All practice questions

Question 381Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = -x^2 + 6x + 7$ has a vertex at $(h, k)$ . If the graph of $f$ is translated $4$ units to the right and $3$ units up to produce the graph of the function $g$ , what is the maximum value of $g$ ?

Show answer & explanation

Answer: 19

Answer

The maximum value of the function

g

is

19

.

The vertex of the original quadratic function

f(x) = -x^2 + 6x + 7

is

(3, 16)

. Since the coefficient of

x^2

is negative, the graph opens downward, making

16

the maximum value of the function. Translating the graph

3

units up shifts all

y

-values up by

3

, which increases the maximum value to

16 + 3 = 19

. The horizontal translation of

4

units to the right shifts the graph horizontally but does not affect the maximum output value.

Step-by-Step Solution

1

Find the vertex of the original quadratic function

f(x) = -x^2 + 6x + 7

.

The vertex of

f(x)

is

(3, 16)

.

Rewrite the function in vertex form,

f(x) = -(x-3)^2 + 16

, to identify the vertex

(h, k)

as

(3, 16)

.

2

Determine the maximum value of the original function

f(x)

.

The maximum value of

f(x)

is

16

.

Since the leading coefficient is negative, the parabola opens downward, and the maximum value occurs at the vertex's

y

-coordinate.

3

Apply the vertical translation to find the maximum value of the new function

g(x)

.

The maximum value of

g

is

19

.

Translating the graph

3

units up increases all

y

-values, including the maximum value, by

3

, so

16 + 3 = 19

. The horizontal translation of

4

units to the right does not affect the maximum value.

Key Concept

Quadratic Functions and Graphs

Question 382Question →

The equation $\sqrt{2x + 12} - x = -6$ has one real solution. What is this solution?

Show answer & explanation

Answer: 12

Answer

12

Substituting 12 into the original equation

\sqrt{2x + 12} - x = -6

yields

\sqrt{2(12) + 12} - 12 = \sqrt{36} - 12 = 6 - 12 = -6

. Since this creates a true statement, 12 is the unique real solution to the equation.

Step-by-Step Solution

1

Isolate the radical term by adding

x

to both sides of the equation.

\sqrt{2x + 12} = x - 6

Isolating the radical on one side allows us to eliminate it by squaring both sides in the next step.

2

Square both sides of the equation to clear the square root.

2x + 12 = (x - 6)^2

Squaring a square root cancels the radical, allowing us to solve the equation algebraically.

3

Expand the squared binomial on the right-hand side.

2x + 12 = x^2 - 12x + 36

Expanding

(x - 6)^2

to

x^2 - 12x + 36

helps set up a standard quadratic form.

4

Subtract

2x

and

12

from both sides to form a quadratic equation equal to zero.

x^2 - 14x + 24 = 0

A quadratic equation must be in the form

ax^2 + bx + c = 0

to solve by factoring or the quadratic formula.

5

Factor the quadratic equation.

(x - 12)(x - 2) = 0

Factoring finds two numbers that multiply to 24 and add to -14, which are -12 and -2.

6

Test the potential solutions

x = 12

and

x = 2

in the original equation to identify any extraneous solutions.

For

x = 2

:

\sqrt{2(2) + 12} - 2 = \sqrt{16} - 2 = 4 - 2 = 2 \neq -6

. For

x = 12

:

\sqrt{2(12) + 12} - 12 = \sqrt{36} - 12 = 6 - 12 = -6

. Thus,

x = 12

is the only valid solution.

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

Key Concept

Solving radical equations by isolating the radical, squaring both sides, solving the resulting quadratic equation, and verifying all potential solutions to check for extraneous values.

Estimated Time:1m 30s

Question 383Question →

A coffee roasting company uses a commercial roasting machine. The temperature of the roasting drum, $T$ , in degrees Fahrenheit ( $^\circ\text{F}$ ), is modeled as a linear function of the roasting time, $t$ , in minutes, where $0 \leq t \leq 10$ . After $2$ minutes of roasting, the temperature of the drum is $280^\circ\text{F}$ . After $5$ minutes of roasting, the temperature of the drum is $385^\circ\text{F}$ . What is the rate of temperature increase, in degrees Fahrenheit per minute, of the roasting drum?

Show answer & explanation

Answer: 35

Answer

The rate of temperature increase of the roasting drum is

35

degrees Fahrenheit per minute.

The temperature,

T

, is modeled as a linear function of time,

t

. The constant rate of temperature increase corresponds to the slope of this linear function. Given two points,

(2, 280)

and

(5, 385)

, the slope is calculated as the change in temperature divided by the change in time:

\frac{385 - 280}{5 - 2} = \frac{105}{3} = 35

degrees Fahrenheit per minute.

Step-by-Step Solution

1

Identify the coordinates representing the relationship between time and temperature.

The two points are

(2, 280)

and

(5, 385)

, where the first coordinate is the time,

t

, in minutes, and the second coordinate is the temperature,

T

, in degrees Fahrenheit.

To find the rate of change of a linear relationship, we first need to determine two points

(t_1, T_1)

and

(t_2, T_2)

from the given context.

2

Calculate the rate of temperature increase as the slope of the line passing through these two points.

The slope

m

is given by

m = \frac{385 - 280}{5 - 2} = \frac{105}{3} = 35

.

The rate of temperature increase per minute is the constant slope of the linear relationship.

Key Concept

Slope of a linear relationship in context

Question 384Question →

The function $f$ is defined by $f(x) = x^3 - 3x^2 - 10x + k$ , where $k$ is a constant. In the $xy$ -plane, the graph of $y = f(x)$ has $x$ -intercepts at $(c, 0)$ and $(2c, 0)$ , where $c$ is a positive constant. What is the value of $k$ ?

Show answer & explanation

Answer: 24

Answer

24

By applying the factor theorem to the two

x

-intercepts

(c,0)

and

(2c,0)

, we establish the system

c^3 - 3c^2 - 10c + k = 0

and

8c^3 - 12c^2 - 20c + k = 0

. Subtracting these equations yields

7c^3 - 9c^2 - 10c = 0

. Since

c > 0

, dividing by

c

gives

7c^2 - 9c - 10 = 0

, which factors as

(7c+5)(c-2)=0

, yielding the positive solution

c=2

. Substituting

c=2

back into the first equation gives

8 - 12 - 20 + k = 0

, which solves to

k = 24

.

Step-by-Step Solution

1

Set up equations for the roots

c

and

2c

using the factor theorem.

f(c) = c^3 - 3c^2 - 10c + k = 0

and

f(2c) = 8c^3 - 12c^2 - 20c + k = 0

An

x

-intercept at

(r, 0)

means that

r

is a root of the polynomial, so

f(r) = 0

.

2

Subtract the first equation from the second to eliminate the constant

k

.

7c^3 - 9c^2 - 10c = 0

Eliminating

k

allows us to solve for the root

c

directly.

3

Solve the polynomial equation for the positive constant

c

.

c = 2

Dividing the equation by

c

(since

c > 0

) yields

7c^2 - 9c - 10 = 0

, which factors as

(7c+5)(c-2) = 0

. Since

c

must be positive,

c = 2

.

4

Substitute

c = 2

back into the equation for

f(c) = 0

to solve for

k

.

k = 24

Substituting the known root value allows us to find the value of the constant coefficient

k

.

Key Concept

The relationship between a polynomial's algebraic factors, its roots, and its

x

-intercepts in the coordinate plane.

Estimated Time:2m 0s

Question 385Question →

A deep space satellite transmits data to a ground station at a constant rate of $1.6$ megabits per second. The transmitted data consists of actual scientific data and protocol overhead. The protocol overhead accounts for $25\%$ of the total transmitted bits, meaning the actual scientific data constitutes only $75\%$ of the total transmitted bits. The ground station needs to receive $5$ scientific data files, each with a size of $1.08$ gigabytes. How many hours will it take to transmit all $5$ files? (Given that $1\\text{ byte} = 8\\text{ bits}$ , $1\\text{ megabit} = 10^6\\text{ bits}$ , and $1\\text{ gigabyte} = 10^9\\text{ bytes}$ .)

Show answer & explanation

Answer: 10

Answer

It will take 10 hours to transmit all 5 files.

The correct answer is 10. To find this, first calculate the total size of the scientific files: 5 files * 1.08 gigabytes per file = 5.4 gigabytes. Convert this to bytes: 5.4 * 10^9 bytes. Convert bytes to bits: 5.4 * 10^9 * 8 = 43.2 * 10^9 bits. Since protocol overhead is 25%, the scientific data is 75% of the total transmitted bits, so the total bits transmitted is 43.2 * 10^9 / 0.75 = 57.6 * 10^9 bits. The transmission rate is 1.6 megabits per second, or 1.6 * 10^6 bits per second. The transmission time in seconds is 57.6 * 10^9 / (1.6 * 10^6) = 36,000 seconds. Converting seconds to hours: 36,000 / 3,600 = 10 hours.

Step-by-Step Solution

1

Calculate the total size of the scientific files in gigabytes.

5.4

gigabytes

To find the total amount of scientific data that must be received.

2

Convert the total scientific data size from gigabytes to bytes and then to bits.

43.2 \\times 10^9

bits

The transmission rate is given in megabits per second, so the data size must be converted to bits for unit consistency. Since

1\\text{ GB} = 10^9\\text{ bytes}

and

1\\text{ byte} = 8\\text{ bits}

, we have

5.4 \\times 10^9 \\times 8 = 43.2 \\times 10^9

bits.

3

Calculate the total number of bits transmitted, including protocol overhead.

57.6 \\times 10^9

bits

Since protocol overhead is

25\%

, the scientific data is only

75\%

of the total bits transmitted. Thus, we divide the scientific bits by

0.75

to find the total bits transmitted:

\\frac{43.2 \\times 10^9}{0.75} = 57.6 \\times 10^9

bits.

4

Calculate the transmission time in seconds.

36,000

seconds

Divide the total bits by the transmission rate in bits per second (

1.6\\text{ Mbps} = 1.6 \\times 10^6

bits per second):

\\frac{57.6 \\times 10^9}{1.6 \\times 10^6} = 36,000

seconds.

5

Convert the transmission time from seconds to hours.

10

hours

Since there are

3,600

seconds in one hour, divide the total seconds by

3,600

:

\\frac{36,000}{3,600} = 10

.

Key Concept

Multi-step dimensional analysis and compound rate conversions incorporating percentage overhead

Alternative Method

Instead of converting units step-by-step, dimensional analysis can be set up as a single product of conversion factors: 5 files * (1.08 GB / 1 file) * (10^9 bytes / 1 GB) * (8 bits / 1 byte) * (1 total bit / 0.75 scientific bits) * (1 second / 1.6 * 10^6 bits) * (1 hour / 3600 seconds) = 10 hours.

Estimated Time:3m 0s

Question 386Question →

If $\frac{4^{x+3}}{8^{x-1}} = 16^{3-x}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 1

Answer

1

The correct answer is 1. By writing all terms with a base of 2, the equation is expressed as

\frac{(2^2)^{x+3}}{(2^3)^{x-1}} = (2^4)^{3-x}

. Applying the power rule to simplify the exponents yields

\frac{2^{2x+6}}{2^{3x-3}} = 2^{12-4x}

. Using the quotient rule, we subtract the exponents on the left-hand side to get

2^{(2x+6)-(3x-3)} = 2^{-x+9}

. Equating the exponents gives the linear equation

-x+9 = 12-4x

. Adding

4x

and subtracting

9

from both sides results in

3x = 3

, which simplifies to

x=1

.

Step-by-Step Solution

1

Express all exponential terms using a common base of 2

\frac{(2^2)^{x+3}}{(2^3)^{x-1}} = (2^4)^{3-x}

Rewriting each base as a power of 2 allows the application of standard exponent rules.

2

Apply the power of a power rule

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

to distribute the exponents

\frac{2^{2x+6}}{2^{3x-3}} = 2^{12-4x}

Multiplying the inner exponent by each term of the outer exponent simplifies the expression.

3

Apply the quotient of powers rule

\frac{a^m}{a^n} = a^{m-n}

to combine the fraction

2^{-x+9} = 2^{12-4x}

Subtracting the exponent in the denominator from the exponent in the numerator simplifies the left-hand side.

4

Equate the exponents since the bases are equal

-x + 9 = 12 - 4x

If two exponential expressions with the same positive base are equal, their exponents must be equal.

5

Solve the linear equation for

x

x = 1

Isolating

x

by algebraic manipulation yields the final answer.

Key Concept

Solving exponential equations by finding a common base and applying the laws of exponents.

Question 387Question →

The graph of a linear equation in the $xy$ -plane has an $x$ -intercept of $(k, 0)$ and a $y$ -intercept of $(0, 3k)$ , where $k$ is a positive constant. If the line passes through the point $(2, 12)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 6

Answer

The value of

k

is

6

.

The correct answer is

6

. The slope of the line can be found using the two intercepts

(k, 0)

and

(0, 3k)

:

m = \frac{3k - 0}{0 - k} = -3

. The equation of the line in slope-intercept form is

y = -3x + 3k

. Since the line passes through the point

(2, 12)

, we substitute

x = 2

and

y = 12

into the equation to get

12 = -3(2) + 3k

. Simplifying the equation yields

12 = -6 + 3k

, which gives

18 = 3k

. Dividing both sides by

3

results in

k = 6

.

Step-by-Step Solution

1

Find the slope of the line using the

x

-intercept

(k, 0)

and the

y

-intercept

(0, 3k)

.

The slope of the line is

-3

.

The slope formula is

m = \frac{y_2 - y_1}{x_2 - x_1}

. Substituting the points

(k, 0)

and

(0, 3k)

gives

m = \frac{3k - 0}{0 - k} = \frac{3k}{-k} = -3

.

2

Write the equation of the line in slope-intercept form.

The equation of the line is

y = -3x + 3k

.

Using the slope-intercept form

y = mx + b

, we substitute the slope

m = -3

and the

y

-intercept value

b = 3k

from the point

(0, 3k)

.

3

Substitute the given point

(2, 12)

into the equation and solve for

k

.

The value of

k

is

6

.

Substituting

x = 2

and

y = 12

into

y = -3x + 3k

yields

12 = -3(2) + 3k

. This simplifies to

12 = -6 + 3k

. Adding

6

to both sides gives

18 = 3k

, and dividing by

3

gives

k = 6

.

Key Concept

Linear equations in two variables, finding equations from intercepts, and constant determination via point substitution.

Question 388Question →

The table below shows the daily high temperatures, in degrees Celsius, recorded in a city over a $9$ -day period.

Daily High Temperature (°C)	Number of Days
$18$	$3$
$19$	$1$
$20$	$2$
$21$	$3$

Based on the table, what was the median daily high temperature, in degrees Celsius, for the $9$ -day period?

Show answer & explanation

Answer: 20

Answer

The median daily high temperature for the 9-day period is

20

°C.

To find the median daily high temperature, we first sum the frequencies to find the total number of days:

3 + 1 + 2 + 3 = 9

. Because

9

is an odd number, the median is the middle value, which is the

5

th value when the data points are ordered from least to greatest. The first

3

values are

18

, the

4

th value is

19

, and the

5

th and

6

th values are

20

. Therefore, the

5

th value in the ordered list is

20

.

Step-by-Step Solution

1

Determine the total number of observations (days) in the data set.

9

days

To find the position of the median value, we must first calculate the total number of data points by adding the frequencies:

3 + 1 + 2 + 3 = 9

.

2

Find the position of the median in the ordered data set.

5

th position

For an odd number of observations

n

, the median is located at the

\frac{n+1}{2}

position. With

n = 9

, this corresponds to the

\frac{9+1}{2} = 5

th value.

3

Identify the temperature at the

5

th position using the frequency table.

20

Listing the temperatures in ascending order gives:

18, 18, 18

(positions 1–3),

19

(position 4), and

20, 20

(positions 5–6). The value at the

5

th position is

20

.

Key Concept

Finding the median of a frequency distribution

Question 389Question →

A library administrator tracked a sample of $120$ books checked out last week, categorizing them by genre (fiction or nonfiction) and format (print or audiobook). The results are summarized in the table below.

Format	Fiction	Nonfiction
Print	$45$	$35$
Audiobook	$15$	$25$

Of the books in the sample that were checked out as audiobooks, what fraction were fiction?

Show answer & explanation

Answer: 0.375

Answer

The fraction of audiobooks that were fiction is 3/8 (which can also be gridded as the decimal .375).

To find the fraction of audiobooks that were fiction, find the total number of audiobooks in the sample by adding the values in the audiobook row:

15 + 25 = 40

. Next, find the number of fiction audiobooks, which is

15

. The fraction of audiobooks that were fiction is the ratio of these two values:

\frac{15}{40}

, which simplifies to

\frac{3}{8}

(or

0.375

in decimal form).

Step-by-Step Solution

1

Calculate the total number of audiobooks in the sample.

The total number of audiobooks is

15 + 25 = 40

.

The question asks for the fraction of books *that were checked out as audiobooks*, which means the denominator of our fraction must be the total number of audiobooks.

2

Identify the number of fiction audiobooks in the sample.

There are

15

fiction audiobooks.

The question asks what fraction of the audiobooks were *fiction*, so the numerator must be the number of books that are both audiobooks and fiction.

3

Write and simplify the fraction representing the conditional probability.

\frac{15}{40} = \frac{3}{8}

Divide the target number of books (15) by the total number of audiobooks (40). Simplify by dividing the numerator and the denominator by their greatest common divisor, which is 5.

Key Concept

Conditional Probability from a Two-Way Table

Estimated Time:1m 15s

Question 390Question →

In the quadratic equation $2x^2 - bx + 18 = 0$ , $b$ is a positive constant. If one of the solutions to the equation is $4$ times the other solution, what is the value of $b$ ?

Show answer & explanation

Answer: 15

Answer

15

By Vieta's formulas, the product of the roots of the quadratic equation

2x^2 - bx + 18 = 0

is

\frac{18}{2} = 9

. Letting the roots be

r_1

and

r_2

with

r_1 = 4r_2

, we have

4r_2^2 = 9

, which yields

r_2 = \frac{3}{2}

(since

b > 0

implies the roots must be positive). Thus,

r_1 = 6

. The sum of the roots is

6 + \frac{3}{2} = \frac{15}{2}

, and by Vieta's formulas, this sum equals

\frac{b}{2}

. Solving for

b

gives

15

.

Step-by-Step Solution

1

Set up the relationships for the product and sum of the roots using Vieta's formulas.

r_1 \cdot r_2 = 9

and

r_1 + r_2 = \frac{b}{2}

Vieta's formulas relate the coefficients of a quadratic equation to the sum and product of its roots.

2

Substitute the given condition that one root is

4

times the other (

r_1 = 4r_2

) into the product equation.

4r_2^2 = 9

This reduces the product equation to a single variable equation in terms of

r_2

.

3

Solve for

r_2

and determine its sign based on the constraint that

b

is positive.

r_2 = \frac{3}{2}

Since

b > 0

, the sum of the roots

5r_2 = \frac{b}{2}

must be positive, which requires

r_2 > 0

.

4

Calculate the second root

r_1

and then use the sum of the roots to find

b

.

b = 15

The sum of the roots is

6 + \frac{3}{2} = \frac{15}{2}

, and since

r_1 + r_2 = \frac{b}{2}

, we have

\frac{b}{2} = \frac{15}{2}

.

Key Concept

Relationship between roots and coefficients of a quadratic equation

Question 391Question →

A digital music playlist contains only rock songs and pop songs. At the beginning of the month, $60\%$ of the songs in the playlist were rock songs. During the month, the number of rock songs in the playlist increased by $10\%$ , and the number of pop songs in the playlist increased by $35\%$ . If there were $120$ rock songs in the playlist at the beginning of the month, what percent of the songs in the playlist at the end of the month were pop songs?

Show answer & explanation

Answer: 45

Answer

The percent of the songs in the playlist at the end of the month that were pop songs is 45.

To find the final percentage of pop songs, we first determine the initial number of songs. Since

120

rock songs represent

60\%

of the playlist, the initial total is

120 / 0.60 = 200

songs. This means there are

200 - 120 = 80

pop songs initially. During the month, the rock songs increase by

10\%

, yielding

120 \times 1.10 = 132

songs. The pop songs increase by

35\%

, yielding

80 \times 1.35 = 108

songs. The final total number of songs is

132 + 108 = 240

. The final percentage of pop songs is

\frac{108}{240} \times 100 = 45\%

. Thus, the correct value to grid in is 45.

Step-by-Step Solution

1

Determine the initial total number of songs and initial pop songs.

The initial total number of songs is 200, and the initial number of pop songs is 80.

Since 120 rock songs represent 60% of the playlist, the initial total is 120 / 0.60 = 200. The remaining songs are pop songs: 200 - 120 = 80.

2

Calculate the updated number of rock songs and pop songs after the percentage increases.

The final rock song count is 132, and the final pop song count is 108.

The number of rock songs increased by 10%, which is 120 * 1.10 = 132. The number of pop songs increased by 35%, which is 80 * 1.35 = 108.

3

Calculate the final total number of songs and the final percentage of pop songs.

The final total is 240 songs, and the final pop song percentage is 45%.

The final total is 132 + 108 = 240. The final pop song percentage is (108 / 240) * 100 = 45%.

Key Concept

Calculating absolute quantities from initial percentages, applying percentage increases to separate groups, and finding a new conditional percentage based on the final total.

Question 392Question →

An industrial printing press uses ink at a constant rate of $0.05$ milliliters per square centimeter of printed paper. The press prints a continuous roll of paper that is $40$ centimeters wide at a constant speed of $2$ meters per second. At this rate, how many liters of ink does the press use during $1$ hour of continuous printing?

Show answer & explanation

Answer: 1440

Answer

1440

The correct answer is 1440. First, convert the speed of the paper from meters per second to centimeters per second:

2 \text{ m/s} \times 100 \text{ cm/m} = 200 \text{ cm/s}

. Next, find the rate at which the paper surface area is printed:

40 \text{ cm} \times 200 \text{ cm/s} = 8,000 \text{ cm}^2/\text{s}

. Multiply this by the ink usage rate to find milliliters of ink per second:

8,000 \text{ cm}^2/\text{s} \times 0.05 \text{ mL/cm}^2 = 400 \text{ mL/s}

. Since there are

3,600

seconds in

1

hour, the press uses

400 \text{ mL/s} \times 3,600 \text{ s} = 1,440,000 \text{ mL}

of ink per hour. Finally, convert milliliters to liters:

1,440,000 \text{ mL} \div 1,000 \text{ mL/L} = 1,440

liters.

Step-by-Step Solution

1

Convert the speed of the paper from meters per second to centimeters per second.

200 \text{ cm/s}

To align the speed unit with the paper width unit (

40 \text{ cm}

).

2

Calculate the surface area of paper printed per second.

8,000 \text{ cm}^2/\text{s}

Multiply the width of the paper (

40 \text{ cm}

) by the converted speed (

200 \text{ cm/s}

).

3

Calculate the volume of ink used per second in milliliters.

400 \text{ mL/s}

Multiply the area printed per second (

8,000 \text{ cm}^2/\text{s}

) by the ink consumption rate (

0.05 \text{ mL/cm}^2

).

4

Convert the ink volume rate from per second to per hour.

1,440,000 \text{ mL/h}

Multiply the rate per second by

3,600

, since there are

3,600

seconds in

1

hour.

5

Convert the final volume from milliliters to liters.

1440

Divide the total milliliters (

1,440,000

) by

1,000

, since

1

liter is equal to

1,000

milliliters.

Key Concept

Multi-step dimensional analysis and compound unit conversion (length, area, volume, and time).

Question 393Question →

A town's population was $12,000$ in the year $2010$ . Since $2010$ , the population of the town has increased by a constant amount of $350$ people each year. Based on this information, what was the population of the town in the year $2018$ ?

Show answer & explanation

Answer: 14800

Answer

14,800

The population starts at

12,000

in

2010

and grows linearly by

350

people per year. The number of years from

2010

to

2018

is

2018 - 2010 = 8

years. The total increase in population is

350 \times 8 = 2,800

people. Adding this to the initial population of

12,000

gives a total population of

12,000 + 2,800 = 14,800

in the year

2018

.

Step-by-Step Solution

1

Calculate the number of years that passed between

2010

and

2018

.

8

years

To determine the total population growth, we need the elapsed time in years.

2

Calculate the total population growth over this

8

-year period.

2,800

people

Since the growth is linear, the total growth is the constant annual rate of change multiplied by the number of years.

3

Add the total population growth to the initial population from

2010

.

14,800

The population in

2018

is the initial population plus the total growth over the

8

-year period.

Key Concept

Linear growth represents a quantity that increases by a constant amount per unit of time.

Question 394Question →

An online retailer offers free shipping on orders of $75 or more. A customer has placed a shirt costing$ 22.50 and a pair of pants costing $34.80 in their shopping cart. They want to buy some pairs of socks that cost$ 4.50 per pair to qualify for free shipping. What is the minimum number of pairs of socks the customer must add to their cart to qualify for free shipping?

Show answer & explanation

Answer: 4

Answer

The minimum number of pairs of socks the customer must add to the cart is 4.

The customer needs to spend at least

75 to qualify for free shipping. The current subtotal of the cart is

22.50 +

34.80 =

57.30. The remaining amount needed to reach

75 is

75.00 -

57.30 =

17.70. Let

s

represent the number of pairs of socks purchased at

4.50 per pair. This can be represented by the inequality

4.50s \geq 17.70

. Dividing both sides of the inequality by

4.50

gives

s \geq 3.93$. Since the customer cannot purchase a fraction of a pair of socks, the minimum number of pairs of socks they must purchase is 4.

Step-by-Step Solution

1

Calculate the total cost of the items already in the shopping cart.

22.50 +

34.80 = $57.30

To determine how much has already been spent before adding socks.

2

Set up an inequality to represent the total cost of the order including

s

pairs of socks meeting the free shipping threshold of $75.

57.30 + 4.50s \geq 75

The total cost of the shirt, pants, and socks must be greater than or equal to $75.

3

Subtract $57.30 from both sides of the inequality to find the minimum amount that needs to be spent on socks.

4.50s \geq 17.70

To isolate the term representing the cost of the socks.

4

Divide both sides of the inequality by $4.50 to find the minimum number of pairs of socks.

s \geq 3.93

To isolate the variable representing the number of pairs of socks.

5

Determine the smallest integer value of

s

that satisfies the inequality.

4

The number of pairs of socks must be a whole number, and 4 is the smallest integer greater than or equal to 3.93.

Key Concept

Solving linear inequalities in one variable to find the minimum integer solution in a real-world context.

Estimated Time:1m 30s

Question 395Question →

A quadratic function $f$ has a vertex at $(h, k)$ , where $h$ and $k$ are constants. In the $xy$ -plane, the graph of $y = f(x)$ contains the points $(-1, 3)$ and $(7, 3)$ . The function $g$ is defined by $g(x) = f(x + 2) - 4$ , and its graph has a vertex at $(p, q)$ . If $p + q = 5$ , what is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

The value of k is 8.

The axis of symmetry of the quadratic function f is located at the midpoint of the x-values of any two points on the graph that have the same y-value. Since the graph of f contains (-1, 3) and (7, 3), the x-coordinate of its vertex, h, is equal to the midpoint (-1 + 7) / 2 = 3. Thus, the vertex of f is (3, k). The definition of g(x) = f(x + 2) - 4 indicates a horizontal translation of 2 units to the left and a vertical translation of 4 units down. Applying these translations to the vertex (3, k) of f gives the vertex of g as (3 - 2, k - 4) = (1, k - 4). Since the vertex of g is defined as (p, q), we have p = 1 and q = k - 4. Substituting these values into the equation p + q = 5 yields 1 + (k - 4) = 5, which simplifies to k - 3 = 5, or k = 8.

Step-by-Step Solution

1

Find the x-coordinate of the vertex of the function f

h = 3

Because the points (-1, 3) and (7, 3) have the same y-coordinate, the axis of symmetry (and thus the x-coordinate of the vertex) must lie exactly halfway between x = -1 and x = 7.

2

Determine the vertex of the function g(x)

(p, q) = (1, k - 4)

The function g(x) = f(x + 2) - 4 represents a translation of the graph of f by 2 units to the left and 4 units down. Since the vertex of f is (3, k), the vertex of g is (3 - 2, k - 4).

3

Solve for k using the given sum p + q = 5

k = 8

Substitute p = 1 and q = k - 4 into the equation p + q = 5 to get 1 + k - 4 = 5, which simplifies to k = 8.

Key Concept

Quadratic functions and graphs, including symmetry and translations.

Question 396Question →

A quadratic function $f$ is defined by $f(x) = a(x - 4)(x - 10)$ , where $a$ is a positive constant. In the $xy$ -plane, the graph of $f$ has a vertex with a $y$ -coordinate of $-18$ . What is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

The value of

a

is 2.

The vertex of a parabola lies on the axis of symmetry, which is located midway between its x-intercepts. Since the function is

f(x) = a(x - 4)(x - 10)

, the x-intercepts are at

x = 4

and

x = 10

. The midpoint of these values is

\frac{4 + 10}{2} = 7

. Thus, the x-coordinate of the vertex is 7. Since the y-coordinate of the vertex is given as

-18

, the vertex is at

(7, -18)

. Substituting these coordinates into the function equation yields

-18 = a(7 - 4)(7 - 10)

, which simplifies to

-18 = a(3)(-3)

, or

-18 = -9a

. Solving for

a

gives

a = 2

.

Step-by-Step Solution

1

Find the x-coordinate of the vertex using the symmetry of the parabola.

The x-coordinate of the vertex is 7.

The x-coordinate of the vertex lies on the axis of symmetry, which is the midpoint of the x-intercepts

x = 4

and

x = 10

.

2

Identify the coordinates of the vertex.

The vertex is at

(7, -18)

.

The y-coordinate of the vertex is given as

-18

, and its x-coordinate was calculated as 7.

3

Substitute the vertex coordinates into the function definition to solve for

a

.

a = 2

Substituting

(7, -18)

into

f(x) = a(x - 4)(x - 10)

gives

-18 = a(7 - 4)(7 - 10)

, which simplifies to

-18 = -9a

.

Key Concept

Finding the vertex of a quadratic function from its factored form and solving for a leading coefficient.

Alternative Method

Alternatively, the function

f(x) = a(x - 4)(x - 10)

can be expanded to standard form:

f(x) = a(x^2 - 14x + 40)

. The x-coordinate of the vertex can be found using the formula

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

, which gives

x = -\frac{-14a}{2a} = 7

. Then, substitute

x = 7

and

f(7) = -18

to solve for

a

.

Estimated Time:1m 30s

Question 397Question →

A quadratic function $f$ has its vertex at $(4, -3)$ in the coordinate plane. The point $(2, 5)$ is on the graph of $y = f(x)$ . The function $g$ is defined by $g(x) = -f(x - 3) + 8$ . What is the value of $g(8)$ ?

Show answer & explanation

Answer: 9

Answer

The correct answer is 9.

The correct answer is 9. To find this, we first establish the vertex form of the quadratic function

f(x) = a(x - 4)^2 - 3

. Substituting the point

(2, 5)

gives

5 = a(2 - 4)^2 - 3

, which simplifies to

8 = 4a

, so

a = 2

. Thus,

f(x) = 2(x - 4)^2 - 3

. To find the value of

g(8)

, we substitute

8

into the definition of

g(x)

, yielding

g(8) = -f(8 - 3) + 8 = -f(5) + 8

. Evaluating

f(5)

gives

2(5 - 4)^2 - 3 = -1

. Finally, substituting

-1

back into the expression for

g(8)

gives

-(-1) + 8 = 9

.

Step-by-Step Solution

1

Write the quadratic function

f(x)

in vertex form using its vertex

(4, -3)

.

f(x) = a(x - 4)^2 - 3

The vertex form of a quadratic function with vertex

(h, k)

is

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

.

2

Substitute the coordinates of the point

(2, 5)

into the vertex form to find the value of the constant

a

.

a = 2

, so

f(x) = 2(x - 4)^2 - 3

Since the point

(2, 5)

lies on the graph of

f

, substituting

x = 2

and

f(2) = 5

allows us to solve for the vertical stretch factor

a

.

3

Use the definition of

g(x)

to express

g(8)

in terms of

f

.

g(8) = -f(5) + 8

Substituting

x = 8

into the equation

g(x) = -f(x - 3) + 8

yields

g(8) = -f(8 - 3) + 8 = -f(5) + 8

.

4

Evaluate

f(5)

using the equation for

f(x)

found in Step 2.

f(5) = -1

Substituting

x = 5

into

f(x) = 2(x - 4)^2 - 3

gives

2(5 - 4)^2 - 3 = 2(1) - 3 = -1

.

5

Substitute the value of

f(5)

into the expression for

g(8)

to find the final result.

g(8) = 9

Substituting

f(5) = -1

into

g(8) = -f(5) + 8

yields

-(-1) + 8 = 1 + 8 = 9

.

Key Concept

Finding the equation of a quadratic function from its vertex and a point, and evaluating transformed functions using function notation.

Question 398Question →

In the equation $\frac{25^x}{5^{x-3}} = 125^2$ , what is the value of $x$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

By writing all terms with a base of 5, the equation

\frac{25^x}{5^{x-3}} = 125^2

becomes

\frac{5^{2x}}{5^{x-3}} = 5^6

. Applying the quotient rule of exponents, the left side simplifies to

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

. Setting the exponents equal gives

x+3 = 6

, which yields

x = 3

.

Step-by-Step Solution

1

Express all terms with a common base of 5.

25^x = (5^2)^x = 5^{2x}

and

125^2 = (5^3)^2 = 5^6

.

Expressing all exponential terms with the same base allows the exponents to be equated directly once simplified.

2

Substitute these expressions back into the original equation and simplify the left side using the quotient rule of exponents.

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

. The equation becomes

5^{x+3} = 5^6

.

The quotient rule states that dividing exponential terms with the same base requires subtracting the exponent of the denominator from the exponent of the numerator:

\frac{b^m}{b^n} = b^{m-n}

.

3

Equate the exponents and solve for

x

.

x + 3 = 6 \implies x = 3

.

Since the bases on both sides of the equation are equal, their exponents must also be equal: if

b^y = b^z

where

b > 0

and

b \neq 1

, then

y = z

.

Key Concept

Solving exponential equations by expressing terms with a common base and applying exponent rules.

Estimated Time:1m 30s

Question 399Question →

A biologist measures the heights, in centimeters, of five seedlings in a laboratory. The heights are $12$ , $15$ , $8$ , $21$ , and $14$ . What is the range, in centimeters, of these heights?

Show answer & explanation

Answer: 13

Answer

13

To find the range of a dataset, subtract the minimum value from the maximum value. In the given dataset of seedling heights (

12

,

15

,

8

,

21

, and

14

), the maximum height is

21

centimeters and the minimum height is

8

centimeters. Calculating the difference gives

21 - 8 = 13

centimeters.

Step-by-Step Solution

1

Identify the maximum and minimum values in the dataset.

The maximum value is

21

and the minimum value is

8

.

The range is defined as the difference between the largest (maximum) and smallest (minimum) values in a data distribution.

2

Subtract the minimum value from the maximum value to calculate the range.

21 - 8 = 13

Subtracting the minimum value of

8

from the maximum value of

21

yields the range of the dataset.

Key Concept

The range of a dataset measures the spread of the data and is calculated by subtracting the minimum value from the maximum value.

Question 400Question →

An express delivery truck travels from Warehouse A to Warehouse B at a constant speed, and then returns from Warehouse B to Warehouse A along the same route at a different constant speed. On the first trip, the ratio of the outbound travel time to the inbound travel time is $3$ to $5$ . On a second trip along the same route, the truck's outbound speed is $50\%$ greater than its outbound speed on the first trip, and its inbound speed is $25\%$ less than its inbound speed on the first trip. If the total round-trip travel time for the second trip is $13$ hours, what is the total round-trip travel time, in hours, for the first trip?

Show answer & explanation

Answer: 12

Answer

The total round-trip travel time for the first trip is 12 hours.

The correct answer is

12

. By setting the outbound and inbound travel times of the first trip as

t_1

and

t_2

with

t_1 = 0.6t_2

, we apply the inverse relationships of speed and time to express the second trip's times as

t_1' = \frac{2}{3}t_1

and

t_2' = \frac{4}{3}t_2

. Substituting these into the second trip's total time equation

\frac{2}{3}t_1 + \frac{4}{3}t_2 = 13

yields

t_2 = 7.5

and

t_1 = 4.5

, summing to

12

hours.

Step-by-Step Solution

1

Set up the ratio of outbound to inbound travel times for the first trip.

t_1 = 0.6t_2

(or

t_1 = \frac{3}{5}t_2

)

The ratio of the outbound travel time (

t_1

) to the inbound travel time (

t_2

) is given as

3

to

5

.

2

Express the new travel times for the second trip (

t_1'

and

t_2'

) in terms of the original travel times.

t_1' = \frac{2}{3}t_1

and

t_2' = \frac{4}{3}t_2

For a constant distance, travel time is inversely proportional to speed. A

50\%

speed increase multiplies time by

\frac{1}{1.5} = \frac{2}{3}

, and a

25\%

speed decrease multiplies time by

\frac{1}{0.75} = \frac{4}{3}

.

3

Create an equation for the total travel time of the second trip.

\frac{2}{3}t_1 + \frac{4}{3}t_2 = 13

The sum of the outbound and inbound travel times on the second trip is given as

13

hours.

4

Substitute the first trip's relationship into the equation and solve for

t_2

.

t_2 = 7.5

Substituting

t_1 = \frac{3}{5}t_2

gives

\frac{2}{3}\left(\frac{3}{5}t_2\right) + \frac{4}{3}t_2 = 13 \implies \frac{26}{15}t_2 = 13 \implies t_2 = 13 \times \frac{15}{26} = 7.5

.

5

Determine

t_1

and calculate the total travel time for the first trip.

t_1 + t_2 = 12

Since

t_2 = 7.5

, then

t_1 = 0.6(7.5) = 4.5

. The total round-trip time is

4.5 + 7.5 = 12

.

Key Concept

Inverse proportionality of speed and time, and solving linear systems involving ratios.

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

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

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