All practice questions

Question 2001Question →

A solid sphere has a volume of $36\pi$ cubic centimeters. What is the radius, in centimeters, of the sphere?

Show answer & explanation

Answer: 3

Answer

The radius of the sphere is 3 centimeters.

The volume of a sphere is given by the formula

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

. Substituting

36\pi

for

V

gives

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

. Dividing both sides by

\pi

results in

36 = \frac{4}{3}r^3

. Multiplying both sides by

\frac{3}{4}

yields

27 = r^3

. Taking the cube root of both sides gives

r = 3

. Thus, the radius of the sphere is 3 centimeters.

Step-by-Step Solution

1

State the sphere volume formula.

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

The volume of a sphere is calculated using this standard formula relating volume to radius.

2

Substitute the given volume value into the formula.

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

The problem specifies the volume is

36\pi

cubic centimeters.

3

Solve for the radius cubed (

r^3

).

r^3 = 27

Dividing both sides of the equation by

\pi

yields

36 = \frac{4}{3}r^3

, and multiplying both sides by

\frac{3}{4}

isolates

r^3

to give 27.

4

Take the cube root of both sides to find the radius (

r

).

r = 3

The cube root of 27 is 3, since

3 \times 3 \times 3 = 27

.

Key Concept

Using the volume of a sphere formula to solve for an unknown radius.

Question 2002Question →

A fitness center surveyed a random sample of its members about their primary workout time and their membership type. The results of the survey are partially shown in the table below.

Primary Workout Time	Basic Membership	Premium Membership	Total
Morning	$60$	$a$
Afternoon	$b$	$20$	$60$
Evening	$60$	$140$
Total	$180$	$300$

Given that a randomly selected member from this survey has a Basic membership, what is the probability that this member's primary workout time is in the evening?

\frac{4}{15}

\frac{4}{9}

\frac{4}{7}

\frac{2}{9}

Show answer & explanation

Answer: $\frac{4}{9}$

Answer

\frac{4}{9}

To find the probability that a randomly selected member's primary workout time is in the evening given that they have a Basic membership, we restrict the denominator to the total number of Basic members, which is

180

. The number of Basic members who work out in the afternoon is

60 - 20 = 40

. Subtracting the morning (

60

) and afternoon (

40

) Basic members from the total Basic members (

180

) yields

180 - 60 - 40 = 80

Basic members who work out in the evening. The probability is

\frac{80}{180}

, which simplifies to

\frac{4}{9}

.

Step-by-Step Solution

1

Find the total number of members with a Premium membership.

The total number of Premium members is

120

.

The grand total of surveyed members is

300

, and the total number of Basic members is

180

. Subtracting the Basic members from the grand total gives the Premium members:

300 - 180 = 120

.

2

Find the value of

b

, which is the number of Basic members whose primary workout time is in the afternoon.

b = 40

The total number of members who work out in the afternoon is

60

, and

20

of them have a Premium membership. Subtracting the Premium members from the afternoon total gives the Basic members:

60 - 20 = 40

.

3

Find the number of Basic members whose primary workout time is in the evening.

There are

80

Basic members who work out in the evening.

The total number of Basic members is

180

. Subtracting the Basic members who work out in the morning (

60

) and afternoon (

40

) from the total number of Basic members gives the remaining Basic members who work out in the evening:

180 - 60 - 40 = 80

.

4

Calculate the conditional probability that a randomly selected member works out in the evening, given they have a Basic membership.

The probability is

\frac{4}{9}

.

The condition restricts the sample space to the

180

members with a Basic membership. Of these

180

members,

80

work out in the evening. The probability is the ratio of these two values:

\frac{80}{180} = \frac{4}{9}

.

Key Concept

Conditional Probability in Two-Way Tables

Alternative Method

Instead of subtracting from the column total to find the missing Basic Evening value, one could first find all Premium values: the Premium Total is

300 - 180 = 120

. Then, the Premium Morning value is

120 - 20 - 60 = 40

. This allows calculation of the Morning Total (

60 + 40 = 100

). The Evening Total is given as

140

, and the Premium Evening value is

60

, which directly gives the Basic Evening value as

140 - 60 = 80

. Finally, the probability is

\frac{80}{180} = \frac{4}{9}

.

Estimated Time:2m 0s

Question 2003Question →

At the beginning of 2015, City A and City B each had a daily water consumption of $1,200,000$ gallons. The daily water consumption of City A decreased linearly by $24,000$ gallons each year, and the daily water consumption of City B decreased exponentially by a constant percentage each year. At the beginning of 2025, the daily water consumption of City A was equal to the daily water consumption of City B. If these trends continue, what will be the daily water consumption of City B, in gallons, at the beginning of 2035?

Show answer & explanation

Answer: 768000

Answer

768,000

To find the daily water consumption of City B at the beginning of 2035, we first determine the consumption of both cities at the beginning of 2025 (

t = 10

years after the beginning of 2015). For City A, which decreases linearly, the consumption is

1,200,000 - 10 \times 24,000 = 960,000

gallons. Since City B's consumption is equal to City A's at this time and decreases exponentially, we set up the equation

1,200,000 \cdot b^{10} = 960,000

, where

b

is the annual decay factor. This gives

b^{10} = 0.8

. The consumption of City B at the beginning of 2035 (

t = 20

) is given by

1,200,000 \cdot b^{20}

. Using exponent rules, we can rewrite

b^{20}

as

(b^{10})^2

. Substituting

b^{10} = 0.8

gives

1,200,000 \cdot (0.8)^2 = 1,200,000 \cdot 0.64 = 768,000

.

Step-by-Step Solution

1

Calculate the consumption of City A at the beginning of 2025.

960,000

gallons

City A decreases linearly by a constant

24,000

gallons per year for

10

years starting from

1,200,000

gallons.

2

Set up the exponential model for City B at the beginning of 2025.

b^{10} = 0.8

City B's consumption decreases exponentially, so it is modeled by

W_B(t) = 1,200,000 \cdot b^t

. Since it equals City A's consumption at

t = 10

,

1,200,000 \cdot b^{10} = 960,000

.

3

Determine the consumption of City B at the beginning of 2035.

768,000

gallons

At the beginning of 2035 (

t = 20

), City B's consumption is

1,200,000 \cdot b^{20}

. Since

b^{20} = (b^{10})^2

, we substitute

0.8

for

b^{10}

to get

1,200,000 \cdot (0.8)^2 = 1,200,000 \cdot 0.64 = 768,000

.

Key Concept

Distinguishing between linear and exponential models and applying exponent properties to solve exponential growth/decay problems.

Alternative Method

Alternatively, you can solve for the annual decay factor

b

directly using a calculator. Since

b^{10} = 0.8

, we take the 10th root of both sides to get

b = 0.8^{0.1} \approx 0.97793

. The water consumption at

t = 20

is then calculated as

1,200,000 \cdot (0.97793)^{20} \approx 768,000

gallons. Recognizing the algebraic shortcut

(b^{10})^2 = b^{20}

avoids decimal approximations and is much faster.

Estimated Time:2m 30s

Question 2004Question →

The population of a species of fish in a lake can be modeled by the function $P(t) = P_0 \cdot 2^{\frac{t}{d}}$ , where $P_0$ is the initial population when the population was first measured, $t$ represents the time in years since it was first measured, and $d$ is a constant representing the doubling time in years. If the population of the fish doubles every 6 years, and the population after 18 years is 3,200, what was the initial population of the fish?

Show answer & explanation

Answer: 400

Answer

The initial population of the fish was 400.

By substituting the doubling time

d = 6

and the final population of 3,200 at

t = 18

into the exponential model

P(t) = P_0 \cdot 2^{\frac{t}{d}}

, we get

3,200 = P_0 \cdot 2^{\frac{18}{6}}

. Simplifying the exponent gives

3,200 = P_0 \cdot 2^3

, which simplifies further to

3,200 = 8P_0

. Dividing both sides by 8 yields

P_0 = 400

.

Step-by-Step Solution

1

Identify the values for the known variables from the word problem.

d = 6

years and at

t = 18

years,

P(18) = 3,200

.

To substitute these values into the exponential growth function model.

2

Substitute the known values into the exponential function

P(t) = P_0 \cdot 2^{\frac{t}{d}}

.

3,200 = P_0 \cdot 2^{\frac{18}{6}}

To set up an equation to solve for the unknown parameter

P_0

.

3

Simplify the exponent and calculate the growth factor.

3,200 = P_0 \cdot 2^3 \Rightarrow 3,200 = 8P_0

Reducing the fractional exponent simplifies the equation.

4

Solve for the initial population

P_0

by dividing both sides of the equation by 8.

P_0 = 400

Isolating

P_0

gives the initial population of the fish.

Key Concept

Using an exponential function to model real-world growth and solving for the initial value.

Question 2005Question →

A right circular cylindrical tank with a base radius of $4\text{ inches}$ contains water. A solid metal sphere with a radius of $3\text{ inches}$ is completely submerged in the water, causing the water level to rise. If no water overflows from the tank, what is the increase, in inches, in the water level of the tank?

0.5625

9.00

2.25

6.75

Show answer & explanation

Answer: 2.25

Answer

2.25

The correct answer is 2.25. The volume of the submerged sphere is calculated using the formula

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

. With a radius of

3\text{ inches}

, the volume is

V = \frac{4}{3}\pi (3)^3 = 36\pi\text{ cubic inches}

. The volume of the displaced water is represented by a cylinder of radius

4\text{ inches}

and height

h

, which represents the increase in water level. The volume of this displaced cylinder is

V = \pi (4)^2 h = 16\pi h

. Since the volume of the displaced water is equal to the volume of the submerged sphere, we set them equal:

16\pi h = 36\pi

. Dividing both sides by

16\pi

yields

h = 2.25

.

Step-by-Step Solution

1

Calculate the volume of the solid metal sphere using the formula

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

with radius

r = 3\text{ inches}

.

V_{\text{sphere}} = \frac{4}{3}\pi (3)^3 = 36\pi\text{ cubic inches}

The volume of the sphere represents the total volume of water that will be displaced when the sphere is completely submerged.

2

Express the volume of the displaced water in the cylinder as a function of the rise in water level,

h

.

V_{\text{displaced}} = \pi (4)^2 h = 16\pi h\text{ cubic inches}

The displaced water forms a cylindrical shape with the same base radius as the tank (

4\text{ inches}

) and a height equal to the change in the water level (

h

).

3

Equate the volume of the displaced water to the volume of the sphere and solve for

h

.

16\pi h = 36\pi \Rightarrow h = \frac{36}{16} = 2.25\text{ inches}

By Archimedes' principle, the volume of the displaced water must equal the volume of the fully submerged solid sphere.

Key Concept

Equating the volume of a submerged object to the volume of displaced liquid in a cylindrical container.

Estimated Time:2m 0s

Question 2006Question →

If $8^{x + 1} = \left(\frac{1}{4}\right)^{2x - 3}$ , what is the value of $x$ ?

\frac{3}{7}

9

\frac{9}{7}

-\frac{9}{7}

Show answer & explanation

Answer: $\frac{3}{7}$

Answer

\frac{3}{7}

To solve the equation, we rewrite both bases using the common base

2

. Because

8 = 2^3

, the left side simplifies to

(2^3)^{x + 1} = 2^{3x + 3}

. Because

\frac{1}{4} = 2^{-2}

, the right side simplifies to

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

. Since the bases are now identical, their exponents must be equal:

3x + 3 = -4x + 6

. Solving this linear equation by adding

4x

to both sides gives

7x + 3 = 6

. Subtracting

3

from both sides gives

7x = 3

. Dividing by

7

yields the value of

\frac{3}{7}

.

Step-by-Step Solution

1

Express both sides of the equation using a common base of

2

.

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

Since

8 = 2^3

and

\frac{1}{4} = 2^{-2}

, we can rewrite the terms with the same base to solve the exponential equation.

2

Apply the distributive property to simplify the exponents.

2^{3x + 3} = 2^{-4x + 6}

Multiplying the outer exponent by each term inside the parentheses simplifies the expression.

3

Set the exponents equal to each other and solve the linear equation for

x

.

3x + 3 = -4x + 6 \implies 7x = 3 \implies x = \frac{3}{7}

When bases are equal and positive (and not 1), their exponents must be equal.

Key Concept

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

Question 2007Question →

A technology company surveyed a random sample of $128$ of its employees to determine their primary operating system and their job role. The table below summarizes the results of the survey.

Job Role	macOS	Windows	Total
Developer	$30$	$50$	$80$
Designer	$30$	$18$	$48$
Total	$60$	$68$	$128$

If an employee from the survey whose job role is developer is selected at random, what is the probability that the employee's primary operating system is Windows?

Show answer & explanation

Answer: 0.625

Answer

5/8 (or 0.625)

The question asks for the probability that a randomly selected employee uses Windows, given that their job role is developer. This restricts the sample space to only the developers. From the table, there are

80

total developers, which serves as the denominator. Within this group of developers,

50

use Windows, which serves as the numerator. The probability is

\frac{50}{80}

, which simplifies to

\frac{5}{8}

or

0.625

. Both

5/8

and

.625

are correct formats that fit the standard SAT grid-in requirements.

Step-by-Step Solution

1

Identify the subgroup defined by the condition.

The total number of developers is

80

.

The probability is conditional on the employee being a developer, which restricts the denominator of the probability to the total number of developers.

2

Identify the favorable outcomes within the subgroup.

There are

50

developers whose primary operating system is Windows.

This value represents the numerator for our conditional probability calculation.

3

Calculate the conditional probability.

The probability is

\frac{50}{80} = \frac{5}{8}

(or

0.625

).

Dividing the favorable outcomes by the total outcomes in the restricted sample space gives the correct conditional probability.

Key Concept

Conditional Probability in Two-Way Tables

Question 2008Question →

In the equation $(2x - 3)(x + 4) = k$ , $k$ is a constant. If $x = 2$ is a solution to the equation, what is the other solution to the equation?

-\frac{9}{2}

\frac{9}{2}

\frac{1}{2}

-9

Show answer & explanation

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

Answer

-\frac{9}{2}

Substituting the known solution

x = 2

into the equation

(2x - 3)(x + 4) = k

yields

(2(2) - 3)(2 + 4) = k

, which simplifies to

(1)(6) = k

, so

k = 6

. Substituting this value back into the equation gives

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

. Expanding the left side yields

2x^2 + 5x - 12 = 6

. Subtracting 6 from both sides places the quadratic equation in standard form:

2x^2 + 5x - 18 = 0

. Since

x = 2

is a root,

(x - 2)

is a factor. Factoring the quadratic gives

(x - 2)(2x + 9) = 0

. Setting the second factor equal to zero,

2x + 9 = 0

, yields the other solution,

x = -\frac{9}{2}

.

Step-by-Step Solution

1

Substitute the known solution

x = 2

into the equation to solve for

k

.

k = 6

Since

x = 2

is a solution, it must satisfy the equation

(2(2) - 3)(2 + 4) = k

, which simplifies to

(1)(6) = k

.

2

Substitute

k = 6

back into the original equation, expand the binomial product, and write the quadratic equation in standard form.

2x^2 + 5x - 18 = 0

Expanding the binomials gives

2x^2 + 5x - 12 = 6

. Subtracting 6 from both sides yields the standard form quadratic equation

2x^2 + 5x - 18 = 0

.

3

Factor the quadratic equation to find the other root.

(x - 2)(2x + 9) = 0

, yielding

x = 2

and

x = -\frac{9}{2}

Since

x = 2

is a solution,

(x - 2)

must be a factor. Dividing the quadratic by

(x - 2)

yields the other factor,

(2x + 9)

.

Key Concept

Solving quadratic equations by substituting a known root to determine constants, then rewriting and factoring the equation.

Alternative Method

Another way to find the other solution is to use the relationship between the coefficients of a quadratic equation and its roots. Once the equation is written in standard form as

2x^2 + 5x - 18 = 0

, the sum of the roots is given by

-\frac{b}{a} = -\frac{5}{2}

. Since one root is

2

, the other root

r

must satisfy

2 + r = -\frac{5}{2}

, which simplifies to

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

.

Estimated Time:1m 30s

Question 2009Question →

An equation is shown below.

\frac{x+2}{x-1} - \frac{4}{x} = \frac{4}{x^2 - x}

What is the real solution to the equation above?

Show answer & explanation

Answer: 2

Answer

The only real solution to the equation is

2

.

To solve the rational equation, we multiply all terms by the common denominator

x(x-1)

, assuming

x \neq 0

and

x \neq 1

. This results in the equation

x(x+2) - 4(x-1) = 4

. Expanding and simplifying this gives

x^2 + 2x - 4x + 4 = 4

, which simplifies to

x^2 - 2x = 0

. Factoring the quadratic expression yields

x(x-2) = 0

, which gives the potential solutions

x = 0

and

x = 2

. However, substituting

x = 0

back into the original equation causes a division by zero. Therefore,

x = 0

is an extraneous solution, and the only valid real solution is

2

.

Step-by-Step Solution

1

Find the common denominator for the terms in the rational equation.

The common denominator is

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

, which requires

x \neq 0

and

x \neq 1

.

Multiplying by the common denominator allows us to eliminate the fractions.

2

Multiply every term in the equation by the common denominator

x(x-1)

and simplify.

x(x+2) - 4(x-1) = 4

This clears the denominators and converts the rational equation into a polynomial equation.

3

Expand and simplify the resulting equation to standard quadratic form.

x^2 + 2x - 4x + 4 = 4

, which simplifies to

x^2 - 2x = 0

.

Grouping like terms is necessary to solve the quadratic equation.

4

Factor the quadratic equation to determine the potential solutions.

x(x-2) = 0

, giving potential solutions of

x = 0

and

x = 2

.

Factoring allows us to find the roots of the quadratic expression.

5

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

x = 0

is extraneous because it leads to division by zero. Thus, the only real solution is

x = 2

.

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

Key Concept

Solving rational equations and identifying extraneous solutions

Question 2010Question →

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

Fill in the blanks below

In the mid-twentieth century, the emergence of Abstract Expressionism was often characterized by critics as a sudden and complete rupture from traditional representation. However, art historian Clement Greenberg argued that the movement was actually a logical of Western painting’s historical trajectory toward flatness, rather than a rejection of it.

Show answer & explanation

Answer

The correct word to fill the blank is 'outgrowth' (or similar synonyms like 'continuation', 'evolution', or 'extension').

The text sets up a contrast between critics who saw Abstract Expressionism as a 'rupture' (a break) from the past, and Clement Greenberg, who argued that it was a natural development of Western painting's trajectory. The blank requires a word that indicates a natural development or result arising from a previous state. Words like 'outgrowth,' 'continuation,' 'evolution,' and 'extension' logically complete the text by indicating that Abstract Expressionism was a direct result of earlier artistic trends.

Step-by-Step Solution

1

Analyze the context of the sentence containing the blank to determine the logical relationship.

The first sentence describes Abstract Expressionism as a 'rupture' from tradition, but the second sentence uses 'However' to contrast this view, presenting Greenberg's argument that it is actually a logical progression.

This establishes that the word in the blank must mean a development or continuation, not a break.

2

Identify the word that completes the text with the most logical and precise meaning.

The word 'outgrowth' (or related words like 'continuation', 'evolution', 'extension') perfectly fits, describing a natural development from an existing trajectory.

It maintains consistency with the idea that the movement was a development of Western painting rather than a rejection of it.

Key Concept

Words in Context: High-Utility Vocabulary

Estimated Time:1m 0s

Question 2011Question →

An electric delivery van's remaining battery capacity, $B$ , in kilowatt-hours (kWh), after driving $d$ miles is modeled by a linear equation. After driving $45$ miles, the remaining battery capacity is $68$ kWh. After driving a total of $120$ miles, the remaining battery capacity is $38$ kWh. According to this model, how many miles can the van drive on a full charge before the battery capacity reaches $0$ kWh?

Show answer & explanation

Answer: 215

Answer

215

To find the maximum distance the van can drive on a full charge, we model the relationship between the remaining battery capacity,

B

, and the distance driven,

d

, using the linear equation

B = md + B_0

. First, calculate the rate of consumption (slope,

m

) using the points

(45, 68)

and

(120, 38)

:

m = \frac{38 - 68}{120 - 45} = \frac{-30}{75} = -0.4

kWh per mile. Next, find the initial battery capacity (

B_0

) by substituting the values from one of the points:

68 = -0.4(45) + B_0

, which simplifies to

68 = -18 + B_0

, so

B_0 = 86

kWh. Finally, determine the distance

d

when the battery is fully depleted (

B = 0

):

0 = -0.4d + 86

, which yields

0.4d = 86

, and

d = 215

miles.

Step-by-Step Solution

1

Calculate the rate of change of the battery capacity per mile driven.

The rate of change (slope) is

-0.4

kWh per mile.

The slope

m

represents the energy consumption rate and is calculated using the two data points

(45, 68)

and

(120, 38)

with the formula

m = \frac{38 - 68}{120 - 45} = \frac{-30}{75} = -0.4

.

2

Determine the initial battery capacity when the distance driven is 0 miles.

The initial battery capacity is

86

kWh.

Using the slope-intercept form

B = md + B_0

, substitute

m = -0.4

and the point

(45, 68)

to get

68 = -0.4(45) + B_0

. This simplifies to

68 = -18 + B_0

, so

B_0 = 86

.

3

Solve for the distance driven

d

when the battery capacity

B

is

0

kWh.

The distance is

215

miles.

Set

B = 0

in the linear model

B = -0.4d + 86

to get

0 = -0.4d + 86

. Solving for

d

gives

0.4d = 86

, which results in

d = \frac{86}{0.4} = 215

.

Key Concept

Interpreting and calculating slope, y-intercept, and x-intercept of a linear function in a real-world context.

Question 2012Question →

For the inequality $-2(3x - 4) + 5 > -3x - 12$ , which inequality represents all possible values of $x$ ?

x > \frac{25}{3}

x < \frac{25}{9}

x < 3

x < \frac{25}{3}

Show answer & explanation

Answer: $x < \frac{25}{3}$

Answer

x < \frac{25}{3}

The correct inequality is obtained by first distributing

-2

to the terms inside the parentheses to get

-6x + 8 + 5 > -3x - 12

. Combining the constants on the left yields

-6x + 13 > -3x - 12

. Adding

3x

to both sides results in

-3x + 13 > -12

, and subtracting 13 from both sides gives

-3x > -25

. Finally, dividing by

-3

and flipping the inequality sign results in the correct solution.

Step-by-Step Solution

1

Distribute

-2

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

-6x + 8 + 5 > -3x - 12

Applying the distributive property simplifies the expression.

2

Combine the constant terms on the left side.

-6x + 13 > -3x - 12

Simplifying the constant terms makes it easier to isolate the variable.

3

Add

3x

to both sides to move all variable terms to the left side.

-3x + 13 > -12

This isolates the variable terms on one side of the inequality.

4

Subtract 13 from both sides to isolate the variable term.

-3x > -25

This isolates the term containing

x

.

5

Divide both sides by

-3

and reverse the direction of the inequality sign.

x < \frac{25}{3}

Dividing by a negative number requires reversing the inequality sign.

Key Concept

Solving linear inequalities in one variable using the distributive property and sign reversal rules.

Question 2013Question →

A right square pyramid has a height of $9\text{ centimeters}$ and a square base with a side length of $5\text{ centimeters}$ . What is the volume, in cubic centimeters, of the pyramid?

15

45

75

225

Show answer & explanation

Answer: 75

Answer

75

The volume of a right square pyramid is given by the formula

V = \frac{1}{3}Bh

, where

B

is the area of the base and

h

is the height. Since the base is a square with side length

5\text{ centimeters}

, its area is

B = 5^2 = 25\text{ square centimeters}

. Substituting the base area and the height of

9\text{ centimeters}

into the formula gives

V = \frac{1}{3} \times 25 \times 9 = 75\text{ cubic centimeters}

.

Step-by-Step Solution

1

Calculate the area of the square base (

B

).

B = 5^2 = 25\text{ square centimeters}

The base is a square with side length

5\text{ centimeters}

, and the area of a square is calculated by squaring its side length.

2

Apply the volume formula for a pyramid:

V = \frac{1}{3}Bh

.

V = \frac{1}{3} \times 25 \times 9 = 75\text{ cubic centimeters}

Substituting the base area of

25\text{ square centimeters}

and the height of

9\text{ centimeters}

into the pyramid volume formula yields the volume.

Key Concept

The volume of a pyramid is calculated using the formula

V = \frac{1}{3}Bh

, where

B

is the area of the base and

h

is the height.

Estimated Time:45s

Question 2014Question →

A technology company monitored the annual carbon emissions of its data centers. In 2025, the data centers' carbon emissions decreased by $12\%$ compared to the emissions in 2024. In 2026, the emissions were $387.2$ metric tons, which represented a $10\%$ increase compared to the emissions in 2025. What were the data centers' carbon emissions, in metric tons, in 2024?

380

394

396

400

Show answer & explanation

Answer: 400

Answer

400

The correct answer is 400. To find the carbon emissions in 2024, we can set up the relationship using the variable

E

for the 2024 emissions. A

12\%

decrease in 2025 means the emissions were

0.88E

. A

10\%

increase in 2026 means the emissions were

1.10 \times 0.88E = 0.968E

. Since the 2026 emissions were

387.2

metric tons, we solve

0.968E = 387.2

to get

E = 400

.

Step-by-Step Solution

1

Set up an equation for the 2025 emissions in terms of the 2024 emissions (

E

).

2025\text{ emissions} = E \times (1 - 0.12) = 0.88E

The emissions in 2025 decreased by

12\%

compared to the emissions in 2024.

2

Set up an equation for the 2026 emissions in terms of the 2025 emissions.

2026\text{ emissions} = 2025\text{ emissions} \times (1 + 0.10) = 1.10 \times 0.88E = 0.968E

The emissions in 2026 represented a

10\%

increase compared to the emissions in 2025.

3

Equate the expression for 2026 emissions to the given value and solve for

E

.

0.968E = 387.2 \implies E = \frac{387.2}{0.968} = 400

The 2026 emissions are given as

387.2

metric tons.

Key Concept

Solving multi-step percent change problems by working backward using the correct base values.

Alternative Method

Instead of setting up an algebraic equation, we can work backward step-by-step. First, find the 2025 emissions by reversing the

10\%

increase from 2025 to 2026:

\frac{387.2}{1.10} = 352

metric tons. Next, find the 2024 emissions by reversing the

12\%

decrease from 2024 to 2025:

\frac{352}{0.88} = 400

metric tons.

Estimated Time:1m 15s

Question 2015Question →

Two startup companies, Company X and Company Y, begin operations at the same time. At the start of operations ( $m = 0$ ), both companies have a monthly revenue of $R$ dollars.

* The monthly revenue of Company X increases by a constant amount of $0.12R$ dollars every $3$ months.
* The monthly revenue of Company Y increases exponentially, growing by a constant percent of $6\%$ every $2$ months.

Let $X(m)$ and $Y(m)$ represent the monthly revenue of Company X and Company Y, respectively, after $m$ months of operation. Which of the following functions represents the ratio $\frac{Y(m)}{X(m)}$ for any positive integer $m$ that is a multiple of $6$ ?

\frac{1 + 0.03m}{1 + 0.04m}

\frac{(1.06)^{\frac{m}{2}}}{1 + 0.12m}

\frac{(1.06)^m}{1 + 0.04m}

\frac{(1.06)^{\frac{m}{2}}}{1 + 0.04m}

Show answer & explanation

Answer: $\frac{(1.06)^{\frac{m}{2}}}{1 + 0.04m}$

Answer

The expression

\frac{(1.06)^{\frac{m}{2}}}{1 + 0.04m}

The correct answer is the expression

\frac{(1.06)^{\frac{m}{2}}}{1 + 0.04m}

. This is found by representing the linear growth of Company X as

X(m) = R(1 + 0.04m)

because its revenue increases by a constant amount of

0.04R

per month (since

0.12R

every 3 months is

\frac{0.12R}{3} = 0.04R

). The exponential growth of Company Y is represented as

Y(m) = R(1.06)^{\frac{m}{2}}

because its revenue increases by

6\%

every 2 months, which yields a growth factor of

1.06

applied

\frac{m}{2}

times over

m

months. The ratio

\frac{Y(m)}{X(m)}

is then

\frac{R(1.06)^{\frac{m}{2}}}{R(1 + 0.04m)}

, which simplifies to the correct expression after canceling the common factor

R

.

Step-by-Step Solution

1

Determine the expression for Company X's monthly revenue,

X(m)

.

X(m) = R(1 + 0.04m)

Since Company X's revenue increases by a constant amount of

0.12R

every 3 months, it grows linearly. The rate of increase per month is

\frac{0.12R}{3} = 0.04R

dollars. Thus, after

m

months, the revenue is

X(m) = R + 0.04R \cdot m = R(1 + 0.04m)

.

2

Determine the expression for Company Y's monthly revenue,

Y(m)

.

Y(m) = R(1.06)^{\frac{m}{2}}

Since Company Y's revenue grows by a constant percent of

6\%

every 2 months, it grows exponentially. The growth factor for each 2-month period is

1 + 0.06 = 1.06

. In

m

months, the number of 2-month compounding periods is

\frac{m}{2}

. Therefore, the revenue is

Y(m) = R(1.06)^{\frac{m}{2}}

.

3

Find the ratio of

Y(m)

to

X(m)

.

\frac{Y(m)}{X(m)} = \frac{(1.06)^{\frac{m}{2}}}{1 + 0.04m}

Divide the expression for

Y(m)

by the expression for

X(m)

:

\frac{Y(m)}{X(m)} = \frac{R(1.06)^{\frac{m}{2}}}{R(1 + 0.04m)}

. The constant initial revenue factor

R

cancels out, simplifying to the final ratio.

Key Concept

Linear and Exponential Growth

Estimated Time:2m 30s

Question 2016Question →

A solid metal right circular cylinder has a height of $h$ and a base radius of $r$ , where $h > 2r$ . From one flat end of the cylinder, a hemisphere of radius $r$ is carved out. From the other flat end, a right circular cone of base radius $r$ and height $h - 2r$ is carved out. If the volume of the remaining solid is equal to the volume of a sphere with radius $R$ , which of the following equations represents $R$ in terms of $r$ and $h$ ?

R = \sqrt[3]{\frac{r^2 h}{2}}

R = \sqrt{\frac{r^2 h}{2}}

R = \sqrt[3]{r^2 h}

R = \frac{r^2 h}{2}

Show answer & explanation

Answer: $R = \sqrt[3]{\frac{r^2 h}{2}}$

Answer

The correct equation is

R = \sqrt[3]{\frac{r^2 h}{2}}

The volume of the cylinder is

\pi r^2 h

. Subtracting the hemisphere's volume (

\frac{2}{3}\pi r^3

) and the cone's volume (

\frac{1}{3}\pi r^2(h - 2r) = \frac{1}{3}\pi r^2 h - \frac{2}{3}\pi r^3

) leaves a remaining volume of

\frac{2}{3}\pi r^2 h

. Setting this equal to the volume of a sphere of radius

R

(

\frac{4}{3}\pi R^3

) gives

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

. Simplifying this equation yields

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

, and taking the cube root gives the correct relation.

Step-by-Step Solution

1

Write the formula for the volume of the original cylinder.

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

The volume of a cylinder with radius

r

and height

h

is given by the formula

\pi r^2 h

.

2

Write the volume formulas for the carved-out shapes.

V_{\text{hemisphere}} = \frac{2}{3}\pi r^3

and

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

A hemisphere is half a sphere, so its volume is

\frac{2}{3}\pi r^3

. The volume of a cone with radius

r

and height

h - 2r

is

\frac{1}{3}\pi r^2 (h - 2r)

.

3

Subtract the volumes of the carved-out shapes from the cylinder's volume.

V_{\text{remaining}} = \pi r^2 h - \frac{2}{3}\pi r^3 - \frac{1}{3}\pi r^2 (h - 2r) = \frac{2}{3}\pi r^2 h

Expanding the cone's volume gives

\frac{1}{3}\pi r^2 h - \frac{2}{3}\pi r^3

. Subtracting this and the hemisphere's volume from the cylinder's volume simplifies to

\frac{2}{3}\pi r^2 h

because the

\pi r^3

terms cancel out.

4

Equate the remaining volume to the volume of a sphere with radius

R

and solve for

R

.

\frac{4}{3}\pi R^3 = \frac{2}{3}\pi r^2 h \implies R^3 = \frac{r^2 h}{2} \implies R = \sqrt[3]{\frac{r^2 h}{2}}

Equating the remaining volume to the volume of a sphere of radius

R

and dividing both sides by

\frac{2}{3}\pi

gives

2 R^3 = r^2 h

. Dividing by 2 and taking the cube root isolates

R

.

Key Concept

Calculating and comparing volumes of combined geometric solids and algebraically isolating variables.

Estimated Time:3m 0s

Question 2017Question →

Acoustic ecologist Bernie Krause coined the term "biophony" to describe the collective sound landscape produced by all living organisms in a given habitat. In a healthy ecosystem, animal species partition their vocalizations across different frequencies and time slots to avoid overlapping with one another ______ acoustic niche hypothesis suggests that vocal space is as limited and contested as physical space.

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

, this

this

; this

, but this

Show answer & explanation

Answer: ; this

Answer

; this

The correct option correctly uses a semicolon to separate two independent clauses. The clause preceding the blank and the clause succeeding the blank are both independent, meaning they contain their own subjects and verbs and can stand alone as complete thoughts. A semicolon is a grammatically standard way to link such closely related clauses.

Step-by-Step Solution

1

Identify the clause boundaries within the sentence.

The text contains two independent clauses: 'In a healthy ecosystem, animal species partition their vocalizations across different frequencies and time slots to avoid overlapping with one another' and 'this acoustic niche hypothesis suggests that vocal space is as limited and contested as physical space.'

Locating the boundaries of the independent clauses helps determine what punctuation or conjunction is necessary to connect them.

2

Analyze the logical relationship between the two independent clauses.

The second clause explains and supports the observation made in the first clause. There is no contrast between them.

Understanding the logical connection prevents the selection of inappropriate coordinators or subordinators.

3

Determine the grammatically correct way to link the independent clauses.

Independent clauses must be separated by a period, a semicolon, a colon, or a comma followed by a coordinating conjunction (such as 'and'). A semicolon followed by the pronoun 'this' correctly links them.

Applying standard English conventions to eliminate grammatical errors like comma splices and run-ons.

Key Concept

Clause Boundaries and Linking

Question 2018Question →

For a real number $x$ , the equation $\sqrt{3x + 10} = x + 2$ is given. What is the value of the expression $x - 1$ ?

2

1

-4

-3

Show answer & explanation

Answer: 1

Answer

The correct answer is the value 1, which is obtained by evaluating the expression for the only valid solution to the equation.

To solve the given radical equation, square both sides to obtain a quadratic equation, which simplifies to

x^2 + x - 6 = 0

. Factoring this equation yields potential solutions of

x = 2

and

x = -3

. Substituting

x = 2

back into the original equation produces a true statement, confirming it as a valid solution. Substituting

x = -3

results in an inequality, identifying it as an extraneous solution. Evaluating the expression for the valid solution yields

2 - 1 = 1

.

Step-by-Step Solution

1

Eliminate the radical by squaring both sides of the equation.

3x + 10 = (x + 2)^2

, which expands to

3x + 10 = x^2 + 4x + 4

.

Squaring both sides removes the square root, allowing the equation to be solved algebraically.

2

Move all terms to one side to set the quadratic equation to zero.

x^2 + x - 6 = 0

.

A quadratic equation must be in the form

ax^2 + bx + c = 0

to solve it by factoring.

3

Factor the quadratic equation.

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

, giving potential solutions of

x = -3

and

x = 2

.

Factoring allows us to find the roots that satisfy the quadratic relationship.

4

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

For

x = 2

,

\sqrt{3(2) + 10} = 2 + 2 \Rightarrow 4 = 4

(valid). For

x = -3

,

\sqrt{3(-3) + 10} = -3 + 2 \Rightarrow 1 = -1

(invalid). Thus,

x = -3

is extraneous and

x = 2

is the only valid solution.

Squaring both sides of an equation can introduce extraneous roots that do not satisfy the original radical equation.

5

Evaluate the requested expression using the valid solution.

x - 1 = 2 - 1 = 1

.

The question asks for the value of the expression rather than the value of the variable itself.

Key Concept

Radical and Rational Equations

Question 2019Question →

A quality control analyst records the battery life, in hours, of 11 distinct prototype laptop batteries. The shortest battery life recorded is 10 hours, the median battery life is 50 hours, and the range of the battery lives is 74 hours. If each battery life is a whole number of hours, what is the maximum possible mean battery life, in hours, of the 11 prototypes?

50

60

61

65

Show answer & explanation

Answer: 60

Answer

The maximum possible mean battery life of the 11 prototypes is 60 hours.

The maximum possible mean is achieved by maximizing the sum of the 11 distinct integers under the given constraints. By setting the elements below the median to the largest possible distinct integers less than 50 (46, 47, 48, 49) and the elements above the median to the largest possible distinct integers less than the maximum value of 84 (80, 81, 82, 83), we find the maximum sum to be 660. Dividing this sum by the 11 elements yields a maximum mean of 60.

Step-by-Step Solution

1

Define the variable terms representing the sorted battery lives.

Let the 11 battery lives in sorted order be

x_1 < x_2 < x_3 < x_4 < x_5 < x_6 < x_7 < x_8 < x_9 < x_{10} < x_{11}

.

Establishing the order of elements helps apply the median and distinct integer constraints.

2

Identify the values of the minimum, maximum, and median elements.

The minimum value is

x_1 = 10

. The median of 11 elements is the 6th element, so

x_6 = 50

. The range is 74, which means the maximum value is

x_{11} = 10 + 74 = 84

.

These fixed values act as boundaries for the remaining elements.

3

Maximize the remaining elements under the distinct integer constraint.

To maximize the mean, we must maximize the sum

S = x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 + x_{10} + x_{11}

. The elements below the median (

x_2, x_3, x_4, x_5

) must be distinct integers strictly less than 50, so their maximum values are 46, 47, 48, and 49. The elements above the median (

x_7, x_8, x_9, x_{10}

) must be distinct integers strictly less than 84, so their maximum values are 80, 81, 82, and 83.

Maximizing the individual elements maximizes the total sum, which in turn maximizes the mean.

4

Calculate the maximum sum and the resulting maximum mean.

The maximum sum is

10 + 46 + 47 + 48 + 49 + 50 + 80 + 81 + 82 + 83 + 84 = 660

. The maximum possible mean is

\frac{660}{11} = 60

.

Dividing the maximum possible sum by the number of elements gives the maximum possible mean.

Key Concept

Maximizing the mean of a dataset with constraints on distinct values, median, and range.

Question 2020Question →

A 3D printer uses a plastic filament at a constant rate of $2.5\text{ grams}$ per minute of printing time. The filament has a density of $1.25\text{ grams per cubic centimeter}$ . If the printer runs continuously for $4\text{ hours}$ , what is the total volume, in cubic centimeters, of the filament used?

8

480

600

750

Show answer & explanation

Answer: 480

Answer

480

The correct answer is 480. First, convert the printing time from hours to minutes: 4 hours multiplied by 60 minutes per hour equals 240 minutes. Next, calculate the total mass of the filament used by multiplying the printing time by the usage rate: 240 minutes multiplied by 2.5 grams per minute equals 600 grams. Finally, convert the mass to volume by dividing the mass by the density: 600 grams divided by 1.25 grams per cubic centimeter equals 480 cubic centimeters.

Step-by-Step Solution

1

Convert the printing time from hours to minutes.

240\text{ minutes}

Since the filament consumption rate is given in grams per minute, the total time must be in minutes. Multiplying

4\text{ hours}

by

60\text{ minutes per hour}

yields

240\text{ minutes}

.

2

Calculate the total mass of the filament used.

600\text{ grams}

Multiplying the total printing time of

240\text{ minutes}

by the consumption rate of

2.5\text{ grams per minute}

gives a total mass of

600\text{ grams}

.

3

Calculate the total volume of the filament used.

480\text{ cubic centimeters}

Divide the total mass of

600\text{ grams}

by the density of

1.25\text{ grams per cubic centimeter}

to find the volume:

\frac{600}{1.25} = 480\text{ cubic centimeters}

.

Key Concept

Multi-unit dimensional analysis involving rates, time, and density.

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Key Concept

Common Mistakes

Alternative Method

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Alternative Method

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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