vertex

V(x, y)

.

Translating

3

units left and

4

units up maps

V(x, y)

to an intermediate point

V'(x - 3, y + 4)

.

Horizontal translation left subtracts from the x-coordinate, and vertical translation up adds to the y-coordinate.

2

Apply the second transformation, reflecting the intermediate point

V'

across the

x

-axis.

Reflecting

V'(x - 3, y + 4)

across the

x

-axis negates the y-coordinate, resulting in the final image

V''(x - 3, -(y + 4)) = (x - 3, -y - 4)

.

A reflection across the x-axis transforms any point

(a, b)

to

(a, -b)

.

3

Set the algebraic coordinates of the final image equal to the given coordinates

(2, -5)

and solve for

x

and

y

.

x - 3 = 2 \implies x = 5

and

-y - 4 = -5 \implies -y = -1 \implies y = 1

.

This determines the coordinates of the original pre-image vertex

V(5, 1)

.

Key Concept

Performing composite transformations in reverse order to find the pre-image coordinates.

Question 2535Question →

For all real values of $x$ , which of the following inequalities represents the complete solution set to $7 - 3(x - 4) \ge 5x + 31$ ?

x \ge -\frac{3}{2}

x \le -\frac{9}{2}

x \le -\frac{3}{2}

x \ge -\frac{9}{2}

x \le -\frac{25}{4}

Show answer & explanation

Answer: $x \le -\frac{3}{2}$

Answer

The inequality that represents the complete solution set is

x \le -\frac{3}{2}

The correct answer is the inequality stating that

x

is less than or equal to negative three-halves. Simplifying the expression yields

-8x \ge 12

, and dividing by

-8

correctly reverses the inequality direction to produce

x \le -\frac{3}{2}

.

Step-by-Step Solution

1

Distribute

-3

to both terms inside the parentheses on the left side of the inequality.

7 - 3x + 12 \ge 5x + 31

To eliminate the parentheses and prepare the inequality for simplification.

2

Combine the constant terms

7

and

12

on the left side.

19 - 3x \ge 5x + 31

To group like terms on the left side.

3

Subtract

5x

from both sides of the inequality to collect variable terms on one side.

19 - 8x \ge 31

To isolate the variable term on the left side.

4

Subtract

19

from both sides of the inequality.

-8x \ge 12

To isolate the variable term completely before division.

5

Divide both sides of the inequality by

-8

and flip the inequality sign.

x \le -\frac{12}{8}

, which simplifies to

x \le -\frac{3}{2}

Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality sign.

Key Concept

Solving multi-step linear inequalities, specifically reversing the inequality sign when multiplying or dividing by a negative number.

Question 2536Question →

A regular polygon has an interior angle that is $140^\circ$ greater than its exterior angle. How many sides does this polygon have?

8

9

12

18

36

Show answer & explanation

Answer: 18

Answer

The regular polygon has 18 sides.

The correct answer is 18. By setting up the equations for the interior angle

I

and exterior angle

E

, we have

I = E + 140^\circ

and

I + E = 180^\circ

. Substituting the first equation into the second gives

(E + 140^\circ) + E = 180^\circ

, which simplifies to

2E + 140^\circ = 180^\circ \implies 2E = 40^\circ \implies E = 20^\circ

. The number of sides

n

of a regular polygon is given by

n = 360^\circ / E

. Thus,

n = 360^\circ / 20^\circ = 18

.

Step-by-Step Solution

1

Set up the algebraic relationship between the interior angle (

I

) and the exterior angle (

E

).

I = E + 140^\circ

The problem states that the interior angle is

140^\circ

greater than the exterior angle.

2

Use the supplementary relationship between any interior angle and its corresponding exterior angle.

I + E = 180^\circ

An interior angle and its adjacent exterior angle form a linear pair, which sums to

180^\circ

.

3

Substitute the expression for

I

from Step 1 into the equation from Step 2 and solve for

E

.

(E + 140^\circ) + E = 180^\circ \implies 2E + 140^\circ = 180^\circ \implies 2E = 40^\circ \implies E = 20^\circ

Substituting allows us to solve for a single variable representing the exterior angle.

4

Calculate the number of sides (

n

) using the sum of the exterior angles of a convex polygon.

n = \frac{360^\circ}{E} = \frac{360^\circ}{20^\circ} = 18

The sum of the exterior angles of any convex polygon is

360^\circ

, and in a regular polygon, all

n

exterior angles are equal.

Key Concept

Relationship between interior and exterior angles of a regular polygon

Estimated Time:1m 30s

Question 2537Question →

The interior angles of a quadrilateral are in the ratio $2:3:3:4$ . What is the measure, in degrees, of the largest interior angle of this quadrilateral?

60

90

120

180

240

Show answer & explanation

Answer: 120

Answer

120

The correct answer is 120. The sum of the interior angles of any convex quadrilateral is 360 degrees. Given the ratio of the interior angles is

2:3:3:4

, the total number of parts is

2 + 3 + 3 + 4 = 12

. Dividing the total angle sum by the total parts gives the value of one part:

360^\circ / 12 = 30^\circ

. The largest angle has 4 parts, so its measure is

4 \times 30^\circ = 120^\circ

.

Step-by-Step Solution

1

Calculate the sum of the parts in the given ratio of the angles.

2 + 3 + 3 + 4 = 12

parts

To find the fractional share of each angle, we must first determine the total number of equal parts in the ratio.

2

Determine the value in degrees of a single part of the ratio using the sum of interior angles of a quadrilateral.

360^\circ / 12 = 30^\circ

per part

The sum of the interior angles of any convex quadrilateral is 360 degrees. Dividing this sum by the total number of parts gives the angle measure of one part.

3

Multiply the value of one part by the number of parts of the largest angle.

4 \times 30^\circ = 120^\circ

The largest angle corresponds to the largest number in the ratio, which is 4.

Key Concept

Sum of interior angles of a quadrilateral and ratio distribution

Question 2538Question →

In the standard $(x, y)$ coordinate plane, quadrilateral $ABCD$ is a trapezoid with $AB$ parallel to $CD$ . The vertices are $A(2, 9)$ , $B(5, 3)$ , and $C(2, 1)$ . The diagonals $AC$ and $BD$ intersect at point $E$ , and the line segment $BD$ is perpendicular to $AC$ . If the ratio of the area of $\triangle ABE$ to the area of $\triangle CDE$ is $9:1$ , what is the $x$ -coordinate of vertex $D$ ?

-7

-1

1

\frac{5}{3}

3

Show answer & explanation

Answer: $1$

Answer

The correct answer is

1

.

The vertical diagonal

AC

lies on

x = 2

, and the perpendicular diagonal

BD

lies on

y = 3

. Their intersection point is

E(2, 3)

. Since the bases

AB

and

CD

of the trapezoid are parallel, the triangles

\triangle ABE

and

\triangle CDE

formed by the diagonals are similar. The ratio of their areas is

9:1

, which means the ratio of their corresponding sides is the square root of the area ratio, which is

3:1

. The horizontal segment

BE

has a length of

5 - 2 = 3

, meaning the segment

DE

must have a length of

3 \div 3 = 1

. Since

D

must lie to the left of the intersection to keep

CD

parallel to

AB

, its

x

-coordinate is

2 - 1 = 1

.

Step-by-Step Solution

1

Find the equation of the line containing diagonal

AC

.

The line is

x = 2

.

Since vertices

A(2, 9)

and

C(2, 1)

share the same

x

-coordinate of

2

, the diagonal

AC

is a vertical line segment.

2

Find the equation of the line containing diagonal

BD

and locate the intersection

E

.

The line is

y = 3

, and the intersection point is

E(2, 3)

.

Since

BD

is perpendicular to the vertical diagonal

AC

, it must be horizontal. Thus, all points on

BD

share the same

y

-coordinate as

B(5, 3)

. The intersection of

x = 2

and

y = 3

is

E(2, 3)

.

3

Determine the similarity ratio of

\triangle ABE

and

\triangle CDE

.

The ratio of the corresponding sides is

3:1

.

Since

AB \parallel CD

, the alternate interior angles

\angle EAB = \angle ECD

and

\angle EBA = \angle EDC

make

\triangle ABE

similar to

\triangle CDE

. The ratio of the areas of similar triangles is the square of the ratio of their corresponding side lengths, so

\frac{BE}{DE} = \sqrt{9} = 3

.

4

Calculate the length of

BE

and find the length of

DE

.

The length of

BE

is

3

, and the length of

DE

is

1

.

Using the coordinates of

B(5, 3)

and

E(2, 3)

, the horizontal distance is

BE = 5 - 2 = 3

. Since

\frac{BE}{DE} = 3

, we have

DE = 1

.

5

Determine the coordinates of vertex

D

and verify that

AB \parallel CD

.

The

x

-coordinate of

D

is

1

.

Since

DE = 1

and

D

lies on the line

y = 3

,

D

can be at

(3, 3)

or

(1, 3)

. The slope of

AB

is

\frac{3 - 9}{5 - 2} = -2

. If

D

is

(1, 3)

, the slope of

CD

is

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

, which is parallel. If

D

is

(3, 3)

, the slope is

2

, which is not parallel. Thus, the

x

-coordinate of

D

must be

1

.

Key Concept

Properties of Quadrilaterals

Estimated Time:3m 0s

Question 2539Question →

A quadratic equation is defined by $x^2 - bx + 18 = 0$ , where $b$ is a positive constant. If the difference between the two real solutions of this equation is 3, what is the value of $b$ ?

Show answer & explanation

Answer: 9

Answer

The value of the positive constant

b

is 9.

By applying the quadratic formula, the roots of the equation are

x = \frac{b \pm \sqrt{b^2 - 72}}{2}

. The difference between these roots is

\sqrt{b^2 - 72}

. Setting this equal to 3 gives

\sqrt{b^2 - 72} = 3

. Squaring both sides yields

b^2 - 72 = 9

, which simplifies to

b^2 = 81

. Taking the positive root since

b

is a positive constant gives

b = 9

.

Step-by-Step Solution

1

Express the roots of the quadratic equation

x^2 - bx + 18 = 0

using the quadratic formula.

The roots are

x = \frac{b \pm \sqrt{b^2 - 4(1)(18)}}{2} = \frac{b \pm \sqrt{b^2 - 72}}{2}

.

This provides a formulaic representation of the two solutions in terms of the unknown parameter

b

.

2

Subtract the smaller root from the larger root to represent the difference between the solutions, and set this expression equal to 3.

Difference

= \frac{b + \sqrt{b^2 - 72}}{2} - \frac{b - \sqrt{b^2 - 72}}{2} = \sqrt{b^2 - 72} = 3

.

The problem specifies that the difference between the two real solutions is 3.

3

Square both sides of the equation to eliminate the radical, and solve for the positive constant

b

.

b^2 - 72 = 9 \Rightarrow b^2 = 81 \Rightarrow b = 9

(since

b

is positive).

Squaring both sides allows us to isolate

b^2

and find the value of

b

that satisfies the initial condition.

Key Concept

Solving for quadratic coefficients using the difference of roots derived from the quadratic formula.

Alternative Method

Use Vieta's formulas. Let the roots be

r_1

and

r_2

. We know that

r_1 + r_2 = b

and

r_1 r_2 = 18

. We are given that the difference between the roots is 3, so

|r_1 - r_2| = 3

. We can use the algebraic identity

(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1 r_2

. Substituting the known values gives

3^2 = b^2 - 4(18)

, which simplifies to

9 = b^2 - 72

, leading to

b^2 = 81

. Since

b > 0

, we find

b = 9

.

Estimated Time:1m 30s

Question 2540Question →

The following passage describes the history of printing in Europe. Read the passage and consider the historical context. Which form of the verb 'take' correctly completes the sentence to maintain consistent verb tense throughout the passage?

Fill in the blanks below

In the mid-fifteenth century, Johannes Gutenberg introduced movable type printing to Europe, a technology that rapidly transformed the continent's intellectual landscape. Before this innovation, scribes painstakingly copied manuscripts by hand, a process that months or even years for a single book.

Show answer & explanation

Answer

took (or had taken)

The verbs throughout the passage, such as 'introduced', 'transformed', and 'copied', are in the past tense because they describe completed historical actions. Therefore, the verb completing the sentence must also be in the past tense ('took' or 'had taken') to maintain grammatical consistency.

Step-by-Step Solution

1

Analyze the established tense of the passage by identifying surrounding verbs.

The verbs 'introduced', 'transformed', and 'copied' are all in the past tense, establishing a past-tense narrative context.

Maintaining verb tense consistency is necessary to ensure the passage's timeline remains clear and logical.

2

Determine the appropriate tense for the verb 'take' to complete the description of the historical scribal process.

The simple past tense 'took' (or past perfect 'had taken') keeps the verb consistent with the other past-tense actions described in the passage.

Using a present-tense form like 'takes' or a future-tense form like 'will take' would create an illogical shift in tense.

Key Concept

Verb Tense and Consistency

Answer

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