All practice questions

Question 2041Question →

A right circular cylinder has a height of $12\text{ centimeters}$ . If the total surface area of the cylinder is $170\pi\text{ square centimeters}$ , what is the volume, in cubic centimeters, of the cylinder?

120\pi

300\pi

2040\pi

3468\pi

Show answer & explanation

Answer: $300\pi$

Answer

The volume of the cylinder is

300\pi

cubic centimeters.

The correct answer is the volume of

300\pi

cubic centimeters. We find this by using the cylinder's surface area equation

2\pi r h + 2\pi r^2 = 170\pi

with

h = 12

, which reduces to

r^2 + 12r - 85 = 0

. Factoring this equation gives the radius

r = 5

. Plugging

r = 5

and

h = 12

into the volume formula

V = \pi r^2 h

gives

300\pi

.

Step-by-Step Solution

1

Set up the formula for the total surface area of a cylinder and plug in the given values.

2\pi r h + 2\pi r^2 = 170\pi

, where

h = 12

. This simplifies to

2\pi r(12) + 2\pi r^2 = 170\pi

, or

24\pi r + 2\pi r^2 = 170\pi

.

We need to find the radius of the cylinder to calculate its volume.

2

Divide the entire equation by

2\pi

to simplify it into a standard quadratic form.

r^2 + 12r - 85 = 0

.

Simplifying the equation makes it easier to solve for the radius

r

.

3

Factor the quadratic equation to find the positive value of the radius

r

.

(r + 17)(r - 5) = 0

, which gives

r = 5

since the radius must be positive (

r > 0

).

Factoring allows us to solve for the variable

r

.

4

Use the radius

r = 5

and height

h = 12

to compute the volume of the cylinder.

V = \pi r^2 h = \pi (5^2)(12) = 25 \times 12 \pi = 300\pi

cubic centimeters.

The volume formula for a cylinder is

V = \pi r^2 h

.

Key Concept

Calculating volume from total surface area for a cylinder by solving a quadratic relation.

Question 2042Question →

What is the solution set for the inequality $5 - 2(3x - 1) \geq -4x + 11$ ?

x \leq -2

x \geq -2

x \leq -4

x \geq -4

Show answer & explanation

Answer: $x \leq -2$

Answer

The correct solution is

x

is less than or equal to

-2

.

To solve the inequality, first distribute the negative two to the terms inside the parentheses to get

5 - 6x + 2 \geq -4x + 11

. Combining the constants on the left side yields

7 - 6x \geq -4x + 11

. Adding four

x

to both sides results in

7 - 2x \geq 11

, and subtracting seven from both sides isolates the variable term, giving

-2x \geq 4

. Dividing both sides by negative two and reversing the inequality sign yields the correct solution,

x

is less than or equal to negative two.

Step-by-Step Solution

1

Distribute the

-2

to the terms inside the parentheses.

5 - 6x + 2 \geq -4x + 11

To simplify the expression by removing the parentheses.

2

Combine the constant terms on the left side of the inequality.

7 - 6x \geq -4x + 11

To group like terms together before isolating the variable.

3

Add

4x

to both sides of the inequality.

7 - 2x \geq 11

To move all variable terms to one side of the inequality.

4

Subtract

7

from both sides of the inequality.

-2x \geq 4

To isolate the variable term.

5

Divide both sides by

-2

and reverse the inequality sign.

x \leq -2

Dividing an inequality by a negative number requires reversing the direction of the inequality sign to keep the statement true.

Key Concept

Solving linear inequalities in one variable using the distributive property and applying the sign reversal rule when dividing by a negative number.

Estimated Time:1m 30s

Question 2043Question →

The mass of a sample of a chemical compound in a reaction decays exponentially. The mass, in grams, of the sample $t$ hours after the reaction starts can be modeled by the function $M(t) = a \cdot b^t$ , where $a$ and $b$ are positive constants. If the mass of the sample is $18$ grams after $2$ hours and $8$ grams after $4$ hours, what is the initial mass, in grams, of the sample?

Show answer & explanation

Answer: 40.5

Answer

40.5

The initial mass of the sample is

40.5

grams (which can also be entered as the fraction

81/2

). This is found by setting up the two equations from the given points:

a \cdot b^2 = 18

and

a \cdot b^4 = 8

. Dividing the second equation by the first eliminates

a

and gives

b^2 = \frac{4}{9}

. Substituting

b^2 = \frac{4}{9}

back into the first equation yields

a \cdot \frac{4}{9} = 18

. Multiplying both sides by

\frac{9}{4}

results in

a = 40.5

. Since

M(0) = a \cdot b^0 = a

, the initial mass of the sample is

40.5

grams.

Step-by-Step Solution

1

Set up the system of exponential equations using the given coordinates.

a \cdot b^2 = 18

and

a \cdot b^4 = 8

This represents the mass of the sample at

t = 2

and

t = 4

using the model

M(t) = a \cdot b^t

.

2

Divide the second equation by the first equation to eliminate the constant

a

and solve for

b^2

.

b^2 = \frac{4}{9}

Dividing the equations yields

\frac{a \cdot b^4}{a \cdot b^2} = \frac{8}{18}

, which simplifies to

b^2 = \frac{4}{9}

.

3

Substitute the value of

b^2

back into the first equation to solve for the initial mass

a

.

a = 40.5

Substituting

b^2

gives

a \cdot \frac{4}{9} = 18

. Multiplying both sides by

\frac{9}{4}

yields

a = 18 \cdot \frac{9}{4} = 40.5

.

Key Concept

Solving systems of exponential equations to determine the initial value and decay factor.

Question 2044Question →

At an agricultural research station, the nitrogen level of two soil plots, Plot A and Plot B, is monitored over several weeks. The nitrogen level of Plot A increases linearly at a constant rate, and the nitrogen level of Plot B increases exponentially at a constant percentage rate. At week $t = 0$ , Plot A contains $150\text{ grams}$ of nitrogen and Plot B contains $80\text{ grams}$ of nitrogen. At week $t = 2$ , both plots contain the same amount of nitrogen, which is $180\text{ grams}$ . At week $t = 4$ , Plot B contains $N\text{ grams}$ of nitrogen and Plot A contains $A\text{ grams}$ of nitrogen. What is the value of $N - A$ ?

Show answer & explanation

Answer: 195

Answer

195

The correct answer is

195

. At

t = 0

, Plot A contains

150\text{ grams}

and Plot B contains

80\text{ grams}

. Since Plot A increases linearly and reaches

180\text{ grams}

at

t = 2

, its weekly rate of change is

\frac{180 - 150}{2} = 15\text{ grams per week}

. Thus, at week

t = 4

, Plot A contains

150 + 15(4) = 210\text{ grams}

. Since Plot B increases exponentially and reaches

180\text{ grams}

at

t = 2

, its weekly growth factor

b

satisfies

80b^2 = 180

, which gives

b^2 = 2.25

and

b = 1.5

. Thus, at week

t = 4

, Plot B contains

80(1.5)^4 = 405\text{ grams}

. The difference

N - A

is

405 - 210 = 195

.

Step-by-Step Solution

1

Find the nitrogen level of Plot A at week

t = 4

using a linear model.

A = 210

Plot A grows linearly from an initial

150\text{ grams}

at

t = 0

to

180\text{ grams}

at

t = 2

. The rate of increase is

\frac{180 - 150}{2} = 15\text{ grams per week}

. Thus, at

t = 4

, the nitrogen level is

150 + 15(4) = 210\text{ grams}

.

2

Find the nitrogen level of Plot B at week

t = 4

using an exponential model.

N = 405

Plot B grows exponentially from an initial

80\text{ grams}

at

t = 0

to

180\text{ grams}

at

t = 2

. The weekly growth factor

b

satisfies

80b^2 = 180

, so

b^2 = 2.25

and

b = 1.5

. Thus, at

t = 4

, the nitrogen level is

80(1.5)^4 = 80(5.0625) = 405\text{ grams}

.

3

Calculate the difference between the two nitrogen levels at week

t = 4

.

195

Subtracting

A

from

N

yields

N - A = 405 - 210 = 195

.

Key Concept

Modeling linear growth (constant rate of change) and exponential growth (constant percentage rate of change or growth factor) over time.

Question 2045Question →

A trapezoid has parallel bases of length $6$ centimeters and $10$ centimeters, and a height of $5$ centimeters. What is the area, in square centimeters, of the trapezoid?

Show answer & explanation

Answer: 40

Answer

The area of the trapezoid is

40

square centimeters.

The area of a trapezoid is calculated using the formula

A = \frac{a + b}{2} h

, where

a

and

b

are the lengths of the parallel bases and

h

is the height. Substituting the given values

a = 6

,

b = 10

, and

h = 5

into the formula yields

A = \frac{6 + 10}{2} \times 5 = 8 \times 5 = 40

.

Step-by-Step Solution

1

Recall the area formula for a trapezoid.

A = \frac{a + b}{2}h

The area of a trapezoid is the average of its parallel bases multiplied by its height.

2

Substitute the base values of

6

and

10

, and the height value of

5

into the formula.

A = \frac{6 + 10}{2} \times 5

This sets up the calculation with the given measurements.

3

Simplify the expression to find the final area.

A = 40

Adding the bases gives

16

, dividing by

2

gives

8

, and multiplying by the height of

5

results in

40

.

Key Concept

Area of a Trapezoid

Estimated Time:45s

Question 2046Question →

The table below shows the distribution of a group of $180$ commuters surveyed about their primary mode of transportation and their daily commuting distance.

Commuting Distance	Public Transit	Personal Vehicle
Under 10 miles	45	35
10 miles or more	60	40

Based on the table, if a commuter who primarily uses a personal vehicle is selected at random, what is the probability that this commuter has a daily commuting distance of under 10 miles?

\frac{7}{36}

\frac{7}{16}

\frac{7}{15}

\frac{5}{12}

Show answer & explanation

Answer: $\frac{7}{15}$

Answer

The probability that the selected commuter has a daily commuting distance of under 10 miles, given that they primarily use a personal vehicle, is

\frac{7}{15}

.

To find the probability that a commuter has a daily commuting distance of under 10 miles given that they primarily use a personal vehicle, we limit our sample space to only the commuters who use a personal vehicle. The total number of commuters who primarily use a personal vehicle is

35 + 40 = 75

. Within this group, the number of commuters with a daily commuting distance of under 10 miles is 35. Therefore, the probability is the fraction

\frac{35}{75}

, which simplifies to

\frac{7}{15}

.

Step-by-Step Solution

1

Identify the total number of commuters who satisfy the condition of primarily using a personal vehicle.

The total number of commuters who primarily use a personal vehicle is

35 + 40 = 75

.

This sum represents the total number of possible outcomes (the denominator) because the selection is restricted to this group.

2

Identify the number of commuters within that subset who have a daily commuting distance of under 10 miles.

The number of personal vehicle commuters with a daily commuting distance of under 10 miles is 35.

This represents the number of favorable outcomes (the numerator).

3

Divide the number of favorable outcomes by the total number of possible outcomes and simplify the resulting fraction.

The probability is

\frac{35}{75} = \frac{7}{15}

.

Dividing the numerator and denominator by their greatest common divisor, 5, gives the simplified probability.

Key Concept

Calculating conditional probability from a two-way table by identifying the correct subset of the population to use as the denominator.

Estimated Time:1m 30s

Question 2047Question →

If $-5(2x - 9) + 3 \geq 18$ , what is the maximum possible value of $x$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

Distributing the

-5

gives

-10x + 45 + 3 \geq 18

. Combining constants yields

-10x + 48 \geq 18

. Subtracting 48 from both sides gives

-10x \geq -30

. Finally, dividing both sides by

-10

and reversing the inequality sign gives

x \leq 3

. The maximum possible value is therefore 3.

Step-by-Step Solution

1

Distribute

-5

to the terms inside the parentheses.

-10x + 45 + 3 \geq 18

Simplify the expression by expanding the parentheses.

2

Combine the constant terms

45

and

3

on the left side.

-10x + 48 \geq 18

Group like terms together.

3

Subtract

48

from both sides of the inequality.

-10x \geq -30

Isolate the variable term.

4

Divide both sides by

-10

and reverse the inequality symbol.

x \leq 3

Dividing by a negative number reverses the direction of the inequality.

Key Concept

Solving multi-step linear inequalities in one variable, applying the distributive property, and reversing the inequality sign when multiplying or dividing by a negative number.

Question 2048Question →

A commercial cargo ship is unloading shipping containers at a port. The total mass of the ship and its remaining cargo, $M$ , in kilotonnes (kt), is a linear function of the number of hours, $h$ , since the unloading process began. After $3$ hours of unloading, the total mass of the ship and its cargo is $116$ kt. After $8$ hours of unloading, the total mass is $98$ kt. According to this model, what is the mass, in kilotonnes, of the cargo that is unloaded each hour?

Show answer & explanation

Answer: 3.6

Answer

The mass of the cargo unloaded each hour is

3.6

kilotonnes (or the equivalent fraction

\frac{18}{5}

).

The correct answer is

3.6

. Since the relationship between the total mass of the ship and cargo,

M

, and the elapsed time,

h

, is linear, the rate at which cargo is unloaded corresponds to the magnitude of the slope of the linear function. Using the two data points

(3, 116)

and

(8, 98)

, the slope

m

can be calculated as

\frac{98 - 116}{8 - 3} = \frac{-18}{5} = -3.6

. The negative sign indicates that the mass is decreasing, meaning that

3.6

kilotonnes of cargo are unloaded each hour.

Step-by-Step Solution

1

Set up the linear relationship model.

M = mh + b

Since the relationship between total mass

M

and time

h

is linear, it can be modeled by a linear equation where

m

is the slope (unloading rate) and

b

is the y-intercept (initial mass).

2

Determine the two data points from the given information.

(3, 116)

and

(8, 98)

After

3

hours of unloading, the mass is

116

kt, and after

8

hours, the mass is

98

kt.

3

Calculate the slope (rate of change) of the linear relationship.

m = -3.6

The slope is calculated as

m = \frac{98 - 116}{8 - 3} = \frac{-18}{5} = -3.6

.

4

Interpret the slope's value in context to find the amount of cargo unloaded per hour.

3.6

The slope is

-3.6

kilotonnes per hour, which means the total mass decreases by

3.6

kilotonnes each hour. Therefore, the mass of the cargo unloaded each hour is

3.6

kilotonnes.

Key Concept

Interpreting the slope of a linear relationship in context

Estimated Time:1m 30s

Question 2049Question →

In cognitive psychology, change blindness refers to the tendency of observers to fail to notice significant changes in a visual scene. Researchers study this phenomenon by inserting a brief disruption between two versions of an ______ many participants fail to detect the alteration because the brain does not store every visual detail.

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

image;

image,

image

image, but

Show answer & explanation

Answer: image;

Answer

image;

The semicolon correctly links the two independent clauses without a coordinating conjunction, conforming to the conventions of Standard English.

Step-by-Step Solution

1

Identify the clause boundaries in the sentence.

The sentence contains two independent clauses: 'Researchers study this phenomenon by inserting a brief disruption between two versions of an image' and 'many participants fail to detect the alteration because the brain does not store every visual detail.'

Both clauses contain a subject and a verb and can express a complete thought independently.

2

Determine the appropriate punctuation or conjunction to link these clauses.

Two independent clauses must be joined by a period, a semicolon, a colon (if the second explains the first), or a comma followed by a coordinating conjunction (FANBOYS).

Joining them with only a comma creates a comma splice, and joining them with no punctuation creates a run-on sentence.

3

Evaluate the choices to find the grammatically and logically correct option.

The option containing a semicolon correctly links the two clauses. The option with a comma creates an ungrammatical comma splice. The option with no punctuation creates an ungrammatical run-on. The option with the comma and coordinator 'but' is grammatically possible but logically incorrect since there is no contrast between the clauses.

The semicolon is the only choice that maintains grammatical correctness and logical flow.

Key Concept

Semicolons are used to connect two closely related independent clauses that are not joined by a coordinating conjunction.

Question 2050Question →

The quadratic equation $x^2 - 8x - 9 = 0$ can be written in the equivalent form $(x - a)^2 - b = 0$ , where $a$ and $b$ are positive constants. What is the value of $a + b$ ?

11

13

29

21

Show answer & explanation

Answer: 29

Answer

The value of

a + b

is 29.

Completing the square on the quadratic equation

x^2 - 8x - 9 = 0

yields

(x - 4)^2 - 25 = 0

. Comparing this to the form

(x - a)^2 - b = 0

shows that

a = 4

and

b = 25

. The sum of these values is

4 + 25 = 29

.

Step-by-Step Solution

1

Identify the coefficient of the linear term and find half of its value.

The coefficient of the linear term

-8x

is

-8

. Half of

-8

is

-4

.

This is the first step in completing the square.

2

Square the value obtained in the first step and add/subtract it in the equation to form a perfect square trinomial.

(-4)^2 = 16

. The equation becomes

(x^2 - 8x + 16) - 16 - 9 = 0

.

Adding and subtracting

16

maintains the equality while allowing us to group the first three terms as a perfect square.

3

Rewrite the perfect square trinomial and combine the remaining constant terms.

(x - 4)^2 - 25 = 0

.

This simplifies the equation into the desired equivalent form

(x - a)^2 - b = 0

.

4

Compare the equation to the target form

(x - a)^2 - b = 0

to identify the constants

a

and

b

, and calculate

a + b

.

a = 4

and

b = 25

. Therefore,

a + b = 4 + 25 = 29

.

This answers the question by finding the sum of the positive constants

a

and

b

.

Key Concept

Completing the square to rewrite a quadratic equation

Question 2051Question →

In right triangle $ABC$ , the measure of angle $B$ is $90^\circ$ . If $\sin(A) = \frac{5}{13}$ , what is the value of $\cos(C)$ ?

\frac{5}{13}

\frac{12}{13}

\frac{13}{5}

\frac{8}{13}

Show answer & explanation

Answer: $\frac{5}{13}$

Answer

\frac{5}{13}

The correct answer is

\frac{5}{13}

. In any right triangle

ABC

where the right angle is at

B

, the two acute angles

A

and

C

are complementary, meaning

A + C = 90^\circ

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

\sin(A) = \cos(90^\circ - A) = \cos(C)

. Given that

\sin(A) = \frac{5}{13}

, the value of

\cos(C)

must also be

\frac{5}{13}

.

Step-by-Step Solution

1

Identify the relationship between the acute angles in a right triangle.

Since the measure of angle

B

is

90^\circ

, the sum of the measures of angles

A

and

C

must be

90^\circ

. Thus, angles

A

and

C

are complementary.

The sum of angles in any triangle is

180^\circ

.

2

Apply the co-function identity for complementary angles.

For any two complementary angles

A

and

C

, the identity

\cos(C) = \sin(A)

holds true.

The sine of an angle is the ratio of the opposite side to the hypotenuse, which is the same as the ratio of the adjacent side of its complement to the hypotenuse (cosine of the complement).

3

Substitute the given value to find

\cos(C)

.

Since

\sin(A) = \frac{5}{13}

, it follows that

\cos(C) = \frac{5}{13}

.

Direct substitution into the identity

\cos(C) = \sin(A)

.

Key Concept

Co-function identities for complementary angles in a right triangle

Question 2052Question →

For the quadratic function $f$ , the table shows some values of $x$ and their corresponding values of $f(x)$ .

$x$	$f(x)$
$1$	$15$
$3$	$3$
$5$	$15$

What is the value of $f(0)$ ?

Show answer & explanation

Answer: 30

Answer

30

Since the function values at

x = 1

and

x = 5

are both

15

, the axis of symmetry of the quadratic function must be located at the midpoint of these values, which is

x = 3

. The vertex of the function must therefore have an

x

-coordinate of

3

. From the table,

f(3) = 3

, which means the vertex is

(3, 3)

. The vertex form of the quadratic function is

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

. To find the value of

a

, substitute the point

(1, 15)

into the equation:

15 = a(1 - 3)^2 + 3

, which simplifies to

15 = 4a + 3

, giving

a = 3

. The equation of the function is

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

. Substituting

x = 0

yields

f(0) = 3(0 - 3)^2 + 3 = 3(9) + 3 = 30

.

Step-by-Step Solution

1

Find the axis of symmetry.

The axis of symmetry is

x = 3

.

Since

f(1) = 15

and

f(5) = 15

, the function values are equal at these points. Because a quadratic function is symmetric, the axis of symmetry is the midpoint of these two

x

-values:

\frac{1 + 5}{2} = 3

.

2

Identify the vertex.

The vertex is

(3, 3)

.

The vertex of a quadratic function lies on its axis of symmetry. Since the axis of symmetry is

x = 3

, the

x

-coordinate of the vertex is

3

. From the table,

f(3) = 3

, so the vertex is

(3, 3)

.

3

Write the function in vertex form and solve for the leading coefficient.

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

The vertex form of a quadratic function is

f(x) = a(x - h)^2 + k

, where

(h, k)

is the vertex. Substituting

(3, 3)

, we get

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

. Using the point

(1, 15)

from the table, we substitute

x = 1

and

f(x) = 15

:

15 = a(1 - 3)^2 + 3 \Rightarrow 15 = 4a + 3 \Rightarrow 12 = 4a \Rightarrow a = 3

.

4

Evaluate the function at

x = 0

.

f(0) = 30

Substitute

x = 0

into the function:

f(0) = 3(0 - 3)^2 + 3 = 3(9) + 3 = 27 + 3 = 30

.

Key Concept

Quadratic Functions and Graphs

Question 2053Question →

Read the passage about the ecological effects of gray wolves. Which word or phrase logically completes the text by establishing a cause-and-effect connection?

Fill in the blanks below

In 1995, the reintroduction of gray wolves to Yellowstone National Park initiated a trophic cascade that altered the local ecosystem. By preying on elk, the wolves reduced elk populations and changed their grazing habits, allowing willow and aspen trees along riverbanks to regenerate. erosion decreased significantly, stabilizing the river channels and creating new habitats for beaver and fish species.

Show answer & explanation

Answer

Consequently (or another cause-and-effect transition such as 'as a result', 'therefore', or 'thus')

The passage describes a sequence where the recovery of vegetation (willow and aspen trees) leads directly to the stabilization of soil and a decrease in erosion. Because the reduction in erosion is a direct consequence of the regenerated trees, a cause-and-effect transition like 'consequently', 'therefore', or 'as a result' is logically required to connect the sentences.

Step-by-Step Solution

1

Analyze the relationship between the ideas in the passage.

The sentence before the blank describes how the reintroduction of wolves allowed riverbank vegetation (willow and aspen trees) to regenerate. The sentence containing the blank describes a decrease in soil erosion along the river channels.

Identifying the relationship between these two points is necessary to choose the correct transition.

2

Determine the logical connection.

The regeneration of riverbank vegetation directly caused the decrease in soil erosion (because tree roots stabilize soil). This represents a clear cause-and-effect relationship.

Establishing that the second point is a direct result of the first point requires a cause-and-effect transition.

Key Concept

Transitions: Cause and Effect

Question 2054Question →

A dataset consists of $11$ positive integers. The mean of the dataset is $14$ , and the median is $12$ . The range of the dataset is $20$ . If the maximum possible value of the largest integer in the dataset is $L$ , what is the value of $L$ ?

Show answer & explanation

Answer: 32

Answer

The maximum possible value of the largest integer in the dataset is

32

.

The correct answer is

32

. Let the 11 integers in nondecreasing order be

x_1 \le x_2 \le \dots \le x_{11}

. Since the mean of the 11 integers is

14

, their sum is

11 \times 14 = 154

. Since the median is

12

, the 6th integer is

x_6 = 12

. The range is

20

, so

x_{11} - x_1 = 20

, which gives

x_1 = x_{11} - 20

. To maximize the largest integer

x_{11}

, we need to observe that

x_1 \le x_6 = 12

. Substituting

x_1 = x_{11} - 20

into this inequality yields

x_{11} - 20 \le 12

, which simplifies to

x_{11} \le 32

. We can verify that

x_{11} = 32

is achievable with the dataset:

12, 12, 12, 12, 12, 12, 12, 12, 12, 14, 32

. Therefore, the maximum possible value of the largest integer is

32

.

Step-by-Step Solution

1

Calculate the sum of the 11 integers in the dataset.

The sum of the 11 integers is

154

.

Since the mean of 11 values is 14, their sum must be

11 \times 14 = 154

.

2

Identify the median and establish the order constraint.

The 6th value in the ordered list of 11 integers is

12

.

The median of an odd number of sorted values is the middle value, which is the 6th value (

x_6 = 12

).

3

Relate the minimum and maximum values using the range.

The minimum value

x_1

is related to the maximum value

x_{11}

by

x_1 = x_{11} - 20

.

The range of the dataset is 20, which is the difference between the maximum and minimum values.

4

Find the maximum possible value of the largest integer.

The maximum value is

32

.

Since the dataset is sorted, the minimum value

x_1

cannot exceed the median

x_6 = 12

. Therefore,

x_1 \le 12 \implies x_{11} - 20 \le 12 \implies x_{11} \le 32

.

Key Concept

Using measures of center (mean, median) and spread (range) to determine bounds on data values.

Question 2055Question →

A right circular cylindrical container with a base radius of $R$ inches is partially filled with water. A solid metal sphere with a radius of $r$ inches is placed into the container and becomes completely submerged, causing the water level to rise by $1.5$ inches without any water overflowing. The sphere is then removed, and a solid right circular cone with a base radius of $r$ inches and a height of $12$ inches is placed vertex-down into the container. When the cone is completely submerged, the water level is $1$ inch higher than the container's original water level. What is the value of $R$ , in inches?

Show answer & explanation

Answer: 9

Answer

The radius of the cylinder,

R

, is

9

inches.

The volume of a submerged solid is equal to the volume of the cylinder of water it displaces. By setting up equations for the sphere and the cone, we get

\frac{4}{3}r^3 = 1.5 R^2

and

4r^2 = R^2

. Substituting the second equation into the first yields

r = 4.5

, which then gives

R = 2r = 9

.

Step-by-Step Solution

1

Equate the volume of the sphere to the volume of water it displaces in the cylinder.

\frac{4}{3}\pi r^3 = 1.5\pi R^2 \implies \frac{4}{3}r^3 = 1.5 R^2

The volume of a submerged solid equals the volume of the fluid it displaces. The displaced fluid takes the shape of a cylinder of radius

R

and height equal to the water level rise (

1.5

inches).

2

Equate the volume of the cone to the volume of water it displaces in the cylinder.

\frac{1}{3}\pi r^2 (12) = 1.0\pi R^2 \implies 4r^2 = R^2

Similarly, the volume of the cone is equal to the volume of a cylinder of radius

R

and height equal to the water level rise (

1

inch).

3

Solve the system of equations by substituting

R^2

into the sphere's displacement equation.

\frac{4}{3}r^3 = 1.5(4r^2) \implies \frac{4}{3}r^3 = 6r^2 \implies r = 4.5

By replacing

R^2

with

4r^2

, we reduce the system of equations to a single equation containing only

r

. Since

r \neq 0

, we can divide by

r^2

to solve for

r

directly.

4

Calculate the value of

R

from the relationship between

R

and

r

.

R = 2r = 2(4.5) = 9

Since

R^2 = 4r^2

and radii must be positive quantities,

R = 2r

.

Key Concept

Using water displacement to relate the volumes of three-dimensional geometric solids (cylinders, spheres, cones) and solving non-linear systems of equations.

Question 2056Question →

A container in the shape of a right circular cylinder with a base radius of $4$ inches and a height of $9$ inches is completely filled with water. All of the water from this cylinder is poured into an empty container in the shape of a rectangular prism. The rectangular prism container has a square base of side length $6$ inches and a height of $15$ inches. If the height of the water in the rectangular prism container is $k\pi$ inches, what is the value of $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of

k

is

4

.

To find the height of the water in the rectangular prism, we first calculate the volume of the water using the cylinder's volume formula,

V = \pi r^2 h

. With a radius of

4

inches and a height of

9

inches, the volume is

\pi (4)^2 (9) = 144\pi

cubic inches. When this water is poured into the rectangular prism, the volume of the water can also be represented as the area of the base times the height of the water:

V = \text{Base Area} \times h_w = 6^2 \times h_w = 36 h_w

. Equating the two volumes gives

36 h_w = 144\pi

, which simplifies to

h_w = 4\pi

inches. Since the height of the water in the prism is

k\pi

inches, the value of

k

is

4

.

Step-by-Step Solution

1

Calculate the volume of the water in the filled cylinder using the formula

V = \pi r^2 h

.

The volume of the water is

144\pi

cubic inches.

Since the cylinder is completely filled, the volume of the water is equal to the volume of the cylinder with base radius

4

inches and height

9

inches.

2

Calculate the area of the square base of the rectangular prism container.

The base area is

36

square inches.

The base of the prism is a square with side length

6

inches, so its area is

6 \times 6 = 36

square inches.

3

Set the volume of the water in the rectangular prism equal to the volume of the water from the cylinder, and solve for the water height

h_w

.

36 \times h_w = 144\pi \implies h_w = 4\pi

inches.

Pouring the water into the prism container does not change its volume, which remains

144\pi

cubic inches. The volume of a prism is the base area times its height.

4

Compare the height of the water

4\pi

to the expression

k\pi

to find

k

.

k = 4

.

Since the height of the water is represented as

k\pi

inches and we calculated it to be

4\pi

inches,

k

must equal

4

.

Key Concept

Equating the volumes of a right circular cylinder and a rectangular prism to solve for an unknown dimension.

Question 2057Question →

A solid metal right circular cylinder has a base radius of $12\text{ centimeters}$ and a height of $3\text{ centimeters}$ . A cone with the same base and height as the cylinder is carved out and removed. The remaining metal is melted and recast into a solid sphere. What is the radius, in centimeters, of the sphere?

Show answer & explanation

Answer: 6

Answer

The radius of the sphere is 6 centimeters.

The volume of a cylinder is

V = \pi r^2 h = \pi (12)^2 (3) = 432\pi

. The volume of the carved-out cone is

V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (12)^2 (3) = 144\pi

. Subtracting the cone's volume from the cylinder's volume yields the remaining metal volume:

432\pi - 144\pi = 288\pi

. The volume of the recast sphere is

\frac{4}{3}\pi R^3 = 288\pi

. Dividing by

\pi

and multiplying by

\frac{3}{4}

gives

R^3 = 216

. Taking the cube root of both sides gives the radius

R = 6

.

Step-by-Step Solution

1

Calculate the volume of the original cylinder.

432\pi\text{ cubic centimeters}

This determines the starting volume of the solid metal block before any material is removed.

2

Calculate the volume of the cone that is carved out.

144\pi\text{ cubic centimeters}

This determines how much metal is discarded from the cylinder.

3

Subtract the cone's volume from the cylinder's volume.

288\pi\text{ cubic centimeters}

This gives the volume of the remaining metal that will be melted and recast.

4

Equate the remaining volume to the volume formula of a sphere and solve for the radius.

6\text{ centimeters}

This yields the radius of the newly formed sphere.

Key Concept

Volume of composite solids and conservation of volume during recasting

Question 2058Question →

A municipal research group conducted a survey of a random sample of $n_1$ registered voters in a city to estimate the proportion of voters who support a proposed school bond. In this survey, the proportion of voters who supported the bond was $0.52$ , with an associated margin of error of $0.06$ at a $95\%$ confidence level. The group plans to conduct a follow-up survey of a random sample of $n_2$ registered voters from the same population. Assuming the sample proportion remains approximately the same, and the group wants the margin of error for the follow-up survey to be $0.02$ at the same $95\%$ confidence level, what is the value of the ratio $\frac{n_2}{n_1}$ ?

Show answer & explanation

Answer: 9

Answer

9

The margin of error for a sample proportion is inversely proportional to the square root of the sample size. Since the target margin of error is one-third of the initial margin of error (from

0.06

to

0.02

), the sample size must be increased by a factor of

3^2 = 9

. Thus, the ratio of the new sample size to the original sample size is

9

.

Step-by-Step Solution

1

State the formula for the margin of error of a sample proportion.

E = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}

This formula defines the mathematical relationship between the margin of error, the confidence level (via the critical value

z^*

), the sample proportion

\hat{p}

, and the sample size

n

.

2

Determine the relationship between margin of error and sample size when other variables are constant.

E \propto \frac{1}{\sqrt{n}}

Since the confidence level remains at

95\%

(constant

z^*

) and the sample proportion remains approximately the same (constant

\hat{p}

), the margin of error is inversely proportional to the square root of the sample size.

3

Set up the ratio equation for the two surveys.

\frac{E_1}{E_2} = \sqrt{\frac{n_2}{n_1}}

Dividing the expression for the first margin of error by the expression for the second margin of error simplifies to the square root of the ratio of the second sample size to the first sample size.

4

Substitute the given margins of error and solve for the ratio.

\frac{n_2}{n_1} = 9

Substitute

E_1 = 0.06

and

E_2 = 0.02

to get

\frac{0.06}{0.02} = 3 = \sqrt{\frac{n_2}{n_1}}

. Squaring both sides yields the ratio of the sample sizes, which is

9

.

Key Concept

The relationship between sample size and margin of error in statistical studies.

Question 2059Question →

While planning a presentation, a student takes the following notes:
* Kepler-186f is an exoplanet that orbits a red dwarf star known as Kepler-186.
* It belongs to a category of planets that are similar in size to Earth and orbit within their host star's habitable zone.
* The habitable zone is defined as the orbital region around a star where liquid water could potentially exist on a planet's surface.
* Orbiting within this zone does not guarantee habitability, as a planet's surface temperatures depend heavily on its atmospheric composition.
* For example, Kepler-186f receives only one-third of the stellar energy that Earth does, meaning it requires a thick greenhouse atmosphere to prevent freezing.

The student wants to present a generalization about the category of Earth-sized planets in habitable zones. Which choice most effectively uses information from the given notes to accomplish this goal?

Kepler-186f is an Earth-sized exoplanet orbiting within the habitable zone of the red dwarf star Kepler-186, which provides only one-third of the stellar energy that Earth receives.

Because Kepler-186f receives only one-third of the stellar energy that Earth receives from the Sun, it would require a thick greenhouse atmosphere to avoid freezing.

While Earth-sized planets in habitable zones have the potential to support liquid water, their actual habitability is determined by whether their atmospheres can maintain temperatures above freezing.

Since orbiting within a star's habitable zone guarantees that an Earth-sized planet can support liquid surface water, atmospheric conditions are ultimately secondary to its orbital position.

Show answer & explanation

Answer: While Earth-sized planets in habitable zones have the potential to support liquid water, their actual habitability is determined by whether their atmospheres can maintain temperatures above freezing.

Answer

The option stating that while Earth-sized planets in habitable zones have the potential to support liquid water, their actual habitability is determined by whether their atmospheres can maintain temperatures above freezing.

The correct option successfully presents a generalization about the category of Earth-sized planets in habitable zones. It synthesizes the information from the notes regarding their potential for liquid water and the critical role of atmospheric composition in determining their actual surface temperatures, without limiting the statement to the specific example of Kepler-186f.

Step-by-Step Solution

1

Analyze the prompt to identify the student's rhetorical goal.

The goal is to present a generalization about the category of Earth-sized planets in habitable zones.

This determines which information must be prioritized and how the final statement should be framed.

2

Evaluate the notes to determine the key characteristics of the category.

The category consists of Earth-sized planets in habitable zones (where water could exist), but habitability is not guaranteed and depends on atmospheric composition.

This establishes the factual basis for the correct generalization.

3

Assess the options to find the one that generalizes about the category without focusing solely on the example of Kepler-186f.

The statement describing the potential for water vs. the role of atmosphere in these planets is a valid generalization, whereas other choices focus too narrowly on Kepler-186f or misrepresent the notes.

Selecting the option that matches the category-level generalization ensures the rhetorical goal is met.

Key Concept

Rhetorical Synthesis: Summarization and Generalization

Question 2060Question →

A financial technology platform offers two merchant rewards programs, Program $P$ and Program $E$ , based on a merchant's monthly transaction volume. The table below shows the monthly reward for both programs at a transaction volume of $\$50,000$ .

Program	Growth Model	Reward at $\$50,000$ Volume
Program $P$	Linear	$\$800$
Program $E$	Exponential	$\$800$

For every increase of $\$10,000$ in monthly transaction volume:
- Under Program $P$ , the monthly reward increases by $\$150$ .
- Under Program $E$ , the monthly reward increases by $5\%$ .

If a merchant's monthly transaction volume increases from $\$50,000$ to $\$90,000$ , what is the closest dollar amount to the difference between the monthly rewards the merchant would receive under the two programs?

$159

$440

$428

$1,178

Show answer & explanation

Answer: $428

Answer

$428

The correct answer of

\$428

represents the difference between the rewards under the two models at a monthly transaction volume of

\$90,000

. The number of

\$10,000

increments from

\$50,000

to

\$90,000

is

4

. The linear reward under Program P increases by

\$150

per increment, resulting in a reward of

800 + 4(150) = 1400

. The exponential reward under Program E increases by

5\%

per increment, resulting in a reward of

800 \times (1.05)^4 \approx 972.41

. The difference is

1400 - 972.41 = 427.59

, which is closest to

\$428

.

Step-by-Step Solution

1

Determine the number of

\$10,000

increments from the initial transaction volume to the target volume.

4

increments

The transaction volume increases from

\$50,000

to

\$90,000

, a change of

\$40,000

. Since the growth rates are defined per

\$10,000

increase, the number of increments is

\frac{40,000}{10,000} = 4

.

2

Calculate the monthly reward under the linear model (Program P) after

4

increments.

1400

dollars

Program P adds a constant

\$150

per increment. The reward after

4

increments is

800 + 4(150) = 800 + 600 = 1400

.

3

Calculate the monthly reward under the exponential model (Program E) after

4

increments.

972.41

dollars

Program E increases by a constant percent (

5\%

) per increment, which corresponds to a growth factor of

1.05

. The reward after

4

increments is

800 \times (1.05)^4 \approx 800 \times 1.2155 = 972.41

.

4

Find the difference between the two monthly rewards.

427.59

dollars, which rounds to

428

dollars

Subtract the reward under Program E from the reward under Program P:

1400 - 972.41 = 427.59

.

Key Concept

Distinguishing between linear and exponential growth models and evaluating them at specific values in a real-world context.

Estimated Time:2m 30s

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