All practice questions

Question 1621Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = a(x-3)(x-11)$ , where $a$ is a constant, has a vertex at $(h, k)$ . The graph of the quadratic function $g(x) = -(x-h)^2 + q$ has its vertex at $(h, k+8)$ . If the graphs of $f$ and $g$ intersect at a point with an $x$ -coordinate of $5$ , what is the value of $a$ ?

Show answer & explanation

Answer: 1

Answer

1

The value of

a

is 1. By finding the vertex of

f(x)

at

(7, -16a)

and using the vertex of

g(x)

at

(7, -16a+8)

, we evaluate both functions at

x=5

. Setting

f(5) = g(5)

yields the linear equation

-12a = -16a + 4

, which simplifies to

a = 1

.

Step-by-Step Solution

1

Find the x-coordinate of the vertex of

f(x)

h = 7

The x-coordinate of the vertex of a quadratic function of the form

f(x) = a(x-r_1)(x-r_2)

is the midpoint of its x-intercepts,

r_1 = 3

and

r_2 = 11

.

2

Find the y-coordinate of the vertex of

f(x)

in terms of

a

k = -16a

Evaluate the function at the vertex x-coordinate:

f(7) = a(7-3)(7-11) = -16a

.

3

Write the equation of

g(x)

using the given vertex information

g(x) = -(x-7)^2 + k+8

The vertex of

g(x)

is given as

(h, k+8)

. Since

h = 7

, the vertex form of

g(x)

is

g(x) = -(x-7)^2 + k+8

.

4

Set up the equation for the intersection point at

x = 5

f(5) = g(5)

Since the graphs of

f

and

g

intersect at

x=5

, their y-values must be equal at this point.

5

Substitute expressions and solve for

a

a = 1

Evaluate both sides:

f(5) = a(5-3)(5-11) = -12a

, and

g(5) = -(5-7)^2 + k+8 = -4 + k+8 = k+4

. Substituting

k = -16a

gives

-12a = -16a + 4 \implies 4a = 4 \implies a = 1

.

Key Concept

Quadratic Functions and Graphs

Question 1622Question →

Consider the system of linear equations below:

\begin{aligned} 3x + 4y &= 8 \\ ax + 8y &= 15 \end{aligned}

If the system of equations has no solution, and $a$ is a constant, what is the value of $a$ ?

Show answer & explanation

Answer: 6

Answer

The correct answer is 6.

For a system of linear equations to have no solution, the lines representing the equations must be parallel, which requires them to have the same slope but different y-intercepts. Writing

3x + 4y = 8

in slope-intercept form gives

y = -\frac{3}{4}x + 2

, so its slope is

-\frac{3}{4}

. Writing

ax + 8y = 15

in slope-intercept form gives

y = -\frac{a}{8}x + \frac{15}{8}

, so its slope is

-\frac{a}{8}

. Equating the slopes gives

-\frac{3}{4} = -\frac{a}{8}

. Solving for

a

gives

a = 6

. Since the y-intercepts (

2

and

\frac{15}{8}

) are different, the lines are parallel and have no intersection points.

Step-by-Step Solution

1

Convert the first equation

3x + 4y = 8

to slope-intercept form.

y = -\frac{3}{4}x + 2

To find the slope of the first line, which is

-\frac{3}{4}

.

2

Convert the second equation

ax + 8y = 15

to slope-intercept form.

y = -\frac{a}{8}x + \frac{15}{8}

To find the slope of the second line in terms of

a

, which is

-\frac{a}{8}

.

3

Equate the slopes of the two lines.

-\frac{3}{4} = -\frac{a}{8}

Parallel lines have the same slope, and a system with parallel lines has no solution.

4

Solve the equation for

a

.

a = 6

Isolating the variable

a

yields the value of the constant.

Key Concept

A system of linear equations has no solution if the lines represented by the equations are parallel, meaning they have the same slope but different y-intercepts.

Question 1623Question →

The table below shows some values for the linear function $f$ .

$x$	$f(x)$
$2$	$5$
$5$	$14$
$8$	$23$

If the function $g$ is defined by $g(x) = f(x) + k$ , where $k$ is a constant, and the $x$ -intercept of the graph of $y = g(x)$ in the $xy$ -plane is $-2$ , what is the value of $k$ ?

Show answer & explanation

Answer: 7

Answer

The value of

k

is

7

.

To find the value of

k

, we first determine the linear function

f(x)

using the values in the table. The slope of

f

is calculated as

\frac{14 - 5}{5 - 2} = 3

. Substituting the point

(2, 5)

into the slope-intercept form

f(x) = 3x + b

gives

5 = 3(2) + b

, which simplifies to

b = -1

. Therefore,

f(x) = 3x - 1

. The function

g

is defined as

g(x) = f(x) + k = 3x - 1 + k

. Since the

x

-intercept of the graph of

y = g(x)

is

-2

, we must have

g(-2) = 0

. Substituting

x = -2

into the equation for

g(x)

gives

3(-2) - 1 + k = 0

, which simplifies to

-7 + k = 0

. Solving for

k

yields

k = 7

.

Step-by-Step Solution

1

Determine the slope of the linear function

f

.

The slope of

f

is

3

.

Using the points

(2, 5)

and

(5, 14)

from the table, the slope is calculated as

m = \frac{14 - 5}{5 - 2} = \frac{9}{3} = 3

.

2

Find the equation for

f(x)

.

f(x) = 3x - 1

Using the slope-intercept form

f(x) = 3x + b

and the point

(2, 5)

, we solve for

b

:

5 = 3(2) + b

, which gives

b = -1

.

3

Set up the equation for

g(x)

and use the

x

-intercept to find the value of

k

.

k = 7

Since

g(x) = f(x) + k

, we have

g(x) = 3x - 1 + k

. The

x

-intercept of

g(x)

is

-2

, so

g(-2) = 0

. This gives

3(-2) - 1 + k = 0

, which simplifies to

-7 + k = 0

, meaning

k = 7

.

Key Concept

Finding the equation of a linear function from tabular data and applying translations and intercepts.

Question 1624Question →

The population of a certain species of wildflower in a national park increases by $300\%$ every $8$ years. The population can be modeled by the function $P(t) = P_0(1 + r)^t$ , where $P_0$ is the initial population, $r$ is the constant annual growth rate, and $t$ is the time in years. Which of the following is closest to the value of $r$ ?

0.15

0.19

0.38

1.19

Show answer & explanation

Answer: 0.19

Answer

The correct value of the annual growth rate is

0.19

.

An increase of

300\%

means the population becomes

100\% + 300\% = 400\%

of its initial value, which corresponds to a growth factor of

4

over an

8

-year period. The annual growth rate

r

satisfies the equation

(1 + r)^8 = 4

. Solving for

1 + r

gives

1 + r = 4^{1/8} = (2^2)^{1/8} = 2^{1/4} \approx 1.1892

. Subtracting

1

from both sides yields the annual growth rate

r \approx 0.1892

, which is closest to

0.19

.

Step-by-Step Solution

1

Determine the total growth factor over the 8-year period from the percentage increase.

The population increases by

300\%

, meaning the new population is

100\% + 300\% = 400\%

of the initial population. This corresponds to a growth factor of

4

every

8

years.

An increase of

300\%

adds

3

times the original value to the original value, resulting in a total of

4

times the original value.

2

Set up the equation relating the annual growth rate

r

to the 8-year growth factor.

(1+r)^8 = 4

Since the population is multiplied by

1+r

each year, after

8

years it is multiplied by

(1+r)^8

, which must equal the total growth factor of

4

.

3

Solve the equation for

r

and round to the nearest hundredth.

1+r = 4^{1/8} \approx 1.1892 \implies r \approx 0.1892 \approx 0.19

Taking the eighth root of both sides gives the annual growth factor

1+r \approx 1.19

. Subtracting

1

isolates the annual growth rate

r

.

Key Concept

Relating exponential growth rates and growth factors across different time periods

Alternative Method

Instead of solving

(1+r)^8 = 4

, we can write the function directly using the 8-year growth factor as

P(t) = P_0 \cdot 4^{t/8}

. Rewriting this in the form

P(t) = P_0(4^{1/8})^t

shows that the annual growth factor is

4^{1/8}

. We can approximate

4^{1/8} \approx 1.1892

, which corresponds to an annual growth rate

r = 1.1892 - 1 = 0.1892

, or approximately

0.19

.

Estimated Time:1m 30s

Question 1625Question →

If $4^{2x + 1} = 8^{x - 1}$ , what is the value of $x$ ?

-5

-2

-1

5

Show answer & explanation

Answer: -5

Answer

-5

The correct answer is

-5

. By converting the bases of both sides to

2

, the equation becomes

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

. Equating the exponents gives

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

, which expands to

4x+2 = 3x-3

. Solving for

x

yields

x = -5

.

Step-by-Step Solution

1

Express both bases as powers of 2

4 = 2^2

and

8 = 2^3

To solve an exponential equation, we need to rewrite the terms with a common base.

2

Substitute the bases back into the equation and apply exponent power rules

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

Applying the power of a power property,

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

, allows us to simplify the exponents.

3

Set the exponents equal to each other

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

Since the bases are identical and positive, their exponents must be equal.

4

Distribute and solve the linear equation for

x

4x + 2 = 3x - 3 \Rightarrow x = -5

Distributing the constants gives

4x + 2 = 3x - 3

. Subtracting

3x

from both sides gives

x + 2 = -3

, and subtracting

2

from both sides isolates the variable to find

x = -5

.

Key Concept

Solving exponential equations by expressing both sides with a common base

Question 1626Question →

First developed in the classical world, the astrolabe was a sophisticated analog calculator used by astronomers and navigators to solve problems relating to time and the position of celestial bodies. In the medieval Islamic world, scholars made significant advancements to the instrument's design ______ they added specialized plates that allowed users to determine the precise times of daily prayers and align themselves toward Mecca.

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

design, they

design: they

design; which they

design they

Show answer & explanation

Answer: design: they

Answer

design: they

The correct option correctly uses a colon to connect two independent clauses. The clause preceding the colon is independent, and the clause following it is also independent and directly explains the advancements made to the design. This meets the grammatical criteria for colon usage.

Step-by-Step Solution

1

Analyze the clause before the blank.

The clause 'In the medieval Islamic world, scholars made significant advancements to the instrument's design' is independent because it contains a subject ('scholars') and a verb ('made') and can stand alone as a complete sentence.

Identifying the grammatical nature of the first clause helps determine what punctuation can follow it.

2

Analyze the clause after the blank.

The clause 'they added specialized plates that allowed users to determine the precise times of daily prayers and align themselves toward Mecca' is also independent because it contains a subject ('they') and a verb ('added') and can stand alone.

Knowing that both clauses are independent narrows down the correct punctuation options to those that can join independent clauses.

3

Determine the logical relationship between the two clauses.

The second clause explains and elaborates on the specific 'significant advancements' mentioned in the first clause.

A colon is appropriate here because it introduces an explanation or elaboration of the preceding independent clause.

Key Concept

Colons are used to join two independent clauses when the second clause explains, illustrates, or details the first.

Question 1627Question →

The quadratic equation $x^2 - (k + 3)x + (2k + 3) = 0$ has exactly one real solution, where $k$ is a constant. If $k > 1$ , what is the value of $k$ ?

-1

1

3

5

Show answer & explanation

Answer: 3

Answer

The value of

k

is

3

.

For the quadratic equation to have exactly one real solution, its discriminant must be equal to

0

. Setting the discriminant of

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

to

0

yields

[-(k + 3)]^2 - 4(1)(2k + 3) = 0

. Simplifying this yields

k^2 - 2k - 3 = 0

, which factors as

(k - 3)(k + 1) = 0

. This gives two potential values for

k

:

3

and

-1

. Because the problem specifies that

k > 1

, the value of

k

must be

3

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation and state the condition for having exactly one real solution.

The coefficients of the quadratic equation

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

are

a = 1

,

b = -(k + 3)

, and

c = 2k + 3

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

b^2 - 4ac

must equal

0

.

A discriminant of zero indicates that the quadratic equation has two equal real roots, which is equivalent to exactly one distinct real solution.

2

Set up the discriminant equation in terms of

k

and simplify it.

[-(k + 3)]^2 - 4(1)(2k + 3) = 0

(k^2 + 6k + 9) - (8k + 12) = 0

k^2 - 2k - 3 = 0

This algebraic expansion and simplification transforms the discriminant condition into a solvable quadratic equation for

k

.

3

Solve the quadratic equation

k^2 - 2k - 3 = 0

by factoring.

(k - 3)(k + 1) = 0

, which gives

k = 3

or

k = -1

.

Factoring is the most direct method to find the roots of this quadratic equation.

4

Apply the constraint

k > 1

to select the correct value of

k

.

Since

k

must be greater than

1

, the value

k = -1

is discarded, leaving

k = 3

.

The solution must satisfy all given constraints in the problem statement.

Key Concept

Discriminant of a Quadratic Equation

Alternative Method

Instead of solving the quadratic equation algebraically, you can test the given choices for

k

. For the correct option of

3

, substituting

k = 3

into the original equation yields

x^2 - 6x + 9 = 0

. Factoring this expression gives

(x - 3)^2 = 0

, which clearly has exactly one real solution (

x = 3

). Testing the other options would not produce a perfect square trinomial.

Estimated Time:2m 0s

Question 1628Question →

For the 1900 Paris Exposition, sociologist W.E.B. Du Bois and his research team created dozens of vibrant, hand-drawn charts to document the progress of Black Americans after the Civil War. The ________ meticulous handiwork, which was presented in a dedicated exhibition, challenged prevailing racist theories of the era by showcasing empirical evidence of economic, social, and educational advancement.

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

researchers'

researchers

researcher's

researcher

Show answer & explanation

Answer: researchers'

Answer

researchers'

The context established by the mention of a research team indicates that multiple people collaborated on the charts, meaning a plural noun is needed. Furthermore, because these individuals own the 'meticulous handiwork,' a possessive noun is required. The plural possessive form is 'researchers'.

Step-by-Step Solution

1

Determine if the noun should be singular or plural.

Plural

The context mentions W.E.B. Du Bois and his research team, indicating multiple members contributed to the work.

2

Determine if the noun should be possessive or non-possessive.

Possessive

The noun modifies the noun phrase 'meticulous handiwork,' showing ownership of the handiwork by the team members.

3

Select the correct plural possessive spelling.

researchers'

The plural possessive of 'researcher' is formed by adding an apostrophe after the final 's' of the plural noun.

Key Concept

Plural and Possessive Nouns and Pronouns

Question 1629Question →

Consider the system of equations below:

\begin{aligned} y &= 2x^2 + kx + 9 \\ y &= -2x + 1 \end{aligned}

In the system of equations above, $k$ is a positive constant. If the system has exactly one real solution, what is the value of $k$ ?

Show answer & explanation

Answer: 6

Answer

The value of the positive constant

k

is 6.

By substituting the linear equation into the quadratic equation, we obtain the equation

2x^2 + (k+2)x + 8 = 0

. For the system to have exactly one real solution, the discriminant of this quadratic equation,

\Delta = b^2 - 4ac

, must be equal to zero. Substituting

a = 2

,

b = k+2

, and

c = 8

into the discriminant formula gives

(k+2)^2 - 4(2)(8) = 0

, which simplifies to

(k+2)^2 = 64

. Solving for

k

gives

k+2 = 8

or

k+2 = -8

, resulting in

k = 6

or

k = -10

. Since

k

is specified to be a positive constant, the value of

k

must be 6.

Step-by-Step Solution

1

Substitute the expression for

y

from the linear equation into the quadratic equation.

2x^2 + kx + 9 = -2x + 1

Setting the two expressions for

y

equal to each other allows us to solve for the

x

-coordinates of the intersection points.

2

Rearrange the equation into the standard quadratic form

ax^2 + bx + c = 0

.

2x^2 + (k+2)x + 8 = 0

Grouping the terms by power of

x

lets us identify the coefficients

a = 2

,

b = k+2

, and

c = 8

.

3

Set the discriminant of the quadratic equation to zero.

(k+2)^2 - 4(2)(8) = 0

, which simplifies to

(k+2)^2 - 64 = 0

A quadratic equation has exactly one real solution if and only if its discriminant,

b^2 - 4ac

, is equal to zero.

4

Solve for the positive constant

k

.

k = 6

Solving

(k+2)^2 = 64

yields

k+2 = 8

or

k+2 = -8

, which means

k = 6

or

k = -10

. Since

k

must be a positive constant, we select 6.

Key Concept

Determining the number of solutions of a quadratic-linear system using the discriminant.

Alternative Method

Alternatively, the quadratic equation

2x^2 + (k+2)x + 8 = 0

has exactly one real solution when it is a perfect square trinomial. Factoring out the leading coefficient gives

2(x^2 + \frac{k+2}{2}x + 4) = 0

. For the term inside the parentheses to be a perfect square trinomial of the form

x^2 + 2dx + d^2

, we must have

d^2 = 4

, which means

d = \pm 2

. Thus, the middle coefficient

\frac{k+2}{2}

must equal

2d = \pm 4

. Since

k

is positive, we set

\frac{k+2}{2} = 4

, which solves to

k = 6

.

Estimated Time:1m 30s

Question 1630Question →

What is the solution to the equation $\sqrt{3x + 13} = x + 3$ ?

1

-4

1 and -4

4

Show answer & explanation

Answer: 1

Answer

1

The value

1

is the correct answer because substituting it back into the original equation results in a true mathematical statement:

\sqrt{3(1) + 13} = 1 + 3

, which simplifies to

4 = 4

.

Step-by-Step Solution

1

Square both sides of the equation to eliminate the radical.

3x + 13 = (x + 3)^2

Squaring both sides is the standard method to solve equations containing square roots.

2

Expand the right side of the equation.

3x + 13 = x^2 + 6x + 9

Expanding the squared binomial gives a quadratic expression.

3

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

x^2 + 3x - 4 = 0

Setting the quadratic equation to zero allows us to solve for

x

by factoring.

4

Factor the quadratic equation.

(x + 4)(x - 1) = 0

Finding two numbers that multiply to

-4

and add to

3

gives the binomial factors.

5

Solve for the potential values of

x

.

x = -4

or

x = 1

Setting each factor equal to zero yields the potential solutions.

6

Substitute each potential solution back into the original equation to check for extraneous solutions.

For

x = 1

:

\sqrt{3(1) + 13} = 1 + 3 \Rightarrow 4 = 4

(valid). For

x = -4

:

\sqrt{3(-4) + 13} = -4 + 3 \Rightarrow 1 = -1

(invalid). Thus, the only solution is

1

.

Squaring both sides can introduce extraneous solutions, so checking all potential solutions in the original equation is necessary to verify their validity.

Key Concept

Solving radical equations by squaring both sides and checking for extraneous solutions.

Question 1631Question →

In the mid-twentieth century, archaeologists restoring the frescoes of Pompeii faced a significant dilemma: the paraffin wax coating applied by early conservators to preserve the paintings had instead darkened over time. This obscured the vibrant pigments underneath _______ conservators had to develop novel chemical solvents that could selectively dissolve the wax without damaging the ancient plaster.

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

, consequently,

; consequently,

, but

consequently

Show answer & explanation

Answer: ; consequently,

Answer

; consequently,

The correct choice uses a semicolon to separate two independent clauses, followed by the conjunctive adverb 'consequently' and a comma to logically connect the cause (obscured pigments) and the effect (the need for new solvents).

Step-by-Step Solution

1

Identify the clause boundary in the sentence.

The sentence contains two independent clauses: 'This obscured the vibrant pigments underneath' and 'conservators had to develop novel chemical solvents...'.

Locating the boundary helps determine the type of punctuation or conjunction needed to connect them grammatically.

2

Evaluate the relationship between the two clauses.

The second clause describes the result or consequence of the first clause (the obscured pigments led to the development of new solvents).

This determines the appropriate transitional word to use.

3

Select the option that conforms to standard English conventions for linking two independent clauses.

The option containing a semicolon followed by 'consequently' and a comma is the correct choice, as it avoids run-on or comma splice errors while conveying the correct cause-and-effect relationship.

A semicolon is required to link two independent clauses when not using a coordinating conjunction (like 'and' or 'so').

Key Concept

Linking independent clauses using appropriate punctuation and conjunctive adverbs.

Estimated Time:1m 0s

Question 1632Question →

The function $f$ is defined by $f(x) = -(x - 5)^2 + 4$ . What is the maximum value of the function $f$ ?

-4

4

5

-5

Show answer & explanation

Answer: $4$

Answer

The maximum value of the function is 4.

The quadratic function

f(x) = -(x - 5)^2 + 4

is in vertex form

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

, where the vertex of the parabola is

(h, k) = (5, 4)

. Because the leading coefficient

a = -1

is negative, the parabola opens downward, which means the vertex is the highest point on the graph. The maximum value of the function is the y-coordinate of this vertex, which is 4.

Step-by-Step Solution

1

Identify the form of the quadratic function.

The function

f(x) = -(x - 5)^2 + 4

is written in vertex form,

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

.

Recognizing the vertex form allows us to directly identify the vertex coordinates of the parabola.

2

Identify the vertex

(h, k)

and the direction the parabola opens.

The vertex is

(5, 4)

and the leading coefficient is

a = -1

.

Since the leading coefficient

a

is negative, the parabola opens downward, meaning the vertex is the highest point on the graph.

3

Determine the maximum value of the function.

The maximum value of the function is the y-coordinate of the vertex, which is 4.

The maximum value of a function refers to its largest output (y-value), which corresponds to the y-coordinate of its highest point.

Key Concept

Quadratic Functions and Graphs

Question 1633Question →

The temperature, $T$ , in degrees Fahrenheit ( $^\circ\text{F}$ ), of a laboratory incubator $m$ minutes after a cooling cycle begins is modeled by the equation $T = 95 - 0.8(2m - 5)$ . For a specific experiment, the incubator temperature must be at most $75^\circ\text{F}$ . Which of the following inequalities represents all possible values of $m$ for which the incubator temperature meets this requirement?

m \geq 15

m \leq 15

m \geq 10

m \geq 30

Show answer & explanation

Answer: $m \geq 15$

Answer

The correct inequality is

m \geq 15

.

The correct inequality is

m \geq 15

. To find this, set the temperature model

95 - 0.8(2m - 5)

to be less than or equal to

75

. Subtracting

95

from both sides gives

-0.8(2m - 5) \leq -20

. Dividing both sides by

-0.8

requires flipping the inequality sign, which yields

2m - 5 \geq 25

. Adding

5

to both sides gives

2m \geq 30

, and dividing by

2

yields

m \geq 15

.

Step-by-Step Solution

1

Set up the inequality based on the requirement that the temperature must be at most

75^\circ\text{F}

.

95 - 0.8(2m - 5) \leq 75

The phrase 'at most' corresponds to a less-than-or-equal-to sign.

2

Subtract

95

from both sides of the inequality.

-0.8(2m - 5) \leq -20

This begins the process of isolating the term containing the variable

m

.

3

Divide both sides by

-0.8

and flip the inequality sign.

2m - 5 \geq 25

Dividing both sides of an inequality by a negative number requires reversing the direction of the inequality sign.

4

Add

5

to both sides of the inequality.

2m \geq 30

This isolates the variable term

2m

.

5

Divide both sides by

2

to solve for

m

.

m \geq 15

This yields the solution set for the variable

m

.

Key Concept

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

Question 1634Question →

A researcher monitors the temperature of a chemical compound during an experiment. The temperature, $T$ , in degrees Celsius, of the compound $m$ minutes after heating begins can be modeled by a linear relationship. The temperature increases by $4.5$ degrees Celsius every $2$ minutes. If the temperature of the compound was $-12$ degrees Celsius when heating began, after how many minutes will the temperature of the compound reach $15$ degrees Celsius?

Show answer & explanation

Answer: 12

Answer

The temperature of the compound will reach

15

degrees Celsius after

12

minutes.

The correct answer is

12

because the temperature increases at a constant rate of

2.25

degrees Celsius per minute. Starting from an initial temperature of

-12

degrees Celsius, the equation is

T = 2.25m - 12

. Setting

T = 15

gives

15 = 2.25m - 12

, which simplifies to

27 = 2.25m

, resulting in

m = 12

.

Step-by-Step Solution

1

Calculate the constant rate of temperature increase per minute.

2.25

degrees Celsius per minute

This rate represents the slope of the linear relationship between time and temperature.

2

Formulate the linear equation representing the temperature

T

as a function of the elapsed minutes

m

.

T = 2.25m - 12

The slope is

2.25

and the vertical intercept is the initial temperature of

-12

degrees Celsius.

3

Set

T = 15

and solve the linear equation for

m

.

m = 12

This determines the exact number of minutes needed for the temperature to reach

15

degrees Celsius.

Key Concept

Interpreting and solving linear equations modeled from context

Question 1635Question →

To restore a depleted wetland, conservationists pump water into a basin. The volume of water in the basin, $W$ , in thousands of gallons, $t$ hours after the pumping begins is modeled by the equation $W = 12.5t + 85$ . According to the model, how many hours of pumping are required for the volume of water in the basin to increase by $150$ thousand gallons?

Show answer & explanation

Answer: 12

Answer

12

The linear equation is given in slope-intercept form,

W = mt + b

, where

m = 12.5

is the slope and

b = 85

is the y-intercept. In this context, the slope

m = 12.5

represents the rate at which water is pumped into the basin, which is

12.5

thousand gallons per hour. To find the number of hours required for the volume of water to increase by

150

thousand gallons, divide the total increase by the rate:

\frac{150}{12.5} = 12

hours.

Step-by-Step Solution

1

Identify the rate of water volume increase per hour from the equation.

The rate is

12.5

thousand gallons per hour.

In the linear equation

W = 12.5t + 85

, the coefficient of

t

(the slope) represents the rate of change of the water volume with respect to time.

2

Set up a relation to find the time

t

for a volume increase of

150

thousand gallons.

12.5t = 150

The change in volume is equal to the rate of change multiplied by the time elapsed.

3

Solve the equation for

t

.

t = 12

Dividing both sides of the equation

12.5t = 150

by

12.5

isolates

t

.

Key Concept

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

Estimated Time:1m 30s

Question 1636Question →

In the $xy$ -plane, a system of equations consists of the circle with equation $(x - 7)^2 + (y - 4)^2 = 1$ and the line $y = kx$ , where $k$ is a positive constant. If the system has exactly one real solution and $k > 0.5$ , what is the value of $k$ ?

Show answer & explanation

Answer: 0.75

Answer

The correct answer is 3/4 (or 0.75).

The correct answer is

3/4

(or

0.75

). Substituting the line

y = kx

into the circle equation and setting the discriminant of the resulting quadratic equation to zero yields a quadratic in

k

:

48k^2 - 56k + 15 = 0

. Solving this equation gives two positive tangent slopes:

k = 5/12

and

k = 3/4

. Since the problem specifies that

k > 0.5

, we choose

k = 3/4

, which equals

0.75

. Alternatively, using geometry, the distance from the center

(7,4)

to the line

kx - y = 0

must equal the radius

1

. This gives

\frac{|7k-4|}{\sqrt{k^2+1}} = 1

. Squaring both sides yields

(7k-4)^2 = k^2+1

, which simplifies to

48k^2 - 56k + 15 = 0

, giving the same values of

k

.

Step-by-Step Solution

1

Substitute the linear equation

y = kx

into the circle's quadratic equation

(x - 7)^2 + (y - 4)^2 = 1

.

(x - 7)^2 + (kx - 4)^2 = 1

To find the points of intersection between the line and the circle, substitute the expression for

y

from the linear equation into the circle's equation.

2

Expand the terms and write the equation in the standard quadratic form

Ax^2 + Bx + C = 0

.

(1 + k^2)x^2 - (14 + 8k)x + 64 = 0

Expanding

(x - 7)^2

yields

x^2 - 14x + 49

and expanding

(kx - 4)^2

yields

k^2x^2 - 8kx + 16

. Grouping the terms by powers of

x

and subtracting

1

from both sides gives the standard form.

3

Set the discriminant of the quadratic equation to

0

to satisfy the condition of exactly one real solution.

\Delta = [-(14 + 8k)]^2 - 4(1 + k^2)(64) = 0

A quadratic equation has exactly one real solution if and only if its discriminant,

B^2 - 4AC

, is equal to zero. Geometrically, this means the line is tangent to the circle.

4

Simplify the discriminant equation and solve for

k

.

48k^2 - 56k + 15 = 0

, which factors as

(12k - 5)(4k - 3) = 0

. Thus,

k = \frac{5}{12}

or

k = \frac{3}{4}

.

Simplifying

(14 + 8k)^2 - 256(1 + k^2) = 0

gives

196 + 224k + 64k^2 - 256 - 256k^2 = 0

, which simplifies to

-192k^2 + 224k - 60 = 0

. Dividing the entire equation by

-4

yields the quadratic equation in

k

.

5

Select the value of

k

that satisfies the given condition

k > 0.5

.

k = \frac{3}{4}

(or

0.75

)

Since

\frac{5}{12} \approx 0.417

is less than

0.5

and

\frac{3}{4} = 0.75

is greater than

0.5

, the correct value is

3/4

.

Key Concept

Solving a nonlinear system of equations involving a circle and a line by setting the discriminant of the substituted quadratic equation to zero to find the slope of the tangent lines.

Estimated Time:2m 30s

Question 1637Question →

A circle and a line intersect at two points in the $xy$ -plane. The equations of the circle and the line are:

(x - 3)^2 + y^2 = 25

y = x - 2

If $(x_1, y_1)$ is the intersection point in Quadrant I and $(x_2, y_2)$ is the intersection point in Quadrant III, what is the value of $x_1 + y_2$ ?

3

5

1

10

Show answer & explanation

Answer: 3

Answer

The value of

x_1 + y_2

is 3.

To solve the system, we substitute the linear equation

y = x - 2

into the circle equation, obtaining

(x - 3)^2 + (x - 2)^2 = 25

. Expanding this gives

x^2 - 6x + 9 + x^2 - 4x + 4 = 25

, which simplifies to

2x^2 - 10x - 12 = 0

. Dividing by 2, we get

x^2 - 5x - 6 = 0

, which factors as

(x - 6)(x + 1) = 0

. The solutions are

x = 6

and

x = -1

. Substituting these back into the linear equation gives the points of intersection:

(6, 4)

in Quadrant I (since both coordinates are positive) and

(-1, -3)

in Quadrant III (since both coordinates are negative). Therefore,

x_1 = 6

and

y_2 = -3

, and their sum is

6 + (-3) = 3

. This matches the correct option.

Step-by-Step Solution

1

Substitute the linear equation into the circle equation to obtain an equation in terms of

x

only.

(x - 3)^2 + (x - 2)^2 = 25

This reduces the system to a single quadratic equation in one variable, allowing us to find the x-coordinates of the intersection points.

2

Expand and simplify the quadratic equation into standard form.

x^2 - 6x + 9 + x^2 - 4x + 4 = 25 \implies 2x^2 - 10x - 12 = 0 \implies x^2 - 5x - 6 = 0

Simplifying the equation makes it solvable by standard quadratic factoring methods.

3

Solve the simplified quadratic equation for

x

by factoring.

(x - 6)(x + 1) = 0 \implies x = 6

or

x = -1

The roots of this quadratic equation are the x-coordinates of the two intersection points.

4

Find the corresponding

y

-coordinates by substituting the

x

-values back into the linear equation

y = x - 2

.

For

x = 6

,

y = 6 - 2 = 4

, giving the point

(6, 4)

in Quadrant I. For

x = -1

,

y = -1 - 2 = -3

, giving the point

(-1, -3)

in Quadrant III.

This yields the coordinates of the two points of intersection and identifies which point lies in which quadrant.

5

Identify

x_1

and

y_2

and calculate the sum

x_1 + y_2

.

x_1 = 6

and

y_2 = -3

, so

x_1 + y_2 = 6 + (-3) = 3

.

This answers the specific question using the coordinate components of the two points.

Key Concept

Nonlinear Systems of Equations

Question 1638Question →

In medieval Europe, the master builders of Gothic cathedrals sought to achieve unprecedented heights and incorporate expansive stained-glass windows. To prevent the massive stone vaults from collapsing outward, they developed flying _______ these external arched supports redistributed the lateral thrust of the roof down to heavy masonry piers on the exterior of the building.

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

buttresses,

buttresses;

buttresses

buttresses, but

Show answer & explanation

Answer: buttresses;

Answer

buttresses;

The correct answer is the option containing only a semicolon. The passage contains two independent clauses: the first ends with the development of the buttresses, and the second begins with the explanation of how these supports redistributed the lateral thrust of the roof. A semicolon is the standard punctuation mark used to link two independent clauses without a coordinating conjunction.

Step-by-Step Solution

1

Identify the clause boundaries in the sentence.

The sentence contains two independent clauses: 'they developed flying buttresses' and 'these external arched supports redistributed the lateral thrust of the roof down to heavy masonry piers on the exterior of the building.' Each of these clauses has a subject and a verb and can stand alone as a complete sentence.

Before deciding on punctuation, it is necessary to determine if the clauses being linked are independent or dependent.

2

Evaluate the grammar rules for linking two independent clauses.

Two independent clauses can be joined by a period, a semicolon, a colon (if the second explains the first), or a comma followed by a coordinating conjunction (FANBOYS). They cannot be joined by a comma alone or with no punctuation at all.

This narrows down the correct grammatical structures.

3

Evaluate the choices against the grammatical and logical requirements.

The option containing only a semicolon correctly links the two clauses. The option with a comma creates an ungrammatical comma splice. The option with no punctuation creates a run-on sentence. The option with 'but' is grammatically structured but logically incorrect, as there is no relationship of contrast between the clauses.

This identifies the correct choice and eliminates the distractors.

Key Concept

Clause Boundaries and Linking

Question 1639Question →

If $9^{y - 1} \cdot 3^{2y} = 27^{y + 2}$ , what is the value of $y$ ?

1.5

4

7

8

Show answer & explanation

Answer: 8

Answer

8

To solve the equation, we rewrite each term using the common base of

3

. Since

9 = 3^2

and

27 = 3^3

, the equation

9^{y - 1} \cdot 3^{2y} = 27^{y + 2}

can be written as

(3^2)^{y-1} \cdot 3^{2y} = (3^3)^{y+2}

. Applying the power of a power rule, we get

3^{2(y-1)} \cdot 3^{2y} = 3^{3(y+2)}

, which simplifies to

3^{2y-2} \cdot 3^{2y} = 3^{3y+6}

. Using the product rule of exponents to combine the left side gives

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

. Setting the exponents equal to each other gives the linear equation

4y - 2 = 3y + 6

. Solving for

y

yields

y = 8

.

Step-by-Step Solution

1

Express all bases in the equation as powers of 3.

9^{y-1} = (3^2)^{y-1} = 3^{2y-2}

and

27^{y+2} = (3^3)^{y+2} = 3^{3y+6}

. The original equation becomes

3^{2y-2} \cdot 3^{2y} = 3^{3y+6}

.

Having a common base allows us to combine exponents using the properties of exponents.

2

Simplify the left side of the equation by adding the exponents.

3^{(2y-2) + 2y} = 3^{4y-2}

, so the equation is

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

.

When multiplying exponential expressions with the same base, we add their exponents.

3

Set the exponents equal to each other and solve for

y

.

4y - 2 = 3y + 6 \implies y - 2 = 6 \implies y = 8

.

Since the bases are equal and positive (and not equal to 1), their exponents must be equal.

Key Concept

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

Estimated Time:1m 30s

Question 1640Question →

If $\left(\frac{1}{8}\right)^{2x - 3} = 16^{x - 2}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 1.7

Answer

1.7 (or 17/10)

The correct answer is 1.7 (or 17/10). By writing

\frac{1}{8}

as

2^{-3}

and

16

as

2^4

, the equation becomes

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

. Applying exponent rules yields

2^{-6x + 9} = 2^{4x - 8}

. Equating the exponents gives

-6x + 9 = 4x - 8

, which simplifies to

10x = 17

, resulting in

x = 1.7

.

Step-by-Step Solution

1

Express both bases as powers of

2

.

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

To solve the exponential equation, both sides must have a common base. Both

\frac{1}{8}

and

16

can be rewritten as powers of

2

.

2

Apply the power-of-a-power exponent rule

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

to simplify the exponents.

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

, which simplifies to

2^{-6x + 9} = 2^{4x - 8}

This allows us to write each side as a single exponential term with a base of

2

.

3

Set the exponents equal to each other.

-6x + 9 = 4x - 8

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

4

Solve the linear equation for

x

.

10x = 17

, which gives

x = 1.7

Isolate the variable

x

by algebraic manipulation.

Key Concept

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

Alternative Method

Alternatively, you can apply the definition of logarithms to both sides of the equation. Taking the logarithm base 2 of both sides gives

\log_2\left(\left(\frac{1}{8}\right)^{2x-3}\right) = \log_2\left(16^{x-2}\right)

. Using the power property of logarithms, this simplifies to

(2x - 3)\log_2\left(2^{-3}\right) = (x - 2)\log_2\left(2^4\right)

, which simplifies directly to

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

. Solving this linear equation yields the same result,

x = 1.7

.

Estimated Time:1m 30s

Answer

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