All practice questions

Question 2021Question →

For a third-degree polynomial $p(x)$ , the expression $x^2 - 4x + 4$ is a factor. When $p(x)$ is divided by $x - 3$ , the remainder is $5$ , and when $p(x)$ is divided by $x + 1$ , the remainder is $-27$ . What is the remainder when $p(x)$ is divided by $x - 1$ ?

Show answer & explanation

Answer: 1

Answer

The remainder when the polynomial is divided by

x - 1

is

1

.

The correct answer is

1

. The polynomial is expressed as

p(x) = (x-2)^2(ax + b)

since it is a third-degree polynomial with a factor of

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

. Applying the Remainder Theorem, we evaluate the polynomial at

x = 3

and

x = -1

, giving the system of equations

3a + b = 5

and

-a + b = -3

. Solving this system yields

a = 2

and

b = -1

, meaning the polynomial is

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

. Finally, the remainder when

p(x)

is divided by

x - 1

is

p(1) = (1-2)^2(2(1) - 1) = 1

.

Step-by-Step Solution

1

Express the third-degree polynomial in terms of its known quadratic factor.

p(x) = (x-2)^2(ax + b)

Since the polynomial is of degree 3 and has a quadratic factor of

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

, the remaining factor must be linear, of the form

ax+b

.

2

Apply the Remainder Theorem to set up the system of equations.

p(3) = 5

and

p(-1) = -27

The Remainder Theorem states that the remainder of a polynomial

p(x)

when divided by

x - c

is equal to

p(c)

.

3

Substitute the values of

x

into the polynomial expression.

3a + b = 5

and

9(-a + b) = -27

Evaluating

p(3) = (3-2)^2(3a + b) = 5

yields

3a + b = 5

, and evaluating

p(-1) = (-1-2)^2(-a + b) = -27

yields

9(-a + b) = -27

.

4

Simplify the second equation and solve the system of linear equations.

a = 2

and

b = -1

Dividing the second equation by

9

gives

-a + b = -3

. Subtracting this from the first equation (

3a + b = 5

) gives

4a = 8

, which means

a = 2

. Substituting

a = 2

into the first equation yields

b = -1

.

5

Calculate the remainder when

p(x)

is divided by

x - 1

.

p(1) = 1

According to the Remainder Theorem, the remainder of

p(x)

divided by

x - 1

is

p(1)

. Substituting

x = 1

into

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

yields

(1-2)^2(2(1) - 1) = 1

.

Key Concept

Using the Remainder Theorem and known factors of a polynomial to solve for unknown coefficients.

Question 2022Question →

A forestry service worker randomly selected $80$ oak trees in a state park containing $2,000$ oak trees. The worker found that $12$ of the selected oak trees showed signs of a leaf disease. Based on this sample, what is the estimated number of oak trees in the entire state park that show signs of the leaf disease?

Show answer & explanation

Answer: 300

Answer

300

Because the sample of

80

oak trees was selected at random, it can be assumed to be representative of the population of

2,000

oak trees in the state park. The proportion of diseased trees in the sample is

\frac{12}{80} = 0.15

. Multiplying this proportion by the total population of

2,000

oak trees yields the estimated total number of diseased trees:

0.15 \times 2,000 = 300

.

Step-by-Step Solution

1

Calculate the sample proportion of diseased trees.

0.15

Dividing the

12

diseased trees by the sample size of

80

trees gives the proportion of diseased trees in the sample:

\frac{12}{80} = 0.15

.

2

Generalize the sample proportion to the entire population of oak trees in the park.

300

Multiplying the sample proportion of

0.15

by the total population of

2,000

oak trees yields the estimated total number of diseased trees:

0.15 \times 2,000 = 300

.

Key Concept

Generalizing results from a representative random sample to estimate a population parameter.

Question 2023Question →

A food safety laboratory tested $200$ samples of organic and conventional produce to detect the presence of a specific agricultural residue. The table below summarizes the results.

Produce Type	Residue Detected	Residue Not Detected	Total
Organic	$15$	$65$	$80$
Conventional	$45$	$75$	$120$
Total	$60$	$140$	$200$

If one of the tested samples is selected at random, given that the sample had no residue detected, what is the probability that the selected sample is conventional produce?

\frac{3}{8}

\frac{5}{8}

\frac{15}{28}

\frac{13}{28}

Show answer & explanation

Answer: $\frac{15}{28}$

Answer

The correct probability is

\frac{15}{28}

.

The correct answer is

\frac{15}{28}

. To find the conditional probability that a randomly selected sample is conventional produce given that it has no residue detected, we restrict our focus to the column labeled 'Residue Not Detected'. The total number of samples in this column is

140

(

65

organic +

75

conventional). Out of these

140

samples,

75

are conventional produce. The probability is therefore

\frac{75}{140}

, which simplifies to

\frac{15}{28}

by dividing both the numerator and the denominator by

5

.

Step-by-Step Solution

1

Identify the total number of outcomes that satisfy the given condition.

The total number of samples with no residue detected is

65 + 75 = 140

.

Since the question asks for the probability 'given that the sample had no residue detected', the sample space is restricted to only the column 'Residue Not Detected'.

2

Identify the number of favorable outcomes within the restricted sample space.

Within the group of samples with no residue detected, the number of conventional produce samples is

75

.

We need to find the count where the sample is conventional produce under the condition that no residue was detected.

3

Calculate the conditional probability and simplify the fraction.

\text{Probability} = \frac{75}{140} = \frac{15}{28}

.

The probability is the ratio of favorable outcomes to the total outcomes in the restricted sample space. Dividing the numerator and denominator by their greatest common divisor,

5

, yields

\frac{15}{28}

.

Key Concept

Conditional Probability and Two-Way Tables

Estimated Time:1m 15s

Question 2024Question →

An electric vehicle consumes $0.35$ kilowatt-hours ( $\text{kWh}$ ) of electricity per mile traveled. The vehicle travels at a constant speed of $48$ miles per hour. What is the vehicle's rate of electricity consumption, in $\text{kWh}$ per minute?

Show answer & explanation

Answer: 0.28

Answer

0.28

To find the rate of electricity consumption in kilowatt-hours (

\text{kWh}

) per minute, we must convert the given rate of

0.35\text{ kWh}

per mile into

\text{kWh}

per minute. First, convert the speed of

48\text{ miles per hour}

to miles per minute:

\frac{48\text{ miles}}{60\text{ minutes}} = 0.8\text{ miles per minute}

. Next, multiply the energy consumed per mile by the distance traveled per minute:

0.35\text{ kWh per mile} \times 0.8\text{ miles per minute} = 0.28\text{ kWh per minute}

. Therefore, the vehicle's rate of electricity consumption is

0.28\text{ kWh}

per minute.

Step-by-Step Solution

1

Convert the vehicle's speed from miles per hour to miles per minute.

0.8

miles per minute

Since there are 60 minutes in 1 hour, dividing the speed in miles per hour by 60 gives the speed in miles per minute:

\frac{48\text{ miles}}{1\text{ hour}} \times \frac{1\text{ hour}}{60\text{ minutes}} = 0.8\text{ miles per minute}

.

2

Calculate the rate of electricity consumption in kWh per minute.

0.28

kWh per minute

Multiply the consumption rate per mile by the distance traveled per minute:

0.35\text{ kWh per mile} \times 0.8\text{ miles per minute} = 0.28\text{ kWh per minute}

.

Key Concept

Unit conversion involving rates and compound units.

Alternative Method

First, find the total electricity consumed in one hour of driving by multiplying the consumption rate per mile by the miles traveled in one hour:

0.35\text{ kWh per mile} \times 48\text{ miles} = 16.8\text{ kWh}

. Then, convert this hourly rate to a minute rate by dividing by 60 minutes:

\frac{16.8\text{ kWh}}{60\text{ minutes}} = 0.28\text{ kWh per minute}

.

Estimated Time:1m 15s

Question 2025Question →

A dataset consists of $15$ positive integers. The median of the dataset is $24$ , and the mean is $20$ . The dataset has a unique mode of $32$ , which occurs exactly $5$ times. What is the maximum possible range of the dataset?

Show answer & explanation

Answer: 81

Answer

The maximum possible range of the dataset is 81.

The total sum of the 15 elements is 300. With the median (8th term) equal to 24, we minimize the first 7 terms to 10 by using four 1s and three 2s (since no element other than 32 can repeat 5 or more times). Setting the 9th term to 24 and the 10th through 14th terms to the mode of 32 minimizes the sum of the first 14 terms to 218. This leaves a maximum possible value of 82 for the 15th term. The range is the difference between the maximum and minimum values: 82 - 1 = 81.

Step-by-Step Solution

1

Calculate the total sum of the dataset.

The sum of all 15 elements is

300

.

Since the mean of 15 elements is 20, the sum must be

15 \times 20 = 300

.

2

Identify the median and set up the sorted terms.

The 8th term

x_8 = 24

.

For 15 sorted elements, the median is the 8th term.

3

Minimize the sum of the first 7 terms.

The minimum sum of

x_1

through

x_7

is

10

, with

x_1 = 1

.

To minimize the sum, we use the smallest positive integers. However, no value other than 32 can appear 5 or more times. Thus, we can have at most four 1s and three 2s:

4(1) + 3(2) = 10

.

4

Minimize the 9th term.

x_9 = 24

.

Since the dataset is sorted,

x_9 \geq x_8 = 24

. To minimize the sum of the other terms and maximize

x_{15}

, we set

x_9 = 24

.

5

Account for the mode of 32.

x_{10} = x_{11} = x_{12} = x_{13} = x_{14} = 32

.

The mode 32 appears exactly 5 times. Since it is greater than the median 24, these 5 occurrences must be in the upper half of the sorted list.

6

Calculate the maximum possible value of the 15th term and the range.

x_{15} = 82

, and the range is

81

.

Subtracting the minimum sum of the first 14 terms from the total sum:

300 - (10 + 24 + 24 + 160) = 82

. The range is

x_{15} - x_1 = 82 - 1 = 81

.

Key Concept

Calculating measures of center (mean, median, mode) and variability (range) under constrained datasets.

Estimated Time:3m 0s

Question 2026Question →

In the quadratic equation $x^2 - 10x + c = 0$ , $c$ is a constant. If the two real solutions to the equation have a difference of 6, what is the value of $c$ ?

Show answer & explanation

Answer: 16

Answer

16

The correct answer is 16. By using the quadratic formula, the two solutions of the equation

x^2 - 10x + c = 0

are

5 + \sqrt{25 - c}

and

5 - \sqrt{25 - c}

. The difference between these two solutions is

2\sqrt{25 - c}

. Given that the difference is 6, we set

2\sqrt{25 - c} = 6

, which simplifies to

\sqrt{25 - c} = 3

. Squaring both sides gives

25 - c = 9

, which yields

c = 16

.

Step-by-Step Solution

1

Use the quadratic formula to express the solutions of

x^2 - 10x + c = 0

.

The solutions are

x = 5 \pm \sqrt{25 - c}

.

This expresses the roots of the quadratic equation in terms of the constant

c

.

2

Set the difference between the two solutions equal to 6.

(5 + \sqrt{25 - c}) - (5 - \sqrt{25 - c}) = 6

, which simplifies to

2\sqrt{25 - c} = 6

.

We are given that the two real solutions have a difference of 6.

3

Solve the equation

2\sqrt{25 - c} = 6

for

c

.

\sqrt{25 - c} = 3 \implies 25 - c = 9 \implies c = 16

.

This isolates the constant

c

using standard algebraic operations.

Key Concept

Solving quadratic equations and using properties of roots.

Alternative Method

Alternatively, we can use the relationship between the roots of a quadratic equation. If the roots are

x_1

and

x_2

, then

x_1 + x_2 = 10

and

x_1 x_2 = c

. Using the identity

(x_1 - x_2)^2 = (x_1 + x_2)^2 - 4x_1 x_2

, we substitute the given values:

(6)^2 = (10)^2 - 4c

. This simplifies to

36 = 100 - 4c

, which gives

4c = 64

, or

c = 16

.

Estimated Time:1m 30s

Question 2027Question →

An analyst recorded the number of years of experience for $25$ employees at a software company. The distribution of the employees' experience is shown in the table below:

Years of Experience	Number of Employees
$1$	$5$
$2$	$8$
$3$	$6$
$4$	$4$
$15$	$2$

If the $2$ employees with $15$ years of experience are removed from the dataset, which of the following statements is true?

The mean of the dataset will decrease, and the median of the dataset will remain the same.

Both the mean and the median of the dataset will decrease.

The median of the dataset will decrease, and the mean of the dataset will remain the same.

The mean of the dataset will decrease, and the median of the dataset will increase.

Show answer & explanation

Answer: The mean of the dataset will decrease, and the median of the dataset will remain the same.

Answer

The mean of the dataset will decrease, and the median of the dataset will remain the same.

Removing the two extreme values of

15

years of experience significantly reduces the sum of the data, causing the mean to decrease from

3.4

to approximately

2.39

. However, the median remains

2

because both the

13^{\text{th}}

value of the original

25

-element dataset and the

12^{\text{th}}

value of the new

23

-element dataset fall within the category of

2

years of experience.

Step-by-Step Solution

1

Calculate the initial mean and identify the initial median of the dataset.

Initial mean is

3.4

years, and initial median is

2

years.

To establish the baseline values. The total number of employees is

5 + 8 + 6 + 4 + 2 = 25

. The sum of the years of experience is

(1 \times 5) + (2 \times 8) + (3 \times 6) + (4 \times 4) + (15 \times 2) = 5 + 16 + 18 + 16 + 30 = 85

. The initial mean is

\frac{85}{25} = 3.4

. Since there are

25

data points, the median is the

13^{\text{th}}

data point when ordered. Cumulatively counting the frequencies: the first

5

values are

1

, and the next

8

values (positions

6

to

13

) are

2

. Therefore, the

13^{\text{th}}

value is

2

.

2

Calculate the new mean and identify the new median after removing the two employees with

15

years of experience.

New mean is approximately

2.39

years, and new median is

2

years.

To find the new measures of center after the outliers are removed. The new total number of employees is

25 - 2 = 23

. The new sum of experience is

85 - 30 = 55

. The new mean is

\frac{55}{23} \approx 2.39

. With

23

data points, the median is the

12^{\text{th}}

data point when ordered. Cumulatively, the first

5

values are

1

, and the next

8

values (positions

6

to

13

) are

2

. Therefore, the

12^{\text{th}}

value is still

2

.

3

Compare the initial and new values of the mean and median.

The mean decreases from

3.4

to approximately

2.39

, while the median remains

2

.

To determine which option correctly describes the relationship between the changes.

Key Concept

The mean is sensitive to extreme values (outliers) and changes when they are removed. The median is a resistant measure of center and remains unchanged if the middle position of the ordered data stays within the same value class.

Question 2028Question →

A hydroponics facility uses two nutrient solutions, Solution X and Solution Y. In a standard growth tank, the ratio of the volume of Solution X to the volume of Solution Y is $3$ to $5$ . In a specialized high-yield tank, the ratio of the volume of Solution X to the volume of Solution Y is $4$ to $7$ . A technician mixes $24$ gallons of the mixture from the standard growth tank with $44$ gallons of the mixture from the high-yield tank. What is the ratio of the volume of Solution X to the volume of Solution Y in the final mixture?

7 to 12

25 to 43

43 to 25

5 to 9

Show answer & explanation

Answer: 25 to 43

Answer

25 to 43

To find the combined ratio, we must first calculate the actual volume of each solution in both mixtures. For the standard mixture, the ratio of Solution X to Solution Y is

3:5

, meaning Solution X represents

\frac{3}{3+5} = \frac{3}{8}

of the total volume, and Solution Y represents

\frac{5}{8}

. In

24

gallons of this mixture, there are

\frac{3}{8} \times 24 = 9

gallons of Solution X and

\frac{5}{8} \times 24 = 15

gallons of Solution Y. For the high-yield mixture, the ratio is

4:7

, so Solution X represents

\frac{4}{4+7} = \frac{4}{11}

of the total volume, and Solution Y represents

\frac{7}{11}

. In

44

gallons of this mixture, there are

\frac{4}{11} \times 44 = 16

gallons of Solution X and

\frac{7}{11} \times 44 = 28

gallons of Solution Y. Combining the mixtures yields a total of

9 + 16 = 25

gallons of Solution X and

15 + 28 = 43

gallons of Solution Y. Therefore, the ratio of the volume of Solution X to the volume of Solution Y in the final mixture is

25

to

43

.

Step-by-Step Solution

1

Determine the volume of Solution X and Solution Y in the

24

gallons of standard growth tank mixture.

Solution X volume is

9

gallons; Solution Y volume is

15

gallons.

The ratio of Solution X to Solution Y is

3:5

, which means there are

3 + 5 = 8

total parts. Solution X makes up

\frac{3}{8}

of the mixture, so its volume is

\frac{3}{8} \times 24 = 9

gallons. Solution Y makes up

\frac{5}{8}

of the mixture, so its volume is

\frac{5}{8} \times 24 = 15

gallons.

2

Determine the volume of Solution X and Solution Y in the

44

gallons of specialized high-yield tank mixture.

Solution X volume is

16

gallons; Solution Y volume is

28

gallons.

The ratio of Solution X to Solution Y is

4:7

, which means there are

4 + 7 = 11

total parts. Solution X makes up

\frac{4}{11}

of the mixture, so its volume is

\frac{4}{11} \times 44 = 16

gallons. Solution Y makes up

\frac{7}{11}

of the mixture, so its volume is

\frac{7}{11} \times 44 = 28

gallons.

3

Calculate the total volumes of Solution X and Solution Y in the final combined mixture.

Total Solution X =

25

gallons; Total Solution Y =

43

gallons.

Adding the respective volumes from both mixtures gives: Total Solution X =

9 + 16 = 25

gallons. Total Solution Y =

15 + 28 = 43

gallons.

4

Find the ratio of the total volume of Solution X to the total volume of Solution Y.

The final ratio is

25

to

43

.

Comparing the total volume of Solution X (

25

gallons) to Solution Y (

43

gallons) yields the ratio

25:43

(or

25

to

43

).

Key Concept

Combining mixtures with different component ratios by calculating the absolute amounts of each component using part-to-whole fractions.

Question 2029Question →

In analyzing the balance of state authority, many political theorists argue that the rapid expansion of executive power during national emergencies is an absolute necessity for maintaining public order. However, legal scholar Elena Rostova seeks to ________ this conventional view; while she concedes that swift decision-making is vital in times of crisis, she maintains that such authority must be bound by rigorous legislative oversight to prevent permanent overreach.

Which choice completes the text with the most logical and precise word or phrase?

qualify

satisfy

validate

dismantle

Show answer & explanation

Answer: qualify

Answer

qualify

The correct answer is 'qualify'. In academic prose, 'qualify' means to limit or modify a statement or position. The passage contrasts the view that executive expansion is an absolute necessity with Rostova's perspective. Because Rostova concedes that some executive speed is necessary but argues that it must be bound by oversight, she is limiting the scope of the conventional view rather than accepting or rejecting it entirely.

Step-by-Step Solution

1

Analyze the context of the sentence containing the blank and the surrounding sentences.

The passage sets up a contrast using the word 'However.' On one hand, many theorists argue that executive power expansion is an absolute necessity. On the other hand, Elena Rostova seeks to modify this view. She does not completely reject it, as she concedes that speed is vital, but she insists that this power must be bound by legislative oversight.

Understanding the relationship between the opposing ideas helps determine the tone and meaning of the missing word.

2

Identify the precise meaning required for the blank and evaluate the choices.

The missing word must mean to limit or modify a claim to make it less absolute. The word 'qualify' has a secondary, academic definition meaning to limit or modify a statement. Other options like 'validate' (confirm) or 'dismantle' (destroy) represent extremes that do not fit Rostova's qualified agreement.

Matching the contextual definition to the correct vocabulary choice ensures semantic precision.

Key Concept

Words in Context: High-Utility Vocabulary

Question 2030Question →

A science museum offers two ticketing options for groups. Option A is a flat group rate of $125 plus$ 9.50 per person. Option B is a flat group rate of $50 plus$ 12.50 per person. For a group of $p$ people, Option A is less expensive than Option B. What is the minimum number of people in the group for this to be true?

Show answer & explanation

Answer: 26

Answer

The minimum number of people in the group is 26.

To find when Option A is less expensive than Option B, we set up the inequality representing their respective costs:

125 + 9.5p < 50 + 12.5p

. Subtracting

9.5p

from both sides gives

125 < 50 + 3p

. Subtracting

50

from both sides yields

75 < 3p

. Dividing by

3

gives

p > 25

. Because the group must consist of a whole number of people, the minimum integer value of

p

that is strictly greater than

25

is

26

.

Step-by-Step Solution

1

Write the inequality representing the cost comparison between the two ticketing options.

125 + 9.5p < 50 + 12.5p

Option A's cost must be strictly less than Option B's cost for Option A to be less expensive.

2

Isolate the variable

p

by subtracting

9.5p

and

50

from both sides of the inequality.

p > 25

Subtracting

9.5p

yields

125 < 50 + 3p

. Subtracting

50

yields

75 < 3p

. Dividing by

3

yields

p > 25

.

3

Identify the minimum integer value of

p

that satisfies the inequality.

26

Since the number of people must be a positive integer, the smallest integer strictly greater than

25

is

26

.

Key Concept

Solving linear inequalities in one variable and interpreting the solution set within a discrete real-world context.

Estimated Time:1m 30s

Question 2031Question →

In the high-altitude deserts of the Andes Mountains, specialized extremophile bacteria thrive under intense ultraviolet radiation that would normally be lethal to most organisms. To survive in this incredibly harsh environment, the microbes produce protective carotenoids ______ these vibrant organic pigments shield their delicate cellular structures from severe solar damage. Which choice completes the text so that it conforms to the conventions of Standard English?

carotenoids

carotenoids, but

carotenoids;

carotenoids,

Show answer & explanation

Answer: carotenoids;

Answer

The choice ending with a semicolon properly links the two independent clauses.

The correct option uses a semicolon to separate two independent clauses: 'To survive in this incredibly harsh environment, the microbes produce protective carotenoids' and 'these vibrant organic pigments shield their delicate cellular structures from severe solar damage.' Using a semicolon is a grammatically correct way to link two closely related independent clauses without a coordinating conjunction.

Step-by-Step Solution

1

Identify the clause boundaries by analyzing the sentence structure around the blank.

The text before the blank ('To survive in this incredibly harsh environment, the microbes produce protective carotenoids') is an independent clause. The text after the blank ('these vibrant organic pigments shield their delicate cellular structures from severe solar damage') is also an independent clause.

Understanding whether the clauses are independent or dependent determines the type of punctuation or conjunction needed to connect them.

2

Evaluate the relationship between the two independent clauses to determine if a coordinating conjunction is appropriate.

The second clause explains how the carotenoids help the bacteria survive. There is no contrast between the two ideas, so a contrasting coordinator is incorrect.

This helps eliminate choices that introduce inappropriate transition words.

3

Select the punctuation mark that conforms to standard English conventions for linking two independent clauses without a coordinating conjunction.

A semicolon is a standard way to link two independent clauses.

This ensures the final sentence is grammatically correct and free of comma splices or run-on errors.

Key Concept

Clause Boundaries and Linking

Question 2032Question →

Which form of the verb 'confirm' correctly completes the passage to conform to the conventions of Standard English?

Fill in the blanks below

In 1915, the prominent physicist Albert Einstein presented his general theory of relativity, a breakthrough that revolutionized our understanding of gravity and space. Shortly after this publication, astronomers tested the theory's predictions during a solar eclipse in 1919 and that gravity indeed bends light. This observation provided the first empirical confirmation of Einstein's mathematical model.

Show answer & explanation

Answer

confirmed

The passage establishes a historical narrative set in the past, as shown by the verbs 'presented', 'revolutionized', 'tested', and 'provided'. The missing verb is coordinate with the past-tense verb 'tested' ('astronomers tested... and confirmed'). The simple past tense form 'confirmed' is required to maintain chronological and grammatical consistency.

Step-by-Step Solution

1

Determine the time frame of the passage by examining surrounding verbs.

The verbs 'presented', 'revolutionized', 'tested', and 'provided' are all in the simple past tense, establishing a clear past-time context.

Standard English conventions require verb tenses to remain consistent within a passage unless there is a clear logical reason to shift time frames.

2

Identify the grammatical role of the verb in the blank.

The blank serves as the second part of a compound predicate, joined by the conjunction 'and' to the past-tense verb 'tested' ('astronomers tested... and confirmed').

Verbs sharing the same subject in a compound predicate should maintain parallel tense and aspect.

3

Select the past-tense inflection of the verb 'confirm'.

The simple past tense form is 'confirmed'.

This form matches the tense of 'tested' and preserves the historical timeline of the passage.

Key Concept

Verb Tense Consistency

Estimated Time:45s

Question 2033Question →

Right circular Cylinder $A$ has a base radius of $r$ and a height of $h$ . Right circular Cylinder $B$ has a base radius that is $3$ times the base radius of Cylinder $A$ , and a height that is $\frac{1}{2}$ the height of Cylinder $A$ . What is the ratio of the volume of Cylinder $B$ to the volume of Cylinder $A$ ?

\frac{3}{2}

\frac{9}{4}

\frac{9}{2}

18

Show answer & explanation

Answer: $\frac{9}{2}$

Answer

The ratio of the volume of Cylinder

B

to the volume of Cylinder

A

is

\frac{9}{2}

.

The volume of a cylinder is given by the formula

V = \pi r^2 h

. For Cylinder

A

, the volume is

V_A = \pi r^2 h

. For Cylinder

B

, the radius is

3r

and the height is

\frac{1}{2}h

. Substituting these values gives

V_B = \pi (3r)^2 \left(\frac{1}{2}h\right) = \frac{9}{2}\pi r^2 h

. The ratio of the volume of Cylinder

B

to the volume of Cylinder

A

is therefore the ratio of nine to two.

Step-by-Step Solution

1

Write the volume formula for Cylinder

A

in terms of its radius

r

and height

h

.

V_A = \pi r^2 h

To establish a baseline volume for Cylinder

A

using the standard formula for the volume of a right circular cylinder.

2

Express the radius and height of Cylinder

B

in terms of

r

and

h

, and substitute them into the volume formula.

The radius of Cylinder

B

is

3r

and the height is

\frac{1}{2}h

. Thus,

V_B = \pi (3r)^2 \left(\frac{1}{2}h\right) = \pi (9r^2) \left(\frac{1}{2}h\right) = \frac{9}{2}\pi r^2 h

.

To find the volume of Cylinder

B

expressed in terms of the variables

r

and

h

.

3

Divide the volume of Cylinder

B

by the volume of Cylinder

A

to find the ratio.

\frac{V_B}{V_A} = \frac{\frac{9}{2}\pi r^2 h}{\pi r^2 h} = \frac{9}{2}

To calculate the ratio of the volume of Cylinder

B

to Cylinder

A

.

Key Concept

Dimensional scaling of the volume of a cylinder

Question 2034Question →

An equation is shown below.

\frac{x}{x-3} - \frac{2}{x+1} = \frac{8}{x^2-2x-3}

What is the real solution to the equation above?

Show answer & explanation

Answer: 2

Answer

The only real solution is 2.

To solve the rational equation, multiply both sides by the least common denominator, which is

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

. This yields

x(x+1) - 2(x-3) = 8

. Simplifying this equation gives

x^2 - x - 2 = 0

. Factoring the quadratic expression gives

(x-2)(x+1) = 0

, which yields potential solutions of

x = 2

and

x = -1

. However, substituting

x = -1

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

x = 2

into the original equation is valid, so the only real solution is 2.

Step-by-Step Solution

1

Factor the quadratic denominator on the right side of the equation.

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

This helps identify the least common denominator of the rational terms.

2

Multiply all terms of the equation by the least common denominator,

(x - 3)(x + 1)

, to eliminate the denominators.

x(x + 1) - 2(x - 3) = 8

Multiplying by the LCD clears the rational expressions, converting the equation into a polynomial equation, under the restriction that

x \neq 3

and

x \neq -1

.

3

Expand and simplify the resulting equation.

x^2 + x - 2x + 6 = 8 \Rightarrow x^2 - x - 2 = 0

This puts the equation into standard quadratic form:

ax^2 + bx + c = 0

.

4

Factor the quadratic equation.

(x - 2)(x + 1) = 0

Factoring allows us to find the potential solutions by setting each factor equal to zero.

5

Find the roots of the equation.

x = 2 \text{ or } x = -1

These are the values of

x

that satisfy the factored quadratic equation.

6

Check the potential solutions in the original equation to identify any extraneous solutions.

Substituting

x = -1

results in division by zero in the terms

\frac{2}{x+1}

and

\frac{8}{x^2-2x-3}

, so

x = -1

is extraneous. Substituting

x = 2

yields a valid statement:

-\frac{8}{3} = -\frac{8}{3}

.

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

Key Concept

Solving rational equations and identifying extraneous solutions.

Question 2035Question →

A team of glaciologists monitors the thickness of an alpine glacier during the summer season. The thickness of the glacier, $T$ , in meters, can be modeled by the equation $T = 54.5 - 0.12d$ , where $d$ represents the number of days since the start of the summer season, for $0 \leq d \leq 150$ . Which of the following is the best interpretation of the number $0.12$ in this context?

The initial thickness of the glacier, in meters, at the start of the summer season

The number of days it takes for the thickness of the glacier to decrease by

1

meter

The decrease in the thickness of the glacier, in meters, each day during the summer season

The total number of days from the start of the summer season until the glacier melts completely

Show answer & explanation

Answer: The decrease in the thickness of the glacier, in meters, each day during the summer season

Answer

The correct answer is the option stating that 0.12 is the decrease in the thickness of the glacier, in meters, each day during the summer season.

The linear model is given by

T = 54.5 - 0.12d

, which is in the slope-intercept form

y = mx + b

, where

m

is the slope and

b

is the y-intercept. In this equation, the slope

m

is

-0.12

, which represents the rate of change of the glacier's thickness per day. Since the slope is negative, the thickness decreases by

0.12

meters each day. Therefore, the number

0.12

represents the decrease in the thickness of the glacier, in meters, each day during the summer season.

Step-by-Step Solution

1

Identify the components of the linear equation

T = 54.5 - 0.12d

.

The constant term is

54.5

, and the coefficient of the variable

d

is

-0.12

.

To interpret a linear relationship in context, we must distinguish between the initial value (y-intercept) and the rate of change (slope).

2

Determine the meaning of the slope in this context.

The slope is

-0.12

meters per day, meaning the glacier's thickness

T

decreases by

0.12

meters for each day

d

that passes.

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

3

Relate the number

0.12

to the rate of change.

The positive value

0.12

represents the magnitude of this rate of change, which is the amount of decrease in thickness per day.

Since the slope is negative, the change is a decrease, and the rate of this decrease is

0.12

meters per day.

Key Concept

Interpreting the slope of a linear relationship in a real-world context

Question 2036Question →

The graph of the quadratic function $f$ in the $xy$ -plane has its vertex at $(2, -5)$ . If the graph passes through the point $(5, 13)$ , what is the value of $f(-1)$ ?

-23

-3

13

-11

Show answer & explanation

Answer: 13

Answer

13

The vertex of the parabola is

(2, -5)

, which indicates that the axis of symmetry is the vertical line

x = 2

. The given point has an

x

-coordinate of

5

, which is a distance of

3

units from the axis of symmetry (

5 - 2 = 3

). The target point has an

x

-coordinate of

-1

, which is also a distance of

3

units from the axis of symmetry (

2 - (-1) = 3

). Because a parabola is perfectly symmetric about its axis of symmetry, any two points that are the same horizontal distance from this line must have the same

y

-coordinate. Thus,

f(-1)

must be equal to

f(5)

, which is

13

. Alternatively, one can find the specific equation of the quadratic function by substituting the vertex and the point

(5, 13)

into the vertex form

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

, yielding

a = 2

. Evaluating

f(-1) = 2(-1 - 2)^2 - 5

gives

13

.

Step-by-Step Solution

1

Identify the axis of symmetry from the given vertex.

The axis of symmetry is

x = 2

.

For any quadratic function with a vertex at

(h, k)

, the vertical line

x = h

is the axis of symmetry of its parabolic graph.

2

Determine the horizontal distance from the axis of symmetry to the given point

x = 5

and the target point

x = -1

.

The distance for

x = 5

is

5 - 2 = 3

units. The distance for

x = -1

is

2 - (-1) = 3

units.

Checking if the two

x

-coordinates are symmetric with respect to the line

x = 2

allows us to use the symmetry property of parabolas.

3

Apply the symmetry property to find the function value.

Since both

x = 5

and

x = -1

are equidistant from the axis of symmetry, their function values are equal:

f(-1) = f(5) = 13

.

Points on a parabola that are equidistant from the axis of symmetry have the same

y

-coordinate.

Key Concept

Symmetry of quadratic functions about their vertex axis of symmetry

Question 2037Question →

A square picture has an area of $16$ square inches. If the side length of the picture is doubled, what is the area, in square inches, of the new picture?

8

32

64

128

Show answer & explanation

Answer: $64$

Answer

64

square inches

The area of a square with side length

s

is given by

s^2

. Since the original area is

16

square inches, the original side length is

\sqrt{16} = 4

inches. Doubling this side length gives a new side length of

8

inches. Therefore, the area of the new square is

8^2 = 64

square inches. Alternatively, when all linear dimensions of a two-dimensional shape are multiplied by a scale factor

k

, the area is multiplied by

k^2

. Since the side length is doubled (

k = 2

), the area is multiplied by

2^2 = 4

, resulting in

16 \times 4 = 64

square inches.

Step-by-Step Solution

1

Determine the side length of the original square.

The side length is

\sqrt{16} = 4

inches.

The area of a square is given by

A = s^2

, where

s

is the side length. So,

s = \sqrt{A}

.

2

Calculate the side length of the new square after doubling.

The new side length is

4 \times 2 = 8

inches.

The problem states that the side length is doubled.

3

Calculate the area of the new square.

The new area is

8^2 = 64

square inches.

The area of the new square is the square of its new side length (

8 \text{ inches} \times 8 \text{ inches}

).

Key Concept

When a two-dimensional shape is scaled by a factor of

k

, its area is scaled by a factor of

k^2

.

Question 2038Question →

A solid rectangular prism has a volume of $192$ cubic inches. The ratio of the length of the prism to its width is $3:1$ , and the height of the prism is $4$ inches. What is the surface area, in square inches, of the prism?

Show answer & explanation

Answer: 224

Answer

The surface area of the prism is 224 square inches.

The correct answer is 224. By using the volume formula

V = lwh

with a volume of

192

and a height of

4

, the product of the length and width of the base is

lw = 48

. Given that the ratio of the length to the width is

3:1

, the length can be written as

l = 3w

. Substituting this expression into the product equation gives

3w(w) = 48

, or

3w^2 = 48

. Dividing both sides by 3 yields

w^2 = 16

, which means the width of the prism is

4

inches. Since

l = 3w

, the length is

12

inches. The surface area of the prism can be found using the formula

SA = 2(lw + lh + wh)

. Substituting the dimensions

l = 12

,

w = 4

, and

h = 4

yields

SA = 2(12 \cdot 4 + 12 \cdot 4 + 4 \cdot 4) = 2(48 + 48 + 16) = 2(112) = 224

square inches.

Step-by-Step Solution

1

Use the volume formula for a rectangular prism,

V = lwh

, and substitute the given volume of

192

and height of

4

.

192 = l \cdot w \cdot 4

, which simplifies to

l \cdot w = 48

.

To find the product of the length and width of the base of the prism.

2

Express the length in terms of the width using the ratio of

3:1

.

l = 3w

.

The ratio of the length to the width is given as 3 to 1.

3

Substitute

l = 3w

into the equation

l \cdot w = 48

and solve for

w

.

3w \cdot w = 48 \implies 3w^2 = 48 \implies w^2 = 16 \implies w = 4

.

To solve for the width of the rectangular prism.

4

Find the length of the rectangular prism.

l = 3(4) = 12

inches.

Since the length is 3 times the width and the width is 4 inches, the length must be 12 inches.

5

Use the surface area formula

SA = 2(lw + lh + wh)

with the dimensions

l = 12

,

w = 4

, and

h = 4

.

SA = 2(12 \cdot 4 + 12 \cdot 4 + 4 \cdot 4) = 2(48 + 48 + 16) = 2(112) = 224

.

To calculate the total surface area of the prism.

Key Concept

Volume and Surface Area of Rectangular Prisms

Question 2039Question →

A team of $10$ workers, all working at the same constant rate, can complete a project in $18$ days. After working together for $6$ days, additional workers are hired, all of whom work at the same rate as the original workers. If the remaining portion of the project is completed in $8$ days, how many additional workers were hired?

Show answer & explanation

Answer: 5

Answer

The correct answer is 5. Five additional workers were hired to complete the remaining work in 8 days.

The correct answer is

5

. To find the number of additional workers hired, we first calculate the total work required for the project in worker-days. Since

10

workers can complete the project in

18

days, the total work is

10 \times 18 = 180

worker-days. In the first

6

days, the

10

workers complete

10 \times 6 = 60

worker-days of work, leaving

180 - 60 = 120

worker-days of work. To complete this remaining work in

8

days, the total number of workers required is

\frac{120}{8} = 15

workers. Subtracting the original

10

workers gives

15 - 10 = 5

additional workers.

Step-by-Step Solution

1

Calculate the total amount of work required for the project.

180

worker-days

Since

10

workers can complete the project in

18

days, the total work is the product of the number of workers and the number of days:

10 \times 18 = 180

worker-days.

2

Calculate the work completed in the first

6

days.

60

worker-days

The original

10

workers worked for

6

days, completing

10 \times 6 = 60

worker-days of work.

3

Find the remaining work to be done.

120

worker-days

Subtracting the completed work from the total work gives the remaining work:

180 - 60 = 120

worker-days.

4

Find the total number of workers required to finish the remaining work in

8

days.

15

workers

Dividing the remaining work of

120

worker-days by the target time of

8

days gives the total number of workers needed:

\frac{120}{8} = 15

workers.

5

Calculate the number of additional workers hired.

5

workers

Subtract the original

10

workers from the total

15

workers required for the second phase:

15 - 10 = 5

.

Key Concept

Inverse variation and rate-time-work relationships, specifically using the concept of worker-days to solve multi-stage rate problems.

Estimated Time:2m 30s

Question 2040Question →

A cubic polynomial function $p$ with integer coefficients has exactly two $x$ -intercepts at $(-2, 0)$ and $(1, 0)$ in the $xy$ -plane. The graph of $y = p(x)$ is tangent to the $x$ -axis at one of these intercepts and intersects the $y$ -axis at $(0, -6)$ . What is the remainder when $p(x)$ is divided by $x + 3$ ?

48

-48

-6

-60

Show answer & explanation

Answer: 48

Answer

48

The correct answer is 48. A cubic polynomial

p(x)

with exactly two

x

-intercepts at

(-2, 0)

and

(1, 0)

that is tangent to the

x

-axis at one of these intercepts must have one factor of multiplicity 2. This gives two possible forms:

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

or

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

. Using the

y

-intercept

(0, -6)

, we can solve for the coefficients: if

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

, then

p(0) = 2a = -6

, which gives

a = -3

. If

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

, then

p(0) = -4b = -6

, which gives

b = 1.5

. Since the polynomial must have integer coefficients, the correct function is

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

. By the Remainder Theorem, dividing

p(x)

by

x + 3

yields a remainder equal to

p(-3)

. Substituting -3 into the function gives

p(-3) = -3(-3+2)(-3-1)^2 = -3(-1)(16) = 48

.

Step-by-Step Solution

1

Identify the general form of the cubic polynomial based on its x-intercepts and their multiplicities.

The polynomial must be of the form

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

or

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

.

Since there are exactly two x-intercepts and the graph is tangent to the x-axis at one of them, one of the factors must have a multiplicity of 2.

2

Determine the coefficients using the y-intercept (0, -6) and check which function has integer coefficients.

Evaluating at x = 0 gives either

2a = -6 \Rightarrow a = -3

or

-4b = -6 \Rightarrow b = 1.5

. Thus, the correct polynomial with integer coefficients is

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

.

The y-intercept gives the value of the function at x = 0. The constraint requires integer coefficients, which rules out b = 1.5.

3

Apply the Remainder Theorem to find the remainder when p(x) is divided by x + 3.

The remainder is

p(-3) = -3(-3+2)(-3-1)^2 = -3(-1)(-4)^2 = 48

.

According to the Remainder Theorem, the remainder of a polynomial p(x) divided by x - c is p(c). Here, c = -3.

Key Concept

Polynomial Factors and Graphs

Estimated Time:2m 0s

Answer

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