All practice questions

Question 441Question →

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

Show answer & explanation

Answer: 81

Answer

The maximum possible range of the dataset is 81.

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

Step-by-Step Solution

1

Calculate the total sum of the dataset.

The sum of all 15 elements is

300

.

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

15 \times 20 = 300

.

2

Identify the median and set up the sorted terms.

The 8th term

x_8 = 24

.

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

3

Minimize the sum of the first 7 terms.

The minimum sum of

x_1

through

x_7

is

10

, with

x_1 = 1

.

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

4(1) + 3(2) = 10

.

4

Minimize the 9th term.

x_9 = 24

.

Since the dataset is sorted,

x_9 \geq x_8 = 24

. To minimize the sum of the other terms and maximize

x_{15}

, we set

x_9 = 24

.

5

Account for the mode of 32.

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

.

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

6

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

x_{15} = 82

, and the range is

81

.

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

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

. The range is

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

.

Key Concept

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

Estimated Time:3m 0s

Question 442Question →

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

Show answer & explanation

Answer: 16

Answer

16

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

x^2 - 10x + c = 0

are

5 + \sqrt{25 - c}

and

5 - \sqrt{25 - c}

. The difference between these two solutions is

2\sqrt{25 - c}

. Given that the difference is 6, we set

2\sqrt{25 - c} = 6

, which simplifies to

\sqrt{25 - c} = 3

. Squaring both sides gives

25 - c = 9

, which yields

c = 16

.

Step-by-Step Solution

1

Use the quadratic formula to express the solutions of

x^2 - 10x + c = 0

.

The solutions are

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

.

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

c

.

2

Set the difference between the two solutions equal to 6.

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

, which simplifies to

2\sqrt{25 - c} = 6

.

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

3

Solve the equation

2\sqrt{25 - c} = 6

for

c

.

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

.

This isolates the constant

c

using standard algebraic operations.

Key Concept

Solving quadratic equations and using properties of roots.

Alternative Method

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

x_1

and

x_2

, then

x_1 + x_2 = 10

and

x_1 x_2 = c

. Using the identity

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

, we substitute the given values:

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

. This simplifies to

36 = 100 - 4c

, which gives

4c = 64

, or

c = 16

.

Estimated Time:1m 30s

Question 443Question →

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

Show answer & explanation

Answer: 26

Answer

The minimum number of people in the group is 26.

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

125 + 9.5p < 50 + 12.5p

. Subtracting

9.5p

from both sides gives

125 < 50 + 3p

. Subtracting

50

from both sides yields

75 < 3p

. Dividing by

3

gives

p > 25

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

p

that is strictly greater than

25

is

26

.

Step-by-Step Solution

1

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

125 + 9.5p < 50 + 12.5p

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

2

Isolate the variable

p

by subtracting

9.5p

and

50

from both sides of the inequality.

p > 25

Subtracting

9.5p

yields

125 < 50 + 3p

. Subtracting

50

yields

75 < 3p

. Dividing by

3

yields

p > 25

.

3

Identify the minimum integer value of

p

that satisfies the inequality.

26

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

25

is

26

.

Key Concept

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

Estimated Time:1m 30s

Question 444Question →

An equation is shown below.

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

What is the real solution to the equation above?

Show answer & explanation

Answer: 2

Answer

The only real solution is 2.

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

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

. This yields

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

. Simplifying this equation gives

x^2 - x - 2 = 0

. Factoring the quadratic expression gives

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

, which yields potential solutions of

x = 2

and

x = -1

. However, substituting

x = -1

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

x = 2

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

Step-by-Step Solution

1

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

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

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

2

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

(x - 3)(x + 1)

, to eliminate the denominators.

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

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

x \neq 3

and

x \neq -1

.

3

Expand and simplify the resulting equation.

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

This puts the equation into standard quadratic form:

ax^2 + bx + c = 0

.

4

Factor the quadratic equation.

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

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

5

Find the roots of the equation.

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

These are the values of

x

that satisfy the factored quadratic equation.

6

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

Substituting

x = -1

results in division by zero in the terms

\frac{2}{x+1}

and

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

, so

x = -1

is extraneous. Substituting

x = 2

yields a valid statement:

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

.

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

Key Concept

Solving rational equations and identifying extraneous solutions.

Question 445Question →

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

Show answer & explanation

Answer: 224

Answer

The surface area of the prism is 224 square inches.

The correct answer is 224. By using the volume formula

V = lwh

with a volume of

192

and a height of

4

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

lw = 48

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

3:1

, the length can be written as

l = 3w

. Substituting this expression into the product equation gives

3w(w) = 48

, or

3w^2 = 48

. Dividing both sides by 3 yields

w^2 = 16

, which means the width of the prism is

4

inches. Since

l = 3w

, the length is

12

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

SA = 2(lw + lh + wh)

. Substituting the dimensions

l = 12

,

w = 4

, and

h = 4

yields

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

square inches.

Step-by-Step Solution

1

Use the volume formula for a rectangular prism,

V = lwh

, and substitute the given volume of

192

and height of

4

.

192 = l \cdot w \cdot 4

, which simplifies to

l \cdot w = 48

.

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

2

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

3:1

.

l = 3w

.

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

3

Substitute

l = 3w

into the equation

l \cdot w = 48

and solve for

w

.

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

.

To solve for the width of the rectangular prism.

4

Find the length of the rectangular prism.

l = 3(4) = 12

inches.

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

5

Use the surface area formula

SA = 2(lw + lh + wh)

with the dimensions

l = 12

,

w = 4

, and

h = 4

.

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

.

To calculate the total surface area of the prism.

Key Concept

Volume and Surface Area of Rectangular Prisms

Question 446Question →

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

Show answer & explanation

Answer: 5

Answer

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

The correct answer is

5

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

10

workers can complete the project in

18

days, the total work is

10 \times 18 = 180

worker-days. In the first

6

days, the

10

workers complete

10 \times 6 = 60

worker-days of work, leaving

180 - 60 = 120

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

8

days, the total number of workers required is

\frac{120}{8} = 15

workers. Subtracting the original

10

workers gives

15 - 10 = 5

additional workers.

Step-by-Step Solution

1

Calculate the total amount of work required for the project.

180

worker-days

Since

10

workers can complete the project in

18

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

10 \times 18 = 180

worker-days.

2

Calculate the work completed in the first

6

days.

60

worker-days

The original

10

workers worked for

6

days, completing

10 \times 6 = 60

worker-days of work.

3

Find the remaining work to be done.

120

worker-days

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

180 - 60 = 120

worker-days.

4

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

8

days.

15

workers

Dividing the remaining work of

120

worker-days by the target time of

8

days gives the total number of workers needed:

\frac{120}{8} = 15

workers.

5

Calculate the number of additional workers hired.

5

workers

Subtract the original

10

workers from the total

15

workers required for the second phase:

15 - 10 = 5

.

Key Concept

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

Estimated Time:2m 30s

Question 447Question →

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 448Question →

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 449Question →

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 450Question →

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 451Question →

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 452Question →

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 453Question →

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 454Question →

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 455Question →

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 456Question →

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 457Question →

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 458Question →

In the equation $x^2 - bx + 16 = 0$ , $b$ is a positive integer. If the equation has no real solutions, what is the greatest possible value of $b$ ?

Show answer & explanation

Answer: 7

Answer

The greatest possible value of

b

is

7

.

For the quadratic equation

x^2 - bx + 16 = 0

to have no real solutions, its discriminant must be less than

0

. The discriminant is

(-b)^2 - 4(1)(16) = b^2 - 64

. Solving the inequality

b^2 - 64 < 0

gives

b^2 < 64

. Since

b

is a positive integer, taking the square root of both sides gives

b < 8

. The positive integers less than

8

are

1, 2, 3, 4, 5, 6,

and

7

. The greatest of these values is

7

.

Step-by-Step Solution

1

Set up the inequality for the discriminant to be less than zero.

(-b)^2 - 4(1)(16) < 0

A quadratic equation of the form

ax^2 + bx + c = 0

has no real solutions if and only if its discriminant,

D = b^2 - 4ac

, is negative.

2

Simplify the quadratic inequality.

b^2 < 64

Squaring

-b

yields

b^2

and calculating

4(1)(16)

yields

64

.

3

Solve for the greatest positive integer value of

b

.

b < 8

, so the greatest positive integer is

7

.

Since

b

is a positive integer, the values satisfying

b^2 < 64

are

1, 2, 3, 4, 5, 6,

and

7

. The largest of these is

7

.

Key Concept

Quadratic Discriminant and Number of Solutions

Question 459Question →

If $27^{x-1} = \sqrt{3^{5x+1}}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 7

Answer

The correct answer is 7.

The correct answer is 7. By converting the base of 27 to

3^3

and rewriting the radical on the right side as a fractional exponent of

\frac{1}{2}

, the equation becomes

3^{3(x-1)} = 3^{\frac{5x+1}{2}}

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

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

. Multiplying both sides by 2 yields

6x - 6 = 5x + 1

. Subtracting

5x

and adding 6 to both sides isolates

x

, giving the solution

x = 7

.

Step-by-Step Solution

1

Express both sides of the equation with a common base of 3.

(3^3)^{x-1} = (3^{5x+1})^{\frac{1}{2}}

Expressing terms with a common base allows the exponents to be compared directly.

2

Apply the power of a power exponent rule,

(a^m)^n = a^{mn}

, to simplify the exponents on both sides.

3^{3x-3} = 3^{\frac{5x+1}{2}}

Simplifying the expressions makes it possible to set the exponent expressions equal to each other.

3

Equate the exponents since the bases are identical.

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

If two exponential expressions with the same positive base (other than 1) are equal, their exponents must also be equal.

4

Solve the linear equation for

x

by clearing the fraction and isolating the variable.

x = 7

Multiplying both sides by 2 gives

6x - 6 = 5x + 1

. Subtracting

5x

from both sides and adding 6 to both sides isolates

x

, resulting in

x = 7

.

Key Concept

Solving exponential equations by converting to a common base and applying exponent laws.

Question 460Question →

An aquarium filter recirculates water at a constant rate of $250$ milliliters per second. The filter cartridge must be replaced after it has processed a total of $540$ cubic meters of water. For how many hours of continuous operation can the filter run before the cartridge must be replaced? ( $1\text{ cubic meter} = 1,000\text{ liters}$ )

Show answer & explanation

Answer: 600

Answer

The filter can run for

600

hours of continuous operation before the cartridge must be replaced.

To find the number of hours the filter can run, we first convert the flow rate to cubic meters per hour. The flow rate of

250

milliliters per second is equal to

0.25

liters per second. Multiplying by

3,600

seconds per hour gives a flow rate of

900

liters per hour. Since

1\text{ cubic meter} = 1,000\text{ liters}

,

900

liters per hour is equivalent to

0.9

cubic meters per hour. Finally, dividing the total capacity of

540

cubic meters by the flow rate of

0.9

cubic meters per hour yields a duration of

600

hours.

Step-by-Step Solution

1

Convert the water recirculating rate from milliliters per second to liters per second.

0.25

liters per second

To express the flow rate in terms of liters before converting to cubic meters.

2

Convert the rate from liters per second to liters per hour.

900

liters per hour

To align the time unit of the rate with the requested time unit (hours).

3

Convert the rate from liters per hour to cubic meters per hour.

0.9

cubic meters per hour

To match the volume unit of the filter cartridge capacity (cubic meters).

4

Divide the total cartridge capacity by the hourly flow rate of the filter.

600

hours

To find the total duration of continuous operation before replacement.

Key Concept

Unit Conversions

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution