All practice questions

Question 1601Question →

During a dry season, the volume of water $V$ , in millions of gallons, in a municipal reservoir $d$ days after the start of a monitoring period is modeled by the equation $V = 248 - 1.8d$ , where $0 \leq d \leq 60$ . According to the model, at what rate, in gallons per minute, is water leaving the reservoir?

Show answer & explanation

Answer: 1250

Answer

The reservoir is losing water at a rate of

1,250

gallons per minute.

The slope of the linear equation

V = 248 - 1.8d

is

-1.8

, indicating a decrease of

1.8

million gallons of water per day. To express this rate in gallons per minute, we perform the unit conversion. First, convert

1.8

million gallons to gallons:

1.8 \times 1,000,000 = 1,800,000

gallons. Second, convert

1

day to minutes:

24 \text{ hours/day} \times 60 \text{ minutes/hour} = 1,440

minutes. Finally, divide the total gallons by the total minutes:

\frac{1,800,000 \text{ gallons}}{1,440 \text{ minutes}} = 1,250

gallons per minute.

Step-by-Step Solution

1

Identify the daily rate of water leaving the reservoir from the given linear equation.

The daily rate is

1.8

million gallons per day.

The slope of the linear equation

V = 248 - 1.8d

is

-1.8

, which represents the rate of change of the volume with respect to time in days. The negative sign indicates a decrease, meaning water is leaving at a rate of

1.8

million gallons per day.

2

Convert the volume from million gallons to gallons.

1,800,000

gallons.

To convert from millions of gallons to gallons, multiply the value by

1,000,000

, resulting in

1.8 \times 1,000,000 = 1,800,000

.

3

Convert the time unit from days to minutes.

1,440

minutes.

There are

24

hours in a day and

60

minutes in an hour, so

1 \text{ day} = 24 \times 60 = 1,440 \text{ minutes}

.

4

Calculate the rate in gallons per minute.

1,250

gallons per minute.

Divide the volume in gallons by the time in minutes:

\frac{1,800,000 \text{ gallons}}{1,440 \text{ minutes}} = 1,250

.

Key Concept

Interpreting the slope of a linear relationship in context and performing unit conversions on rates.

Question 1602Question →

A tour guide offers two types of guided tours: a walking tour and a trolley tour. On Saturday, the guide led $3$ walking tours and $2$ trolley tours, which lasted a total of $8$ hours. On Sunday, the guide led $2$ walking tours and $4$ trolley tours, which lasted a total of $12$ hours. If each walking tour has a duration of $w$ hours and each trolley tour has a duration of $t$ hours, what is the total duration, in hours, of $1$ walking tour and $1$ trolley tour?

2.0

3.5

4.0

4.25

Show answer & explanation

Answer: 3.5

Answer

The correct answer is 3.5.

The correct answer is

3.5

. The scenario can be represented by the system of linear equations

3w + 2t = 8

and

2w + 4t = 12

. Multiplying the first equation by

2

gives

6w + 4t = 16

. Subtracting the second equation from this yields

4w = 4

, which simplifies to

w = 1

. Substituting

w = 1

into the first equation yields

3(1) + 2t = 8

, which simplifies to

2t = 5

or

t = 2.5

. The sum of the durations of one walking tour and one trolley tour is

w + t = 1 + 2.5 = 3.5

hours.

Step-by-Step Solution

1

Represent the given scenarios with a system of linear equations using the variables

w

(duration of a walking tour) and

t

(duration of a trolley tour).

The system of equations is:
1)

3w + 2t = 8

2)

2w + 4t = 12

This translates the word problem into mathematical equations.

2

Solve the system of equations for

w

and

t

. For example, multiply the first equation by

2

to align the

t

coefficients, and then subtract the second equation from it.

Multiplying the first equation by

2

gives

6w + 4t = 16

.
Subtracting the second equation (

2w + 4t = 12

) from this yields

4w = 4

, which simplifies to

w = 1

.

This eliminates the variable

t

to find the value of

w

.

3

Substitute

w = 1

back into one of the original equations to solve for

t

.

Using the first equation:

3(1) + 2t = 8 \implies 3 + 2t = 8 \implies 2t = 5 \implies t = 2.5

.

This determines the value of the second variable,

t

.

4

Calculate the sum of

w

and

t

to find the total duration of

1

walking tour and

1

trolley tour.

w + t = 1 + 2.5 = 3.5

.

This directly answers the question asked in the problem stem.

Key Concept

Solving a system of linear equations to find a linear combination of variables

Estimated Time:1m 30s

Question 1603Question →

A solar power system stores electricity in a battery at a constant rate during the day. The total amount of energy in the battery, in kilowatt-hours (kWh), is a linear function of the number of hours after sunrise. Exactly $3$ hours after sunrise, the battery contains $14$ kWh of energy, and exactly $7$ hours after sunrise, it contains $26$ kWh of energy. If the battery's maximum capacity is $35$ kWh, how many hours after sunrise will it take for the battery to be fully charged?

Show answer & explanation

Answer: 10

Answer

10

To find the number of hours after sunrise it takes to fully charge the battery to 35 kWh, we first determine the constant rate of energy storage. The rate of change is given by the difference in energy divided by the difference in hours:

\frac{26 - 14}{7 - 3} = \frac{12}{4} = 3

kWh per hour. The relationship between energy

E

and hours

h

can be written as

E = 3h + b

. Substituting the known point

(3, 14)

gives

14 = 3(3) + b

, which simplifies to

b = 5

. Thus, the linear equation representing this relationship is

E = 3h + 5

. To find the time when the battery is fully charged to

35

kWh, set

E = 35

in the equation:

35 = 3h + 5

. Subtracting 5 from both sides yields

30 = 3h

, and dividing by 3 gives

h = 10

hours.

Step-by-Step Solution

1

Calculate the rate of energy storage per hour.

3 kWh per hour

Since the energy increases at a constant rate, the rate is the change in energy divided by the change in time:

\frac{26 - 14}{7 - 3} = 3

.

2

Determine the initial energy stored at sunrise.

5 kWh

Using the rate of 3 kWh per hour and the fact that the battery had 14 kWh at 3 hours, we subtract the energy gained in those 3 hours:

14 - 3(3) = 5

.

3

Set up the linear equation and solve for the time when the battery reaches 35 kWh.

10 hours

The linear relationship is

E = 3h + 5

. Setting

E = 35

gives

35 = 3h + 5

. Subtracting 5 from both sides gives

30 = 3h

, which simplifies to

h = 10

.

Key Concept

Linear Equations in Two Variables

Question 1604Question →

If $x$ and $y$ are positive numbers, which of the following is equivalent to the expression $\frac{\sqrt{36x^5 y^3}}{3x(xy^3)^{1/2}}$ ?

2x\sqrt{x}

2x

12x

2xy

Show answer & explanation

Answer: $2x$

Answer

The expression is equivalent to

2x

.

To find the equivalent expression, simplify the numerator and the denominator separately. The numerator

\sqrt{36x^5 y^3}

simplifies to

6x^{5/2}y^{3/2}

. The denominator

3x(xy^3)^{1/2}

simplifies to

3x^{3/2}y^{3/2}

. Dividing the simplified numerator by the simplified denominator yields

\frac{6x^{5/2}y^{3/2}}{3x^{3/2}y^{3/2}} = 2x

.

Step-by-Step Solution

1

Convert the radical expression in the numerator to an expression with fractional exponents.

\sqrt{36x^5 y^3} = (36x^5 y^3)^{1/2} = 6x^{5/2}y^{3/2}

The square root of a product is the product of the square roots of each factor, where

\sqrt{a} = a^{1/2}

.

2

Apply exponent rules to simplify the denominator.

3x(xy^3)^{1/2} = 3x^1 \cdot x^{1/2} y^{3/2} = 3x^{3/2}y^{3/2}

When multiplying terms with the same base, add their exponents (

x^a \cdot x^b = x^{a+b}

).

3

Divide the simplified numerator by the simplified denominator.

\frac{6x^{5/2}y^{3/2}}{3x^{3/2}y^{3/2}} = 2x^{5/2 - 3/2}y^{3/2 - 3/2} = 2x^1 y^0 = 2x

When dividing terms with the same base, subtract the exponent of the denominator from the exponent of the numerator (

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

).

Key Concept

Equivalent Algebraic Expressions

Estimated Time:1m 15s

Question 1605Question →

Which of the following expressions is equivalent to $\frac{x^2 - 4x - 12}{2x - 12} - \frac{x}{4}$ for any real number $x > 6$ ?

\frac{x - 4}{4}

\frac{x + 2}{4}

\frac{x + 4}{4}

\frac{3x + 4}{4}

Show answer & explanation

Answer: $\frac{x + 4}{4}$

Answer

The expression stating that the equivalent form is the fraction with

x+4

in the numerator and

4

in the denominator

The correct equivalent expression is found by first factoring the rational expression

\frac{x^2 - 4x - 12}{2x - 12}

into

\frac{(x - 6)(x + 2)}{2(x - 6)}

. Since

x > 6

, the factor

x - 6

is non-zero and can be canceled, leaving

\frac{x + 2}{2}

. Converting

\frac{x + 2}{2}

to have a common denominator of

4

yields

\frac{2x + 4}{4}

. Finally, subtracting

\frac{x}{4}

gives the simplified result of

\frac{x + 4}{4}

.

Step-by-Step Solution

1

Factor the quadratic numerator and the linear denominator of the first rational term.

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

To identify and cancel common algebraic factors in the fraction.

2

Cancel the common factor

x - 6

from the numerator and denominator.

\frac{x + 2}{2}

Since

x > 6

, the term

x - 6

is non-zero and can be canceled.

3

Find a common denominator of

4

to perform subtraction with the second term,

-\frac{x}{4}

.

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

To subtract fractions, their denominators must be identical.

4

Subtract the numerators and combine terms over the common denominator.

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

Subtracting like terms in the numerator yields the final simplified expression.

Key Concept

Equivalent Algebraic Expressions

Alternative Method

Instead of simplifying algebraically, you can substitute a value for

x

that is greater than

6

. For example, if

x = 8

, the original expression evaluates to

\frac{8^2 - 4(8) - 12}{2(8) - 12} - \frac{8}{4} = \frac{20}{4} - 2 = 3

. Evaluating the correct option with

x = 8

yields

\frac{8+4}{4} = 3

, confirming equivalence.

Estimated Time:1m 30s

Question 1606Question →

In the $xy$ -plane, the graph of the linear equation $3x + by = 36$ , where $b$ is a constant, is a line with a slope of $-0.75$ . What is the value of $b$ ?

Show answer & explanation

Answer: 4

Answer

The value of

b

is

4

.

The linear equation

3x + by = 36

can be rewritten in slope-intercept form by subtracting

3x

from both sides to get

by = -3x + 36

, and then dividing all terms by

b

to get

y = -\frac{3}{b}x + \frac{36}{b}

. The slope of this line is the coefficient of

x

, which is

-\frac{3}{b}

. Setting this equal to the given slope of

-0.75

(or

-\frac{3}{4}

) gives

-\frac{3}{b} = -\frac{3}{4}

. Solving for

b

yields

b = 4

.

Step-by-Step Solution

1

Express the given linear equation in slope-intercept form.

y = -\frac{3}{b}x + \frac{36}{b}

To identify the slope of the line in terms of the constant

b

, we rewrite the equation

3x + by = 36

in the form

y = mx + d

.

2

Equate the expression for the slope to the given slope value.

-\frac{3}{b} = -0.75

The coefficient of

x

in the slope-intercept form represents the slope of the line, which is given as

-0.75

.

3

Solve the equation for

b

.

b = 4

Multiply both sides of the equation by

-1

to get

\frac{3}{b} = 0.75

. Since

0.75 = \frac{3}{4}

, we have

\frac{3}{b} = \frac{3}{4}

, which gives

b = 4

.

Key Concept

Converting a linear equation from standard form to slope-intercept form to determine its slope.

Estimated Time:1m 30s

Question 1607Question →

A researcher is monitoring the temperature, $T$ , in degrees Celsius, of a chemical reaction. The target temperature range is maintained by an automated cooling system that activates when the temperature satisfies the inequality $-5(T - 12) - 8(3 - 2T) < 14(T + 3)$ . Which of the following inequalities represents all possible values of $T$ for which the cooling system will activate?

T < -2

T > -2

T > -42

T > 2

Show answer & explanation

Answer: $T > -2$

Answer

The inequality representing all possible values of the temperature is

T > -2

.

The correct inequality is obtained by first expanding both sides of the inequality to get

-5T + 60 - 24 + 16T < 14T + 42

. Simplifying the left side yields

11T + 36 < 14T + 42

. Subtracting

14T

from both sides gives

-3T + 36 < 42

, and subtracting

36

from both sides gives

-3T < 6

. Finally, dividing both sides by

-3

and reversing the inequality sign results in

T > -2

.

Step-by-Step Solution

1

Distribute the constants on both sides of the inequality:

-5(T - 12) - 8(3 - 2T) < 14(T + 3)

.

-5T + 60 - 24 + 16T < 14T + 42

Remove the parentheses to prepare the inequality for simplification.

2

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

11T + 36 < 14T + 42

Simplify the expression before isolating the variable.

3

Subtract

14T

and

36

from both sides of the inequality.

-3T < 6

Group all variable terms on one side and constant terms on the other.

4

Divide both sides of the inequality by

-3

and reverse the inequality sign.

T > -2

Dividing by a negative number reverses the inequality relationship.

Key Concept

Solving multi-step linear inequalities in one variable, distributing coefficients, combining like terms, and reversing the inequality sign when dividing by a negative number.

Estimated Time:2m 0s

Question 1608Question →

Although the thick, sulfur-rich clouds of Venus completely obscure its surface from visible-light telescopes, scientists have successfully mapped the planet's complex topography. To bypass this atmospheric barrier, the Magellan spacecraft utilized synthetic aperture radar ______ a sophisticated technology that transmitted microwave pulses through the cloud layer and measured the reflected signals to construct highly detailed three-dimensional maps of the Venusian terrain.

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

—a

; a

, it was a

a

Show answer & explanation

Answer: —a

Answer

The choice containing a single dash followed by 'a' ('—a') is correct because it appropriately sets off the explanatory appositive phrase that describes 'synthetic aperture radar'.

The correct option uses a single dash to link the independent clause to the explanatory appositive noun phrase ('a sophisticated technology...'). A single dash is an acceptable punctuation mark to introduce explanatory details, lists, or summaries at the end of an independent clause.

Step-by-Step Solution

1

Analyze the structure of the clause preceding the blank.

The clause 'To bypass this atmospheric barrier, the Magellan spacecraft utilized synthetic aperture radar' is an independent clause because it contains a subject and a verb and can stand alone as a sentence.

Determining the grammatical status of the preceding clause helps identify which punctuation marks (such as colons, semicolons, or dashes) are permitted.

2

Analyze the structure of the phrase following the blank.

The phrase starting with 'a sophisticated technology...' is a noun phrase (an appositive) that explains 'synthetic aperture radar'; it is not an independent clause.

Knowing whether the following phrase is an independent or dependent clause rules out punctuation marks that require independent clauses on both sides, such as semicolons.

3

Evaluate the punctuation choices based on grammatical rules.

A single dash is correct for connecting an independent clause to an explanatory noun phrase. A semicolon is incorrect because the second part is not an independent clause. A comma splice is created by inserting a subject and verb ('it was') after a comma, and omitting punctuation entirely creates a run-on.

Matching the structural requirements of the sentence to the conventions of Standard English identifies the correct option.

Key Concept

Using a single dash to set off explanatory details at the end of an independent clause, and distinguishing it from semicolon and comma splice errors.

Question 1609Question →

The value of a certain car model decreases by $12\%$ each year. If the initial value of the car is $\$25,000$ , which of the following functions models the value of the car, $V(t)$ , in dollars, $t$ years after it was purchased?

V(t) = 25,000 - 0.12t

V(t) = 25,000(0.12)^t

V(t) = 25,000(0.88)^t

V(t) = 25,000(1.12)^t

Show answer & explanation

Answer: $V(t) = 25,000(0.88)^t$

Answer

The function

V(t) = 25,000(0.88)^t

The initial value of the car is

25,000

. An annual decrease of

12\%

means that the car retains

100\% - 12\% = 88\%

of its value from the previous year. This results in a decay factor of

1 - 0.12 = 0.88

. Substituting these values into the exponential model

V(t) = a(b)^t

yields

V(t) = 25,000(0.88)^t

.

Step-by-Step Solution

1

Identify the initial value of the car.

The initial value is

25,000

.

This is the value of the car at

t = 0

, which corresponds to the coefficient

a

in the general exponential model

V(t) = a(b)^t

.

2

Determine the growth or decay factor base,

b

.

The base is

1 - 0.12 = 0.88

.

Since the car's value decreases by

12\%

each year, it retains

100\% - 12\% = 88\%

of its value. This is represented as a decay factor of

0.88

.

3

Formulate the exponential decay equation.

The equation is

V(t) = 25,000(0.88)^t

.

By substituting the initial value

a = 25,000

and the decay base

b = 0.88

into

V(t) = a(b)^t

.

Key Concept

Exponential decay models and interpretation of growth/decay factors

Alternative Method

Calculate the value of the car after one year (

t = 1

). A

12\%

decrease on a

\$25,000

car means its value drops by

\$3,000

to

\$22,000

. Substituting

t=1

into the correct function should yield

22,000

. Calculating for

V(t) = 25,000(0.88)^t

at

t = 1

gives

25,000(0.88)^1 = 22,000

, confirming the choice.

Estimated Time:45s

Question 1610Question →

While excavating the ruins of the ancient city of Petra in modern-day Jordan, archaeologists uncovered a highly complex network of subterranean water channels. These channels demonstrate the ________ sophisticated understanding of desert hydraulics, which allowed them to harvest seasonal flash floods and sustain a thriving agricultural civilization in an otherwise arid environment.

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

Nabataeans

Nabataeans'

Nabataean's

Nabataean

Show answer & explanation

Answer: Nabataeans'

Answer

Nabataeans'

The correct option is the plural possessive form 'Nabataeans''. The plural number is required because the passage later refers to the group with the plural pronoun 'them'. The possessive case is required because the noun modifies 'sophisticated understanding,' indicating that the understanding belonged to the Nabataeans.

Step-by-Step Solution

1

Determine the grammatical relationship between the noun in the blank and the noun phrase that follows it ('sophisticated understanding').

The blank must contain a possessive noun because the 'understanding' belongs to the people of the ancient city.

Possession is required when one noun modifies another by showing ownership or association.

2

Identify clues in the passage that indicate whether the possessive noun should be singular or plural.

The subsequent pronoun 'them' is plural, indicating that the noun refers to the people collectively (plural).

Pronouns must agree in number with their antecedents or referents.

3

Combine the plural and possessive requirements to select the correct punctuation.

The plural possessive form of the noun is 'Nabataeans'', which places the apostrophe after the plural '-s' suffix.

Standard English conventions dictate that plural possessive nouns ending in '-s' receive an apostrophe at the end of the word.

Key Concept

Plural and Possessive Nouns and Pronouns

Question 1611Question →

A quadratic function $f$ is defined by $f(x) = 2(x + 3)^2 - 8$ . What are the coordinates of the vertex of the graph of $y = f(x)$ in the $xy$ -plane?

(-3, -8)

(3, -8)

(-3, 8)

(3, 8)

Show answer & explanation

Answer: $(-3, -8)$

Answer

The correct answer is the coordinate pair

(-3, -8)

.

The vertex form of a quadratic function is

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

, where the vertex of the graph of the function in the

xy

-plane is

(h, k)

. For the function

f(x) = 2(x + 3)^2 - 8

, the expression inside the parentheses can be rewritten as

(x - (-3))

, showing that

h = -3

. The constant term outside the parentheses is

-8

, showing that

k = -8

. Therefore, the vertex of the graph of the function is

(-3, -8)

.

Step-by-Step Solution

1

Identify the standard vertex form of a quadratic function.

The vertex form is

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

, where

(h, k)

is the vertex of the parabola.

Comparing the given equation to this template allows us to read the coordinates directly.

2

Rewrite the given equation to align with the negative sign inside the parentheses of the vertex form.

The term

(x + 3)^2

can be rewritten as

(x - (-3))^2

, and the subtraction of

8

can be written as adding

-8

. This yields

f(x) = 2(x - (-3))^2 + (-8)

.

Matching the template exactly prevents sign errors when identifying the values of

h

and

k

.

3

Extract the values of

h

and

k

.

h = -3

and

k = -8

, which means the vertex is located at

(-3, -8)

.

The vertex

(h, k)

coordinates represent the minimum point of this upward-opening parabola.

Key Concept

Identifying the vertex coordinates directly from the vertex form of a quadratic function.

Alternative Method

An alternative method is to expand the function into the standard form

f(x) = ax^2 + bx + c

. Expanding

f(x) = 2(x + 3)^2 - 8

gives

f(x) = 2(x^2 + 6x + 9) - 8

, which simplifies to

f(x) = 2x^2 + 12x + 10

. The

x

-coordinate of the vertex can be found using the formula

x = -\frac{b}{2a} = -\frac{12}{2(2)} = -3

. Substituting

x = -3

back into the function gives

f(-3) = 2(-3 + 3)^2 - 8 = -8

. This confirms the vertex is at

(-3, -8)

.

Estimated Time:45s

Question 1612Question →

In the quadratic equation $x^2 - 15x + k = 0$ , where $k$ is a constant, one of the real solutions is $4$ times the other solution. What is the value of $k$ ?

Show answer & explanation

Answer: 36

Answer

36

The correct answer is 36. By Vieta's formulas, the sum of the roots of the quadratic equation

x^2 - 15x + k = 0

is

15

, and the product of the roots is

k

. Let the roots be

r

and

4r

. Their sum is

r + 4r = 5r = 15

, which gives

r = 3

. The roots are

3

and

4(3) = 12

. The product of the roots is

3 \times 12 = 36

, which represents the constant term

k

.

Step-by-Step Solution

1

Express the relationship between the two roots.

Let the two roots be

r

and

4r

.

We are given that one of the real solutions is 4 times the other solution.

2

Use Vieta's formulas to set up an equation for the sum of the roots.

r + 4r = 15 \implies 5r = 15

For a quadratic equation in the form

x^2 - sx + p = 0

, the sum of the roots is equal to

s

.

3

Solve for the variable

r

and find both roots.

r = 3

, so the roots are

3

and

4(3) = 12

.

Dividing both sides of

5r = 15

by 5 gives

r = 3

. Substituting this back gives the two roots.

4

Calculate the product of the roots to find the value of

k

.

k = 3 \times 12 = 36

For a quadratic equation in the form

x^2 - sx + p = 0

, the product of the roots is equal to

p

, which in this case is

k

.

Key Concept

Using Vieta's formulas to find relationships between the coefficients and roots of a quadratic equation.

Question 1613Question →

\begin{aligned} y &= 2x^2 - kx + 8 \\ y &= 4x - 10 \end{aligned}

In the system of equations above, $k$ is a constant. If the system has exactly one real solution, what is the product of all possible values of $k$ ?

-48

-128

-20

128

Show answer & explanation

Answer: -128

Answer

-128

To find the number of solutions to the system, set the two equations equal to each other:

2x^2 - kx + 8 = 4x - 10

. Moving all terms to one side gives

2x^2 - (k + 4)x + 18 = 0

. For this quadratic equation to have exactly one real solution, its discriminant,

\Delta = B^2 - 4AC

, must equal zero. Substituting the coefficients

A = 2

,

B = -(k + 4)

, and

C = 18

into the discriminant formula gives

(-(k + 4))^2 - 4(2)(18) = 0

, which simplifies to

(k + 4)^2 - 144 = 0

. Solving this equation gives

k + 4 = 12 \implies k = 8

, or

k + 4 = -12 \implies k = -16

. The product of these possible values is

8 \times (-16) = -128

.

Step-by-Step Solution

1

Equate the quadratic and linear equations to find the condition for intersection.

2x^2 - kx + 8 = 4x - 10

To find the points where the two graphs intersect, their

y

-values must be set equal to each other.

2

Rearrange the terms to form a standard quadratic equation of the form

Ax^2 + Bx + C = 0

.

2x^2 - (k + 4)x + 18 = 0

Grouping terms allows us to determine the coefficients

A = 2

,

B = -(k + 4)

, and

C = 18

to use in the discriminant.

3

Set the discriminant

\Delta = B^2 - 4AC

of the quadratic equation to

0

.

[-(k + 4)]^2 - 4(2)(18) = 0 \implies (k + 4)^2 - 144 = 0

A system of a line and a parabola has exactly one real solution (meaning the line is tangent to the parabola) when the resulting quadratic equation has exactly one real root, which corresponds to a discriminant of zero.

4

Solve for the possible values of

k

and calculate their product.

k + 4 = \pm 12 \implies k = 8

or

k = -16

. The product is

8 \times (-16) = -128

.

Extracting the square root yields two possible values for

k

, and multiplying them gives the requested product.

Key Concept

Using the discriminant of a quadratic equation to find parameter values that make a linear-quadratic system have exactly one real solution.

Alternative Method

Once the quadratic equation

(k + 4)^2 - 144 = 0

is simplified to

k^2 + 8k - 128 = 0

, Vieta's formulas can be applied directly to find the product of the possible values of

k

. According to Vieta's formulas, the product of the roots of a quadratic equation

ak^2 + bk + c = 0

is given by

c/a

. Here,

a = 1

and

c = -128

, so the product of the roots is

-128 / 1 = -128

.

Estimated Time:2m 0s

Question 1614Question →

A commercial 3D printer uses a polymer filament to print prototype parts. The remaining mass of the filament spool, $M$ , in grams, is modeled by a linear function of the total printing time, $t$ , in minutes. When the printer has been running for $20$ minutes, the spool has $850$ grams of filament remaining. When the printer has been running for $50$ minutes, the spool has $685$ grams of filament remaining. According to the model, how many grams of filament does the printer use per minute?

Show answer & explanation

Answer: 5.5

Answer

The printer uses 5.5 grams of filament per minute.

The relationship between the printing time

t

and the remaining mass of the filament

M

is linear. The slope of this line represents the change in filament mass per unit of time. Based on the given coordinates

(20, 850)

and

(50, 685)

, the slope is calculated as

\frac{685 - 850}{50 - 20} = \frac{-165}{30} = -5.5

grams per minute. This means that the remaining mass decreases by

5.5

grams for each additional minute the printer runs, which corresponds to a usage rate of

5.5

grams of filament per minute.

Step-by-Step Solution

1

Identify the data points representing time and remaining filament mass from the context.

The two points are

(20, 850)

and

(50, 685)

.

These points allow us to calculate the rate at which the mass of the filament is changing over time.

2

Calculate the change in remaining mass and the change in printing time.

Change in mass is

685 - 850 = -165

grams. Change in time is

50 - 20 = 30

minutes.

Finding the differences in the dependent variable (mass) and independent variable (time) is the standard method to calculate a rate of change.

3

Calculate the rate of filament usage per minute.

\frac{-165\text{ grams}}{30\text{ minutes}} = -5.5

grams per minute. The rate of usage is the positive magnitude, which is

5.5

grams per minute.

The slope of the linear relationship is negative because the mass is decreasing, but the rate of consumption/usage is represented as a positive quantity.

Key Concept

Interpreting the slope of a linear function in context as a rate of change.

Question 1615Question →

An agricultural drone is used to spray fertilizer on a crop field. The volume of fertilizer remaining in the drone's tank, $F$ , in liters, after the drone has been spraying for $t$ minutes is modeled by the equation $F = 240 - 3.2t$ , where $0 \leq t \leq 75$ . What is the best interpretation of the number $3.2$ in this context?

The initial volume of fertilizer, in liters, in the tank before spraying begins.

The number of minutes required for the drone to spray

1

liter of fertilizer.

The volume of fertilizer, in liters, sprayed from the tank each minute.

The total number of minutes the drone can spray before the tank is completely empty.

Show answer & explanation

Answer: The volume of fertilizer, in liters, sprayed from the tank each minute.

Answer

The volume of fertilizer, in liters, sprayed from the tank each minute.

In the linear relationship

F = 240 - 3.2t

, the variable

t

represents time in minutes, and

F

represents the remaining volume in liters. The coefficient of

t

, which is

-3.2

, represents the rate of change of the volume of fertilizer in the tank. A slope of

-3.2

means that the volume decreases by

3.2

liters for every additional minute of spraying. Therefore, the number

3.2

represents the volume of fertilizer, in liters, that is sprayed from the tank each minute.

Step-by-Step Solution

1

Identify the structure of the linear equation.

The equation is of the form

y = mx + b

, or in this case,

F = -3.2t + 240

, where the slope is

-3.2

and the

F

-intercept (initial value) is

240

.

Analyzing the components of a linear function helps determine what each constant represents in context.

2

Determine the meaning of the slope in context.

The slope is the coefficient of the independent variable

t

(time in minutes). The value

-3.2

indicates that for each increase of

1

minute in spraying time, the remaining volume of fertilizer,

F

, decreases by

3.2

liters.

Slope represents the rate of change of the dependent variable relative to the independent variable.

3

Interpret the absolute value of the slope.

The value

3.2

represents the magnitude of this rate, which is the amount of fertilizer sprayed out of the tank per minute (

3.2

liters per minute).

Matching the mathematical rate of change to the corresponding verbal description provides the correct interpretation.

Key Concept

Slope in a linear model represents the constant rate of change of the dependent variable per unit increase of the independent variable.

Estimated Time:1m 15s

Question 1616Question →

The population of a colony of bacteria doubles every $15$ minutes. The population of the colony, $P(t)$ , after $t$ hours can be modeled by the function $P(t) = P_0 \cdot 8^{kt}$ , where $P_0$ is the initial population of the colony and $k$ is a constant. What is the value of $k$ ?

\frac{1}{12}

\frac{3}{4}

\frac{4}{3}

12

Show answer & explanation

Answer: $\frac{4}{3}$

Answer

The correct value of

k

is

\frac{4}{3}

.

Since the population doubles every

15

minutes, it doubles

4

times per hour. Therefore, the population after

t

hours is

P_0 \cdot 2^{4t}

. To express this model in the form

P_0 \cdot 8^{kt}

, we equate the growth factors:

2^{4t} = 8^{kt}

. Since

8 = 2^3

, this can be rewritten as

2^{4t} = (2^3)^{kt} = 2^{3kt}

. Setting the exponents equal gives

4t = 3kt

, which simplifies to

4 = 3k

and yields

k = \frac{4}{3}

.

Step-by-Step Solution

1

Determine the number of doubling periods per hour.

The population doubles

4

times per hour.

Since there are

60

minutes in an hour and the population doubles every

15

minutes, the number of doubling periods in one hour is

\frac{60}{15} = 4

.

2

Formulate the population growth equation in terms of base 2.

P(t) = P_0 \cdot 2^{4t}

If the population doubles

4

times per hour, then after

t

hours, it has doubled

4t

times, representing a growth factor of

2^{4t}

.

3

Equate the growth model to the given function and express using a common base.

2^{4t} = (2^3)^{kt} \implies 2^{4t} = 2^{3kt}

Set the two expressions for the population equal:

P_0 \cdot 2^{4t} = P_0 \cdot 8^{kt}

. Dividing by

P_0

gives

2^{4t} = 8^{kt}

. Since

8 = 2^3

, rewrite the right side to have a base of

2

.

4

Solve for the constant

k

by equating the exponents.

4t = 3kt \implies 4 = 3k \implies k = \frac{4}{3}

Since the bases on both sides of the equation are equal, their exponents must also be equal. Divide both sides by

t

(since

t > 0

) and isolate

k

.

Key Concept

Converting bases in exponential functions and adjusting exponents for different units of time.

Question 1617Question →

A commercial drone delivery service models the total power remaining in a drone's battery, $P(w)$ , as a percentage, after carrying a payload of weight $w$ kilograms for a fixed delivery distance of 10 kilometers. The relationship is modeled by the linear function $P(w) = -8.5w + 92$ . Which of the following is the best interpretation of the value 92 in this context?

The decrease in the remaining battery power, as a percentage, for each increase of 1 kilogram in payload weight.

The remaining battery power, as a percentage, after the drone carries a payload of 1 kilogram for the delivery distance.

The remaining battery power, as a percentage, after the drone carries a payload of 0 kilograms for the delivery distance.

The payload weight, in kilograms, for which the remaining battery power after the delivery is 0 percent.

Show answer & explanation

Answer: The remaining battery power, as a percentage, after the drone carries a payload of 0 kilograms for the delivery distance.

Answer

The remaining battery power, as a percentage, after the drone carries a payload of 0 kilograms for the delivery distance.

The linear equation is written in the slope-intercept form

P(w) = mw + b

, where

b = 92

is the vertical intercept (y-intercept). This intercept represents the value of the dependent variable,

P(w)

, when the independent variable,

w

, is equal to

0

. In this context,

w

is the payload weight in kilograms and

P(w)

is the remaining battery power as a percentage. Thus, the value

92

represents the remaining battery power, as a percentage, when the drone carries a payload of

0

kilograms.

Step-by-Step Solution

1

Identify the structure of the linear function and the component being interpreted.

The linear equation is given in slope-intercept form,

P(w) = mw + b

, where

m = -8.5

is the slope and

b = 92

is the

P(w)

-intercept (y-intercept).

Understanding the components of a linear function allows us to relate them to their contextual definitions.

2

Evaluate the value of the function at the

P(w)

-intercept where the independent variable is zero.

Substitute

w = 0

into the equation:

P(0) = -8.5(0) + 92 = 92

.

The y-intercept represents the value of the dependent variable when the independent variable is equal to 0.

3

Interpret the meaning of

w = 0

and

P(0) = 92

in the given context.

Since

w

represents the payload weight in kilograms and

P(w)

represents the remaining battery power as a percentage,

w = 0

corresponds to a payload of 0 kilograms, and

P(0) = 92

corresponds to a remaining battery power of 92%.

Applying the units and definitions of the variables completes the contextual interpretation of the constant.

Key Concept

Interpreting the y-intercept of a linear function in a real-world context

Question 1618Question →

If $5^{a + 2} = 25^{b}$ and $a + b = 10$ , what is the value of $b$ ?

Show answer & explanation

Answer: 4

Answer

The correct answer is 4.

The correct answer is 4. By rewriting

25^b

as

(5^2)^b = 5^{2b}

, we can set the exponents equal to get

a + 2 = 2b

. Substituting

a = 10 - b

from the second equation into this relation yields

(10 - b) + 2 = 2b

. Simplifying this gives

12 = 3b

, which results in

b = 4

.

Step-by-Step Solution

1

Rewrite the exponential equation using a common base.

5^{a + 2} = 5^{2b}

Since

25 = 5^2

, we write

25^b

as

(5^2)^b = 5^{2b}

to match the base on the left side.

2

Equate the exponents from the base 5 terms.

a + 2 = 2b

When bases are equal, their exponents must be equal for the equation to hold.

3

Express

a

in terms of

b

using the linear equation

a + b = 10

.

a = 10 - b

This allows for substitution into the equation from Step 2 to solve for a single variable.

4

Substitute

a = 10 - b

into

a + 2 = 2b

and solve for

b

.

b = 4

Substituting gives

(10 - b) + 2 = 2b \implies 12 - b = 2b \implies 12 = 3b \implies b = 4

.

Key Concept

Solving systems of equations involving exponential equations by finding a common base.

Question 1619Question →

In the $xy$ -plane, the graph of the linear equation $y - 4 = m(x + 2)$ contains the point $(6, 8)$ , where $m$ is a constant. What is the value of the $x$ -intercept of the line?

-10

-6

-4

5

Show answer & explanation

Answer: -10

Answer

-10

The correct answer is

-10

. By substituting the point

(6, 8)

into the equation

y - 4 = m(x + 2)

, we obtain

8 - 4 = m(6 + 2)

, which simplifies to

4 = 8m

, giving

m = 0.5

. Using this value of

m

, the equation of the line is

y - 4 = 0.5(x + 2)

. The

x

-intercept is the value of

x

when

y = 0

. Substituting

0

for

y

gives

-4 = 0.5(x + 2)

, which simplifies to

-8 = x + 2

. Solving for

x

gives

x = -10

.

Step-by-Step Solution

1

Substitute the given point

(6, 8)

into the equation to find the value of the constant

m

.

8 - 4 = m(6 + 2) \Rightarrow 4 = 8m \Rightarrow m = 0.5

Since the line passes through the point

(6, 8)

, these coordinates must satisfy the equation.

2

Write the completed equation of the line using the value of

m

.

y - 4 = 0.5(x + 2)

Substituting

m = 0.5

back into the original point-slope equation gives the specific line.

3

Find the

x

-intercept by setting

y = 0

and solving for

x

.

0 - 4 = 0.5(x + 2) \Rightarrow -4 = 0.5x + 1 \Rightarrow -5 = 0.5x \Rightarrow x = -10

The

x

-intercept of a graph is the point where the graph crosses the

x

-axis, which occurs where

y = 0

.

Key Concept

Finding the intercepts and slope of a linear equation in two variables

Question 1620Question →

A scientist is measuring the temperature of a chemical compound during an experiment. The temperature of the compound decreases at a constant rate. At $t = 4$ minutes after the experiment begins, the temperature of the compound is $72^\circ\text{C}$ . At $t = 12$ minutes, the temperature of the compound is $48^\circ\text{C}$ . Which of the following functions $T$ models the temperature, in degrees Celsius, of the compound $t$ minutes after the experiment begins, where $0 \leq t \leq 25$ ?

T(t) = -3t + 84

T(t) = -3t + 72

T(t) = -\frac{1}{3}t + \frac{220}{3}

T(t) = -3t + 60

Show answer & explanation

Answer: $T(t) = -3t + 84$

Answer

T(t) = -3t + 84

To find the linear model

T(t) = mt + b

, we first calculate the slope

m

using the given points

(4, 72)

and

(12, 48)

. The slope is the change in temperature divided by the change in time:

m = \frac{48 - 72}{12 - 4} = -3

. Next, we find the y-intercept

b

by substituting one of the points into the equation

T(t) = mt + b

. Substituting

t = 4

and

T(4) = 72

yields

72 = -3(4) + b

, which simplifies to

72 = -12 + b

. Adding

12

to both sides gives

b = 84

. Therefore, the correct linear function modeling the temperature is

T(t) = -3t + 84

.

Step-by-Step Solution

1

Calculate the rate of temperature change (slope) per minute.

The slope

m

is

-3

.

Since the temperature decreases at a constant rate, the relationship is linear. The slope

m

is the change in temperature divided by the change in time:

m = \frac{48 - 72}{12 - 4} = \frac{-24}{8} = -3

.

2

Find the initial temperature (y-intercept) at

t = 0

.

The y-intercept

b

is

84

.

Substitute the point

(4, 72)

and the slope

m = -3

into the slope-intercept equation

T(t) = mt + b

. This gives

72 = -3(4) + b

, which simplifies to

72 = -12 + b

. Adding

12

to both sides yields

b = 84

.

3

Write the final equation.

T(t) = -3t + 84

Combine the slope of

-3

and the y-intercept of

84

to complete the linear model equation.

Key Concept

Writing and interpreting linear functions from two points

Alternative Method

Instead of deriving the function algebraically, substitute the given values of

t = 4

and

t = 12

into each of the given options. Only the correct option yields the corresponding temperatures of 72 and 48 degrees Celsius, respectively.

Estimated Time:1m 30s

Answer

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