All practice questions

Question 1961Question →

In the quadratic equation $3x^2 - 15x + c = 0$ , $c$ is a constant. If one of the solutions to the equation is $2$ , what is the value of the other solution?

3

-3

-7

5

Show answer & explanation

Answer: 3

Answer

The other solution to the equation is

3

.

Substituting

x = 2

into the quadratic equation gives

3(2)^2 - 15(2) + c = 0

, which simplifies to

12 - 30 + c = 0

, or

c = 18

. Substituting

c = 18

back into the original equation yields

3x^2 - 15x + 18 = 0

. Dividing the entire equation by

3

results in

x^2 - 5x + 6 = 0

. Factoring this equation gives

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

, which means the solutions are

x = 2

and

x = 3

. Therefore, the other solution is

3

. Alternatively, the sum of the roots of a quadratic equation in the form

ax^2 + bx + c = 0

is given by

-b/a

. For this equation, the sum of the roots is

-(-15)/3 = 5

. Since one root is

2

, the other root must be

5 - 2 = 3

.

Step-by-Step Solution

1

Substitute the given solution

x = 2

into the quadratic equation to find the value of the constant

c

.

3(2)^2 - 15(2) + c = 0 \implies 12 - 30 + c = 0 \implies c = 18

Since

2

is a solution, it must satisfy the equation when substituted for

x

.

2

Substitute

c = 18

back into the original quadratic equation and simplify by dividing both sides by

3

.

3x^2 - 15x + 18 = 0 \implies x^2 - 5x + 6 = 0

Dividing the equation by the greatest common factor simplifies the expression and makes it easier to factor.

3

Factor the simplified quadratic equation to find both solutions.

(x - 2)(x - 3) = 0 \implies x = 2

or

x = 3

The solutions to the factored equation are the values of

x

that make each factor equal to zero.

Key Concept

Solving quadratic equations by substitution and factoring

Estimated Time:1m 30s

Question 1962Question →

A community library starts a digital collection with $5,000$ e-books. The library's goal is to increase the number of e-books in the collection by $8\%$ each year. Which of the following functions best models the number of e-books, $E(t)$ , in the collection $t$ years after the collection was started?

E(t) = 400t + 5,000

E(t) = 5,000(0.08)^t

E(t) = 5,000(1.08)^t

E(t) = 1.08t + 5,000

Show answer & explanation

Answer: $E(t) = 5,000(1.08)^t$

Answer

E(t) = 5,000(1.08)^t

The correct option is the function that represents exponential growth, where the quantity increases by a constant percentage each year. The general formula for exponential growth is

E(t) = a(1 + r)^t

, where

a

is the initial value,

r

is the growth rate as a decimal, and

t

is the number of time periods. Here, the initial number of e-books is

a = 5,000

and the annual growth rate is

r = 8\% = 0.08

. Therefore, the growth factor is

1 + r = 1.08

. Substituting these values into the formula yields the correct model:

E(t) = 5,000(1.08)^t

.

Step-by-Step Solution

1

Determine if the growth is linear or exponential by analyzing the type of increase described.

The problem states that the collection increases by

8\%

each year. Since the increase is a constant percentage rather than a constant quantity, this represents exponential growth.

Identifying the growth type determines whether to use a linear equation or an exponential equation.

2

State the general form of an exponential growth function.

The general formula is

E(t) = a(1 + r)^t

, where

a

is the initial amount,

r

is the percent growth rate expressed as a decimal, and

1 + r

is the growth factor.

This formula provides the template for constructing the model.

3

Identify the values of

a

and

r

from the given context and substitute them into the template.

The initial value

a = 5,000

. The growth rate

r = 8\% = 0.08

, so the growth factor is

1 + 0.08 = 1.08

. Substituting these gives

E(t) = 5,000(1.08)^t

.

This completes the model construction to find the correct expression.

Key Concept

Linear and Exponential Growth

Question 1963Question →

An investment firm analyzes two retirement portfolios, Portfolio P1 and Portfolio P2, each established with an initial deposit of $D$ dollars at time $t = 0$ years. The value of Portfolio P1 increases linearly by a constant amount each year. The value of Portfolio P2 increases exponentially at a constant annual rate. At $t = 10$ years, the value of both portfolios is equal. At $t = 20$ years, the value of Portfolio P2 is exactly $\frac{9}{8}$ times the value of Portfolio P1. Which of the following expressions represents the value of Portfolio P2, in dollars, at $t = 30$ years in terms of $D$ ?

\frac{5}{2}D

\frac{9}{4}D

\frac{27}{8}D

\frac{9}{2}D

Show answer & explanation

Answer: $\frac{27}{8}D$

Answer

The expression that represents the value of Portfolio P2 at

t = 30

years is

\frac{27}{8}D

.

To find the value of Portfolio P2 at

t = 30

, we model the linear portfolio as

V_1(t) = D + ct

and the exponential portfolio as

V_2(t) = D \cdot g^t

. At

t = 10

,

D + 10c = D \cdot g^{10}

, which gives

c = \frac{D}{10}(g^{10}-1)

. At

t = 20

,

V_2(20) = \frac{9}{8}V_1(20) \implies D \cdot g^{20} = \frac{9}{8}(D + 20c)

. Substituting

c

into the equation and simplifying yields the quadratic equation

8(g^{10})^2 - 18(g^{10}) + 9 = 0

. Solving for

g^{10}

and selecting the value greater than 1 (since the portfolio is growing) gives

g^{10} = 1.5

. Thus, at

t = 30

, the value of Portfolio P2 is

D \cdot g^{30} = D \cdot (g^{10})^3 = D \cdot (1.5)^3 = \frac{27}{8}D

.

Step-by-Step Solution

1

Write the models for both portfolios in terms of the initial deposit

D

.

For Portfolio P1 (linear):

V_1(t) = D + c \cdot t

, where

c

is the constant annual increase.
For Portfolio P2 (exponential):

V_2(t) = D \cdot g^t

, where

g

is the annual growth factor.

Establishing the mathematical definitions of linear and exponential growth models is necessary to set up equations representing the given relationships.

2

Use the condition at

t = 10

to express the linear growth rate

c

in terms of

D

and

g^{10}

.

V_1(10) = V_2(10) \implies D + 10c = D \cdot g^{10} \implies 10c = D(g^{10} - 1) \implies c = \frac{D}{10}(g^{10} - 1)

.

This allows us to eliminate the linear growth rate variable

c

and express all equations in terms of the exponential growth term

g^{10}

.

3

Set up the equation for

t = 20

using the given relationship and substitute

c

.

V_2(20) = \frac{9}{8} V_1(20) \implies D \cdot g^{20} = \frac{9}{8} (D + 20c)

. Substituting

c

gives:

D \cdot g^{20} = \frac{9}{8} \left( D + 20 \cdot \frac{D}{10}(g^{10} - 1) \right) \implies D \cdot g^{20} = \frac{9}{8} D \left( 1 + 2(g^{10} - 1) \right)

. Since

D > 0

, we divide both sides by

D

to get:

g^{20} = \frac{9}{8} (2g^{10} - 1)

.

This sets up a solvable quadratic equation for the variable

g^{10}

.

4

Solve the quadratic equation for

g^{10}

.

Let

x = g^{10}

. The equation is

x^2 = \frac{9}{8}(2x - 1) \implies 8x^2 - 18x + 9 = 0

. Factoring the quadratic yields

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

, giving

x = 1.5

or

x = 0.75

. Since the portfolios are growing, the growth factor

g > 1

, so

g^{10} > 1

. Therefore, we choose

g^{10} = 1.5

.

Determining the value of the 10-year growth factor

g^{10}

is the key step to finding the value at future time intervals.

5

Calculate the value of Portfolio P2 at

t = 30

years.

V_2(30) = D \cdot g^{30} = D \cdot (g^{10})^3 = D \cdot (1.5)^3 = D \cdot \left(\frac{3}{2}\right)^3 = \frac{27}{8}D

.

Using the properties of exponents, we express the value at 30 years in terms of the 10-year growth factor cubed.

Key Concept

Linear vs. Exponential Growth Models

Question 1964Question →

Sociologists studying urban communities have observed that community gardens do more than just provide fresh produce to residents in low-income neighborhoods. These shared green spaces serve as vital hubs for daily social interaction ______ also foster social cohesion, strengthen community bonds, and reduce neighborhood crime rates by encouraging residents to work together.

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

they

, they

, and they

, but because they

Show answer & explanation

Answer: , and they

Answer

The option stating ', and they' is the correct answer because it uses a comma and the coordinating conjunction 'and' to correctly link the two independent clauses.

The option stating ', and they' is correct because the passage contains two independent clauses: 'These shared green spaces serve as vital hubs for daily social interaction' and the clause that begins after the blank. Under standard English conventions, two independent clauses must be linked by a period, a semicolon, or a coordinating conjunction preceded by a comma. The correct option correctly uses the comma and the coordinating conjunction 'and' followed by the pronoun 'they' to join these independent clauses.

Step-by-Step Solution

1

Identify the clause boundaries in the sentence.

The sentence is composed of two independent clauses: 'These shared green spaces serve as vital hubs for daily social interaction' and the clause beginning with the blank, which has the subject 'they' and the verb 'foster'.

Both clauses contain a subject and a verb and can stand alone as complete thoughts.

2

Determine the grammatical rules for linking two independent clauses.

Two independent clauses must be joined by a period, a semicolon, or a coordinating conjunction (FANBOYS) preceded by a comma.

Failing to use punctuation results in a run-on sentence, whereas using only a comma results in a comma splice.

3

Evaluate the choices based on standard English conventions.

The choice ', and they' uses a comma and the coordinating conjunction 'and' followed by the subject pronoun 'they', which grammatically links the two clauses.

This is the only choice that avoids grammatical errors such as a run-on, a comma splice, or a fragment.

Key Concept

Clause Boundaries and Linking

Question 1965Question →

A community organization wanted to estimate the proportion of residents in a town who support the construction of a new public library. The organization surveyed a random sample of 200 residents who were visiting the current public library on a Saturday afternoon. Of those surveyed, 150 supported the construction of a new library. Which of the following is the most appropriate conclusion?

The results cannot be generalized to all residents of the town because the sample of people surveyed at the current library is likely biased and not representative of the entire town's population.

Approximately 75% of all residents in the town support the construction of a new library.

The results can be generalized to all residents of the town because a random sample of 200 people is large enough to represent the entire population.

Because 150 out of 200 is 50%, approximately 50% of the people who visit the library on a Saturday support the construction of a new library.

Show answer & explanation

Answer: The results cannot be generalized to all residents of the town because the sample of people surveyed at the current library is likely biased and not representative of the entire town's population.

Answer

The results cannot be generalized to all residents of the town because the sample of people surveyed at the current library is likely biased and not representative of the entire town's population.

The correct option is the one stating that the results cannot be generalized to all town residents because library visitors are likely to support library-related initiatives more than the general public, making the sample non-representative.

Step-by-Step Solution

1

Analyze the population and the sample collection method.

The target population is all residents of the town, but the sample consists of residents visiting the current public library on a Saturday.

To determine if results can be generalized, the sample must be representative of the target population.

2

Evaluate the representativeness of the sample.

People visiting a library are more likely to support a new library than the general public. Thus, the sample has selection bias and is not representative of all town residents.

Bias in the sampling method prevents generalization to the broader population.

Key Concept

Generalizing results from a sample to a population requires that the sample be representative and randomly selected from that population.

Question 1966Question →

A conservationist monitors the populations of two plant species, Species A and Species B, in a forest reserve. The table below shows the population of each species, $P(t)$ and $Q(t)$ respectively, $t$ years after the monitoring began.

$t$ (years)	Species A population, $P(t)$	Species B population, $Q(t)$
$0$	$100$	$100$
$2$	$150$	$144$
$4$	$200$	$207.36$

The population of Species A can be modeled by a linear function, where $r_P$ represents the constant increase in the number of plants per year. The population of Species B can be modeled by an exponential function, where $r_Q\%$ represents the constant percent increase in the number of plants per year. What is the value of $r_P - r_Q$ ?

5

6

19

30

Show answer & explanation

Answer: $5$

Answer

5

To find

r_P

, we calculate the constant amount by which the linear population

P(t)

increases each year. Over the 2-year interval from

t = 0

to

t = 2

, the population increases by

150 - 100 = 50

. Since the growth is linear, the annual increase is

r_P = \frac{50}{2} = 25

plants per year. To find

r_Q

, we write the exponential model in the form

Q(t) = 100(b)^t

, where

b

is the annual growth factor. Using the value at

t = 2

, we have

144 = 100(b)^2

, which simplifies to

b^2 = 1.44

. Taking the square root gives the annual growth factor

b = 1.2

. A growth factor of

1.2

corresponds to an annual growth rate of

20\%

, so

r_Q = 20

. The difference between the two values is

r_P - r_Q = 25 - 20 = 5

. This matches the option with value 5.

Step-by-Step Solution

1

Calculate the annual rate of change (

r_P

) for the linear population model

P(t)

.

r_P = 25

Since Species A's population grows linearly, its rate of change is constant. The population increases from 100 to 150 over 2 years, which is an increase of

150 - 100 = 50

plants. Dividing this by the 2-year interval gives an annual growth rate of

\frac{50}{2} = 25

plants per year.

2

Determine the annual growth factor and percentage growth rate (

r_Q

) for the exponential population model

Q(t)

.

r_Q = 20

Since Species B's population grows exponentially, its population after

t

years can be written in the form

Q(t) = a(b)^t

, where

a

is the initial population and

b

is the annual growth factor. Using the data in the table:

Q(0) = a = 100

and

Q(2) = 100(b)^2 = 144

. Solving for

b

gives

b^2 = 1.44

, which means

b = \sqrt{1.44} = 1.2

. An annual growth factor of

1.2

corresponds to a constant percent increase of

(1.2 - 1) \times 100\% = 20\%

per year. Thus,

r_Q = 20

.

3

Calculate the difference between the two rates (

r_P - r_Q

).

5

Subtracting

r_Q

from

r_P

yields

25 - 20 = 5

.

Key Concept

Identifying and interpreting linear and exponential growth models from data tables, and converting rate of change across different time intervals.

Estimated Time:2m 0s

Question 1967Question →

In the equation $x(x - 8) = k$ , $k$ is a constant. If the product of the two real solutions to the equation is $-20$ , what is the value of the larger solution?

Show answer & explanation

Answer: 10

Answer

The larger solution to the equation is 10.

To find the larger solution, the equation is first rewritten in standard form as

x^2 - 8x - k = 0

. The product of the roots of a quadratic equation in the form

ax^2 + bx + c = 0

is

\frac{c}{a}

. Here,

a = 1

and

c = -k

, so the product of the roots is

-k

. Given that the product of the roots is

-20

, we can set up the equation

-k = -20

, which gives

k = 20

. Substituting

k = 20

back into the equation yields

x^2 - 8x - 20 = 0

. Factoring the quadratic expression gives

(x - 10)(x + 2) = 0

, which has the solutions

x = 10

and

x = -2

. The larger of these two solutions is 10.

Step-by-Step Solution

1

Rewrite the given equation

x(x - 8) = k

in standard quadratic form

ax^2 + bx + c = 0

.

x^2 - 8x - k = 0

To apply quadratic properties, the equation must be in standard form.

2

Use the product of roots formula to find the value of the constant

k

.

-k = -20

, which simplifies to

k = 20

The product of the roots of a quadratic equation in standard form is the constant term divided by the leading coefficient.

3

Substitute the value of

k

back into the quadratic equation and factor it to find the two solutions.

(x - 10)(x + 2) = 0

, so

x = 10

or

x = -2

Factoring the quadratic equation allows us to find the individual roots.

4

Compare the two solutions to identify the larger value.

10

Comparing 10 and -2, 10 is the greater value.

Key Concept

Using the relationship between coefficients and the product of roots to solve a quadratic equation.

Question 1968Question →

A rectangular prism has a volume of $15\text{ cubic centimeters}$ . If the length, width, and height of the prism are all doubled, what is the volume, in cubic centimeters, of the new rectangular prism?

30

120

60

90

Show answer & explanation

Answer: $120$

Answer

120

The volume of a rectangular prism is proportional to the product of its length, width, and height. Doubling each of these dimensions multiplies the volume by a factor of

2 \times 2 \times 2 = 8

. Multiplying the original volume of

15\text{ cubic centimeters}

by

8

gives

120\text{ cubic centimeters}

.

Step-by-Step Solution

1

Identify how the volume of a rectangular prism relates to its dimensions.

The volume is given by

V = l \times w \times h

.

This establishes the relationship between the linear dimensions and the volume of the solid.

2

Determine the scale factor for the volume when all linear dimensions are doubled.

The new dimensions are

2l

,

2w

, and

2h

, which gives a new volume of

(2l)(2w)(2h) = 8(l \times w \times h) = 8V

. Thus, the volume scale factor is

2^3 = 8

.

Volume scales with the cube of the linear dimensions change.

3

Calculate the volume of the scaled prism using the original volume.

New Volume =

8 \times 15 = 120\text{ cubic centimeters}

.

Multiply the original volume by the volume scale factor to obtain the final answer.

Key Concept

Dimensional scaling of solids (volume scales cubically)

Estimated Time:45s

Question 1969Question →

The total daily cost $C$ , in dollars, for a custom apparel company to manufacture $x$ shirts is modeled by a linear equation. The table below shows the total daily cost for two different numbers of shirts manufactured:

Number of shirts, $x$	Total daily cost, $C$ (dollars)
12	260
20	380

Based on the model, what is the total daily cost, in dollars, to manufacture 35 shirts?

262

536

605

785

Show answer & explanation

Answer: 605

Answer

605

To find the total daily cost to manufacture 35 shirts, we first determine the linear relationship

C = mx + b

between the number of shirts manufactured,

x

, and the total cost,

C

. The slope

m

is the constant rate of change, calculated as

m = \frac{380 - 260}{20 - 12} = \frac{120}{8} = 15

dollars per shirt. Using the point-slope form with the point

(12, 260)

, we get

C - 260 = 15(x - 12)

, which simplifies to

C = 15x + 80

. Substituting

35

for

x

in this equation yields

C = 15(35) + 80 = 525 + 80 = 605

. Therefore, the total daily cost to manufacture 35 shirts is 605 dollars.

Step-by-Step Solution

1

Calculate the slope (rate of change) of the linear relationship using the two points from the table,

(12, 260)

and

(20, 380)

.

Slope

m = \frac{380 - 260}{20 - 12} = \frac{120}{8} = 15

dollars per shirt.

Since the relationship is linear, the rate of change is constant.

2

Find the equation of the line using the point-slope form

C - C_1 = m(x - x_1)

with the point

(12, 260)

and slope

m = 15

.

C - 260 = 15(x - 12) \implies C = 15x - 180 + 260 \implies C = 15x + 80

.

This establishes the linear equation relating the number of shirts

x

and the total daily cost

C

.

3

Substitute

x = 35

into the linear equation to find the total daily cost to manufacture 35 shirts.

C = 15(35) + 80 = 525 + 80 = 605

dollars.

This evaluates the linear model at the desired quantity.

Key Concept

Determining a linear equation in two variables from a table of values and using it to predict a value.

Question 1970Question →

The table below shows the distribution of the number of public library cards owned by a group of families in a certain neighborhood.

Number of library cards	Number of families
1	4
2	5
3	2
4	1

What is the median number of public library cards owned by these families?

Show answer & explanation

Answer: 2

Answer

The median number of public library cards is 2.

To find the median, first determine the total number of families by summing the frequencies:

4 + 5 + 2 + 1 = 12

. Since the number of data points is even, the median is the average of the two middle values (

6^{\text{th}}

and

7^{\text{th}}

values). Sorting the data, the first 4 values are 1, and the next 5 values (positions 5 through 9) are 2. Since both the

6^{\text{th}}

and

7^{\text{th}}

values are 2, the median is 2.

Step-by-Step Solution

1

Find the total number of families by adding the frequencies.

4 + 5 + 2 + 1 = 12

To determine the position of the median in the dataset, we must first find the total number of data points.

2

Identify the positions of the middle values.

The

6^{\text{th}}

and

7^{\text{th}}

values.

For an even number of data points (

n = 12

), the median is the average of the values at positions

\frac{n}{2}

and

\frac{n}{2} + 1

.

3

Find the values at the

6^{\text{th}}

and

7^{\text{th}}

positions using the frequency table.

Both the

6^{\text{th}}

and

7^{\text{th}}

values are 2.

Accumulating the frequencies from top to bottom: the first 4 values are 1, and the next 5 values (positions 5 to 9) are 2.

4

Calculate the average of the two middle values.

\frac{2 + 2}{2} = 2

The median of an even-sized dataset is the arithmetic mean of its two middle values.

Key Concept

Calculating the median from a frequency table

Question 1971Question →

Two runners, Estela and Desmond, start at the same point on a circular track and run in opposite directions at constant speeds. The ratio of Estela's speed to Desmond's speed is $5$ to $4$ . If they start at the same time, they meet for the first time after $40$ seconds. If Desmond increases his speed by $25\%$ immediately after their first meeting, how many seconds after their first meeting will they meet for the second time?

Show answer & explanation

Answer: 36

Answer

36

To find the time between the first and second meetings, we determine the track length in terms of a constant

k

. Given the speed ratio of

5

to

4

, the speeds are

5k

and

4k

. The relative speed in opposite directions is the sum of these speeds, or

9k

. In

40

seconds, they cover the track length

L = 9k \times 40 = 360k

. When Desmond increases his speed by

25\%

, his new speed is

4k \times 1.25 = 5k

, making the new relative speed

5k + 5k = 10k

. The time to meet again is the track length divided by the new relative speed:

360k / 10k = 36

seconds.

Step-by-Step Solution

1

Represent the speeds of both runners using the given ratio of

5

to

4

.

Estela's speed is

5k

and Desmond's speed is

4k

, where

k

is a positive constant.

Using a common variable allows calculation of relative rates without needing their absolute speeds.

2

Calculate the initial relative speed of approach.

Relative speed =

5k + 4k = 9k

.

Since they run in opposite directions around the circular track, their rates of travel add up to determine their rate of approach.

3

Determine the track circumference

L

in terms of

k

.

Track length

L = 9k \times 40 = 360k

.

They meet for the first time when their combined distance traveled is exactly equal to one full lap of the track.

4

Calculate Desmond's updated speed after his speed increase.

Desmond's new speed =

4k \times (1 + 0.25) = 5k

.

His speed increases by

25\%

, which corresponds to multiplying his initial speed of

4k

by

1.25

.

5

Find the new relative speed of approach after the first meeting.

New relative speed =

5k + 5k = 10k

.

Estela's speed remains

5k

and Desmond's speed is now

5k

, so their sum gives the new combined rate.

6

Solve for the time elapsed between the first and second meetings.

Time

t = \frac{360k}{10k} = 36

seconds.

To meet a second time, they must cover the track length of

360k

again from their first meeting point at the new relative speed.

Key Concept

Solving multi-step rate-time-distance problems on circular tracks using ratios and percentage change.

Question 1972Question →

An organic farm harvested a certain amount of wheat in 2024. In 2025, due to favorable weather, the amount of wheat harvested was $25\%$ greater than in 2024. In 2026, due to a dry season, the harvest was $12\%$ less than in 2025. If the farm harvested $880$ bags of wheat in 2026, how many bags of wheat did it harvest in 2024?

Show answer & explanation

Answer: 800

Answer

800

The correct answer is 800. Let the 2024 harvest be

w

bags. A

25\%

increase in 2025 means the harvest became

1.25w

. A subsequent

12\%

decrease in 2026 means the harvest became

0.88

of the 2025 value, which is

1.25w \times 0.88 = 1.1w

. Given that the 2026 harvest was

880

bags, we solve

1.1w = 880

to find

w = 800

.

Step-by-Step Solution

1

Define the variable for the 2024 harvest and write an expression for the 2025 harvest.

The harvest in 2025 is

1.25w

, where

w

is the harvest in 2024.

An increase of 25% is equivalent to multiplying the original value by

1 + 0.25 = 1.25

.

2

Write an expression for the 2026 harvest in terms of the 2024 harvest variable.

The harvest in 2026 is

1.1w

.

A decrease of 12% is equivalent to multiplying the 2025 value by

1 - 0.12 = 0.88

. Multiplying the factors gives

1.25 \times 0.88 = 1.1

.

3

Set up an equation using the 2026 harvest total of 880 and solve for the 2024 harvest variable.

w = 800

Dividing the 2026 harvest value of 880 by the cumulative rate of 1.1 yields the initial 2024 value.

Key Concept

Solving for initial values in multi-step percent change scenarios.

Alternative Method

Work backward from the final year. The 2026 harvest of

880

bags is

12\%

less than the 2025 harvest, meaning it represents

88\%

of the 2025 harvest. Thus, the 2025 harvest was

880 / 0.88 = 1000

bags. The 2025 harvest of

1000

bags was

25\%

greater than the 2024 harvest, meaning it represents

125\%

of the 2024 harvest. Thus, the 2024 harvest was

1000 / 1.25 = 800

bags.

Estimated Time:1m 30s

Question 1973Question →

The volume of a right circular cylinder is $72\pi$ . If the radius of the base of the cylinder is $3$ , what is the height of the cylinder?

Show answer & explanation

Answer: 8

Answer

The height of the cylinder is 8.

The volume

V

of a right circular cylinder is given by the formula

V = \pi r^2 h

, where

r

is the base radius and

h

is the height. By substituting the given volume of

72\pi

and radius of

3

, we obtain the equation

72\pi = \pi (3)^2 h

. Simplifying the right side gives

72\pi = 9\pi h

. Dividing both sides of the equation by

9\pi

isolates the height, giving

h = 8

.

Step-by-Step Solution

1

Use the formula for the volume of a right circular cylinder.

V = \pi r^2 h

This formula relates the volume, radius, and height of a cylinder.

2

Substitute the known values into the volume formula.

72\pi = \pi (3)^2 h

To form an equation with the unknown height.

3

Simplify the squared radius and isolate the variable.

h = 8

Simplifying

3^2

gives

9

. Dividing

72\pi

by

9\pi

isolates the height.

Key Concept

Solving for an unknown dimension using the volume formula of a cylinder.

Estimated Time:45s

Question 1974Question →

A quality control manager at a manufacturing plant selected a random sample of $150$ lightbulbs from a batch of $3,000$ lightbulbs produced on a certain day. The manager found that $6$ of the selected lightbulbs were defective. Based on this sample, what is the estimated number of defective lightbulbs in the entire batch of $3,000$ ?

Show answer & explanation

Answer: 120

Answer

120

The correct answer is

120

. Since the sample of lightbulbs was selected at random, the proportion of defective lightbulbs in the sample can be used to estimate the proportion of defective lightbulbs in the entire batch. The proportion of defective lightbulbs in the sample is

\frac{6}{150} = 0.04

. Multiplying this proportion by the total number of lightbulbs in the batch gives the estimated number of defective lightbulbs:

0.04 \times 3,000 = 120

.

Step-by-Step Solution

1

Calculate the proportion of defective lightbulbs in the sample.

The sample proportion is

\frac{6}{150} = 0.04

.

This establishes the rate of defects in the sample to be generalized to the entire batch.

2

Estimate the total number of defective lightbulbs in the batch.

The estimate is

0.04 \times 3,000 = 120

.

Scaling the sample proportion to the population size yields the expected number of defective lightbulbs in the entire batch.

Key Concept

Generalizing results from a random sample to estimate a population parameter

Question 1975Question →

A vertical farm uses an automated nutrient delivery system for a crop of lettuce. The total volume of nutrient solution, $V$ , in liters, remaining in the system's reservoir is modeled as a linear function of the time $t$ , in hours, since the system started running for the day. The table below shows several values of $t$ and the corresponding values of $V$ .

Time, $t$ (hours)	Volume, $V$ (liters)
3	415
5	385
8	340
12	280

According to the model, what was the initial volume of nutrient solution, in liters, in the reservoir when the system started running?

Show answer & explanation

Answer: 460

Answer

460

The correct answer is 460. The remaining volume of nutrient solution,

V

, is a linear function of time,

t

, which can be written in the form

V = mt + b

, where

m

is the rate of change (slope) and

b

is the initial volume (y-intercept). The rate of change can be found using any two points from the table, for example,

(3, 415)

and

(5, 385)

:

m = \frac{385 - 415}{5 - 3} = \frac{-30}{2} = -15

liters per hour. Using the point

(3, 415)

and substituting

m = -15

,

t = 3

, and

V = 415

into the equation

V = mt + b

yields

415 = -15(3) + b

. Simplifying this expression gives

415 = -45 + b

, and adding 45 to both sides gives

b = 460

. Therefore, the initial volume of nutrient solution in the reservoir was 460 liters.

Step-by-Step Solution

1

Calculate the constant rate of consumption (slope) of the nutrient solution.

The rate of consumption is 15 liters per hour.

The relationship between volume and time is linear. The slope

m

can be calculated from two coordinates from the table,

(3, 415)

and

(5, 385)

, as

m = \frac{385 - 415}{5 - 3} = -15

liters per hour.

2

Determine the initial volume of nutrient solution (the y-intercept) using the rate and a data point.

The initial volume is 460 liters.

Substitute the slope

m = -15

, the time

t = 3

, and the remaining volume

V = 415

into the slope-intercept equation

V = mt + b

. Solving

415 = -15(3) + b

gives

b = 460

.

Key Concept

Interpreting the y-intercept of a linear function in context as the initial value of the dependent variable when the independent variable is 0.

Estimated Time:1m 30s

Question 1976Question →

A certain antique watch increases in value by a constant percent each year. The watch was purchased for $150$ , and its value after $2$ years is $216$ . What is the annual percent increase in the value of the watch?

Show answer & explanation

Answer: 20

Answer

The annual percent increase in the value of the watch is 20.

Because the watch increases in value by a constant percent each year, its value can be modeled by the exponential growth equation

V(t) = P(1+r)^t

, where

P

is the initial purchase price,

r

is the annual growth rate as a decimal, and

t

is the number of years. Substituting the given values

P = 150

,

t = 2

, and

V(2) = 216

yields

150(1+r)^2 = 216

. Dividing both sides by

150

gives

(1+r)^2 = 1.44

. Taking the positive square root of both sides results in

1+r = 1.2

, which simplifies to

r = 0.2

. Converting

0.2

to a percentage gives an annual percent increase of

20\%

. Therefore, the correct numeric answer is

20

.

Step-by-Step Solution

1

Set up the exponential growth equation using the given values.

150(1+r)^2 = 216

Since the value increases by a constant percent each year, the growth is exponential and modeled by

V(t) = P(1+r)^t

.

2

Divide both sides of the equation by

150

to isolate the exponential base term.

(1+r)^2 = 1.44

Dividing

216

by

150

simplifies the equation to solve for the growth factor.

3

Take the positive square root of both sides of the equation.

1 + r = 1.2

Taking the square root of

1.44

isolates the expression

1+r

.

4

Solve for

r

and convert the decimal to a percentage.

r = 0.2

, which is

20\%

Subtracting

1

from both sides gives the annual rate

r = 0.2

, and multiplying by

100

gives the percent increase.

Key Concept

Solving for the growth rate in an exponential growth model

Alternative Method

Alternatively, since the question asks for a simple percent increase, you can test a few straightforward percentages. For example, if the annual increase were

10\%

, the value after 1 year would be

150 \times 1.10 = 165

, and after 2 years would be

165 \times 1.10 = 181.5

, which is too low. Testing

20\%

, the value after 1 year is

150 \times 1.20 = 180

, and after 2 years is

180 \times 1.20 = 216

. This matches the given value exactly.

Estimated Time:45s

Question 1977Question →

An industrial water filtration system filters water at a constant rate of $0.5\text{ liters per second}$ . The system consumes a chemical powder at a rate of $2.5\text{ milligrams}$ per gram of impurities removed from the water. If the water contains an average of $0.12\text{ grams}$ of impurities per liter, how many grams of the chemical powder does the system consume during $4\text{ hours}$ of continuous operation?

Show answer & explanation

Answer: 2.16

Answer

The system consumes

2.16\text{ grams}

of chemical powder during the

4

hours of operation.

The correct answer is

2.16

. First, convert the

4

hours of continuous operation into seconds:

4 \times 3,600 = 14,400

seconds. Next, multiply this duration by the filtration rate to find the total water filtered:

14,400 \times 0.5 = 7,200

liters. Multiply the volume by the impurity concentration to find the total mass of impurities:

7,200 \times 0.12 = 864

grams. Then, find the mass of chemical powder consumed:

864 \times 2.5 = 2,160

milligrams. Finally, convert milligrams to grams by dividing by

1,000

, resulting in

2.16

grams.

Step-by-Step Solution

1

Convert the operating time from hours to seconds.

14,400\text{ seconds}

Since the filtration rate is given in liters per second, the total operating time must be converted from hours to seconds to align units. There are

3,600

seconds in one hour.

2

Calculate the total volume of water filtered during the operation.

7,200\text{ liters}

Multiply the operating time in seconds by the filtration rate of

0.5

liters per second.

3

Calculate the total mass of impurities removed from the filtered water.

864\text{ grams}

Multiply the total volume of filtered water by the concentration rate of

0.12

grams of impurities per liter.

4

Determine the mass of chemical powder consumed in milligrams.

2,160\text{ milligrams}

Multiply the total mass of impurities in grams by the consumption rate of

2.5

milligrams of chemical powder per gram of impurities.

5

Convert the mass of chemical powder from milligrams to grams.

2.16\text{ grams}

Divide the mass in milligrams by

1,000

because there are

1,000

milligrams in one gram.

Key Concept

Dimensional analysis and multi-step unit conversion involving rates

Question 1978Question →

The table below summarizes the results of a survey about extracurricular participation among 11th-grade and 12th-grade students at a high school.

Grade	Participates	Does not participate
11th Grade	$45$	$30$
12th Grade	$55$	$20$

Based on the table, if a student who participates in extracurricular activities is selected at random, what is the probability that the student is in the 11th grade? (Express your answer as a decimal or a fraction.)

Show answer & explanation

Answer: 0.45

Answer

0.45 (or 9/20)

To find the probability that a student is in the 11th grade given that they participate in extracurricular activities, the sample space is restricted to only those students who participate. According to the table, the total number of students who participate is

45 + 55 = 100

. Out of these

100

students,

45

are in the 11th grade. The probability is therefore

\frac{45}{100}

, which can be written as

\frac{9}{20}

or

0.45

.

Step-by-Step Solution

1

Determine the size of the restricted sample space by finding the total number of students who participate in extracurricular activities.

100

The probability is conditional on selecting a student who participates in extracurricular activities, so only the 'Participates' column is considered.

2

Identify the number of students within this restricted group who are in the 11th grade.

45

We count the number of students who are both in the 11th grade and participate in extracurricular activities.

3

Calculate the probability by dividing the number of favorable outcomes by the size of the restricted sample space.

45/100 = 9/20 = 0.45

The probability is the ratio of 11th-grade participants to the total participants.

Key Concept

Conditional Probability from a Two-Way Table

Question 1979Question →

The graph of $y = f(x)$ has a relative minimum at the point $(2, -4)$ in the $xy$ -plane. If the function $g$ is defined by $g(x) = 3 - f(2x - 6)$ , what are the coordinates of the corresponding relative maximum on the graph of $y = g(x)$ ?

(4, 7)

(-1, 7)

(4, -1)

(6, 7)

Show answer & explanation

Answer: $(4, 7)$

Answer

(4, 7)

To find the coordinates of the relative maximum on the graph of

y = g(x)

that corresponds to the relative minimum of

y = f(x)

at

(2, -4)

, we first find the new

x

-coordinate. The input to the function

f

must be

2

, so we set the argument

2x - 6 = 2

, which solves to

x = 4

. Next, we find the new

y

-coordinate by evaluating

g(4) = 3 - f(2)

. Substituting

f(2) = -4

, we get

g(4) = 3 - (-4) = 7

. Thus, the coordinates of the corresponding relative maximum are

(4, 7)

.

Step-by-Step Solution

1

Set the argument of the function

f

in

g(x)

equal to the

x

-coordinate of the known point on the graph of

f

.

2x - 6 = 2

The relative minimum of

f

occurs when its input is 2.

2

Solve the equation for

x

to find the transformed

x

-coordinate.

2x = 8 \implies x = 4

Isolating the variable

x

by adding 6 to both sides and then dividing by 2.

3

Substitute

x = 4

into the definition of

g(x)

to calculate the transformed

y

-coordinate.

g(4) = 3 - f(2(4) - 6) = 3 - f(2)

This evaluates the vertical transformation of the function at the corresponding

x

-value.

4

Substitute the value

f(2) = -4

and simplify.

g(4) = 3 - (-4) = 3 + 4 = 7

Since the minimum of

f

is at

(2, -4)

, we know

f(2) = -4

. The negation in front of

f

reflects the minimum to a maximum.

Key Concept

Function Notation and Transformations

Estimated Time:2m 0s

Question 1980Question →

In cognitive psychology, the peak-end rule describes a heuristic in which individuals evaluate their past experiences. When reflecting, people do not average the utility of every single moment of an event _______ they judge the entire experience based primarily on its most intense point and its conclusion. Consequently, a brief but intense spike of discomfort can disproportionately skew a person's memory of an otherwise pleasant activity.

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

instead they

, instead, they

; instead, they

although they

Show answer & explanation

Answer: ; instead, they

Answer

The choice containing a semicolon, the conjunctive adverb 'instead', and a comma ('; instead, they') correctly links the clauses.

The semicolon is used to separate two independent clauses ('When reflecting, people do not average...' and 'they judge the entire experience...'). The conjunctive adverb 'instead' followed by a comma is appropriate to establish the contrast between these two clauses.

Step-by-Step Solution

1

Identify the clauses surrounding the blank.

The clause before the blank ('When reflecting, people do not average the utility of every single moment of an event') and the clause after the blank ('they judge the entire experience based primarily on its most intense point and its conclusion') are both independent clauses.

This determines what type of punctuation or conjunction is needed to link them.

2

Evaluate the choices based on standard punctuation rules for joining independent clauses.

Two independent clauses must be joined by a period, a semicolon, a colon, or a comma plus a coordinating conjunction (FANBOYS).

Using a comma alone results in a comma splice, and using no punctuation results in a run-on.

3

Select the choice that correctly links the clauses while maintaining logical flow.

The choice containing a semicolon, the conjunctive adverb 'instead', and a comma ('; instead, they') correctly links the clauses.

It separates the two independent clauses grammatically and logically.

Key Concept

Clause Boundaries and Linking

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

Alternative Method

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Key Concept

Common Mistakes

Answer