All practice questions

Question 1461Question →

A water utility company charges a flat monthly connection fee plus a constant rate per gallon of water consumed. In a certain month, a household that consumed $2,800$ gallons of water was billed a total of $62.00$ dollars, and a household that consumed $4,500$ gallons of water was billed a total of $87.50$ dollars. What is the flat monthly connection fee, in dollars, charged by the water utility company?

Show answer & explanation

Answer: 20

Answer

The flat monthly connection fee charged by the water utility company is 20 dollars.

The flat monthly connection fee is 20 dollars. A linear relationship can be modeled by the equation

y = mx + b

, where

x

represents the water consumed in gallons,

y

represents the total bill in dollars,

m

represents the rate per gallon, and

b

represents the flat monthly connection fee. Using the data for the two households, we get the points

(2800, 62)

and

(4500, 87.5)

. The rate per gallon is the slope of the line passing through these points:

m = \frac{87.5 - 62}{4500 - 2800} = \frac{25.5}{1700} = 0.015

dollars per gallon. Substituting this slope and the point

(2800, 62)

into the equation gives

62 = 0.015(2800) + b

, which simplifies to

62 = 42 + b

. Solving for

b

yields

20

dollars.

Step-by-Step Solution

1

Define the linear relationship using the variables

x

for water consumed in gallons and

y

for the total bill in dollars.

The relationship can be written as

y = mx + b

, where

m

represents the cost per gallon of water and

b

represents the flat monthly connection fee.

This establishes the linear framework needed to solve for the unknown constants using the provided data points.

2

Calculate the constant rate of change (slope

m

) using the coordinates of the two households:

(2800, 62)

and

(4500, 87.5)

.

m = \frac{87.5 - 62}{4500 - 2800} = \frac{25.5}{1700} = 0.015

dollars per gallon.

The slope of a linear function represents the constant rate of change between the two variables.

3

Substitute the calculated slope

m = 0.015

and the point

(2800, 62)

into the equation

y = mx + b

to find the flat fee

b

.

62 = 0.015(2800) + b \implies 62 = 42 + b \implies b = 20

.

The flat monthly connection fee corresponds to the vertical intercept (

y

-intercept) of the linear equation.

Key Concept

Linear Equations in Two Variables

Question 1462Question →

For all $x > 0$ and $x \neq 1$ , the expression $\frac{x^{5/2} - x^{3/2} - x + 1}{x^{3/2} - x - x^{1/2} + 1}$ is equivalent to $x + x^a + b$ , where $a$ and $b$ are constants and $a < 1$ . What is the value of $2a + b$ ?

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

Substituting

u = x^{1/2}

transforms the expression into

\frac{u^5 - u^3 - u^2 + 1}{u^3 - u^2 - u + 1}

. Factoring the numerator by grouping yields

(u^3 - 1)(u^2 - 1)

, and factoring the denominator yields

(u^2 - 1)(u - 1)

. Canceling the common factor

(u^2 - 1)

leaves

\frac{u^3 - 1}{u - 1}

. Applying the difference of cubes formula to factor

u^3 - 1

as

(u - 1)(u^2 + u + 1)

allows us to cancel the

(u - 1)

term, leaving

u^2 + u + 1

. Re-substituting

u = x^{1/2}

results in the equivalent expression

x + x^{1/2} + 1

. Comparing this to

x + x^a + b

with

a < 1

gives

a = 0.5

and

b = 1

. Evaluating

2a + b

yields

2(0.5) + 1 = 2

.

Step-by-Step Solution

1

Introduce a variable substitution to eliminate fractional exponents.

Let

u = x^{1/2}

, which implies

x = u^2

. The expression becomes

\frac{u^5 - u^3 - u^2 + 1}{u^3 - u^2 - u + 1}

.

Substitution simplifies working with fractional exponents and makes factoring patterns easier to identify.

2

Factor the numerator by grouping terms.

The numerator factors as

u^3(u^2 - 1) - (u^2 - 1) = (u^3 - 1)(u^2 - 1)

.

Grouping common terms helps simplify polynomials into products of lower-degree factors.

3

Factor the denominator by grouping terms.

The denominator factors as

u^2(u - 1) - (u - 1) = (u^2 - 1)(u - 1)

.

Factoring the denominator reveals common terms that can be canceled with the numerator.

4

Combine the factored forms and cancel the common factor.

The expression becomes

\frac{(u^3 - 1)(u^2 - 1)}{(u^2 - 1)(u - 1)} = \frac{u^3 - 1}{u - 1}

for

u \neq 1

.

Canceling

(u^2 - 1)

simplifies the rational expression since

x \neq 1

ensures

u^2 - 1 \neq 0

.

5

Simplify the remaining rational expression using the difference of cubes formula.

Factoring

u^3 - 1

as

(u - 1)(u^2 + u + 1)

and canceling

(u - 1)

yields

u^2 + u + 1

.

The difference of cubes formula allows cancellation of the linear term in the denominator.

6

Substitute back the original variable and determine the constants.

Substituting

u = x^{1/2}

back gives

x + x^{1/2} + 1

, meaning

a = 0.5

and

b = 1

.

This puts the expression in the final requested form

x + x^a + b

to find

a

and

b

.

7

Calculate the requested value

2a + b

.

2(0.5) + 1 = 2

.

This computes the final numeric answer requested by the question.

Key Concept

Simplifying rational expressions with fractional exponents by substitution and grouping

Question 1463Question →

A bank account is opened with an initial deposit of $800. The account balance increases by$ 5\% $each year. If no other transactions are made, which of the following functions best models the account balance,$ B(t) $, in dollars, after$ t$ years?

B(t) = 800(1.05)^t

B(t) = 800(0.05)^t

B(t) = 800(1.5)^t

B(t) = 800 + 40t

Show answer & explanation

Answer: $B(t) = 800(1.05)^t$

Answer

The function

B(t) = 800(1.05)^t

best models the account balance.

The initial deposit of

800 represents the initial value of the exponential function when

t = 0

. Since the account balance increases by

5\%

each year, the value is multiplied by a growth factor of

1 + 0.05 = 1.05

each year. Thus, the exponential function that models the balance after

t

years is

B(t) = 800(1.05)^t$.

Step-by-Step Solution

1

Identify the initial value of the exponential growth function.

The initial deposit is

800, so the initial value at

t = 0

is

800$.

An exponential model is written in the form

B(t) = a(b)^t

, where

a

represents the initial value.

2

Determine the growth factor based on the annual percentage increase.

The growth rate is

r = 5\% = 0.05

. The growth factor

b

is

1 + r = 1 + 0.05 = 1.05

.

To find the growth factor for an increasing quantity, add the growth rate as a decimal to 1.

3

Write the final exponential function by substituting the initial value and growth factor.

The function is

B(t) = 800(1.05)^t

.

Substitute

a = 800

and

b = 1.05

into the standard exponential form

B(t) = a(b)^t

.

Key Concept

Writing and interpreting exponential growth functions from a real-world context.

Question 1464Question →

A custom t-shirt printing company uses the function $C(n) = 12n + 45$ to determine the total cost, in dollars, for an order of $n$ t-shirts. What is the best interpretation of the value $45$ in this function?

The price, in dollars, charged for each printed t-shirt.

The setup fee, in dollars, charged for the entire order regardless of the number of t-shirts printed.

The total cost, in dollars, for an order of exactly

12

t-shirts.

The minimum number of t-shirts that must be purchased in a single order.

Show answer & explanation

Answer: The setup fee, in dollars, charged for the entire order regardless of the number of t-shirts printed.

Answer

The setup fee, in dollars, charged for the entire order regardless of the number of t-shirts printed.

In the linear function

C(n) = 12n + 45

, the term

45

is a constant that does not depend on the variable

n

(the number of t-shirts). Therefore, it represents the initial or flat setup cost in dollars for placing an order, which is charged regardless of the number of t-shirts printed.

Step-by-Step Solution

1

Identify the structure of the linear function

C(n) = 12n + 45

.

The function is in the slope-intercept form

y = mx + b

, where

m = 12

is the slope and

b = 45

is the y-intercept.

Understanding the components of a linear function helps determine which part of the equation corresponds to which real-world quantity.

2

Determine the meaning of the constant term (y-intercept) in the context of the problem.

The constant term

45

represents the value of

C(n)

when

n = 0

, which is the cost before any t-shirts are added, representing a flat setup fee.

Evaluating the function at

n = 0

isolates the base cost or initial charge.

Key Concept

Interpreting Linear Relationships in Context

Question 1465Question →

A landscape designer purchased a total of

35

plants, consisting of boxwood shrubs and fern plants, for a total of \

510$. Each boxwood shrub cost \

18

, and each fern plant cost \$

12

. If the designer purchased

b

boxwood shrubs and

f

fern plants, what is the value of

f - b$?

15

5

27

29

Show answer & explanation

Answer: 5

Answer

The correct answer is 5, representing the difference between the 20 fern plants and 15 boxwood shrubs.

To find the value of

f - b

, we can set up a system of linear equations based on the given information. The total number of plants purchased is

35

, which gives the equation

b + f = 35

. The total cost of the plants is \

510$, with boxwood shrubs costing \

18

each and fern plants costing \$

12

each, giving the equation

18b + 12f = 510

. Solving the first equation for

f

gives

f = 35 - b

. Substituting this into the second equation yields

18b + 12(35 - b) = 510

. Distributing and simplifying gives

18b + 420 - 12b = 510

, which simplifies to

6b + 420 = 510

. Subtracting

420

from both sides gives

6b = 90

, so

b = 15

. Substituting

b = 15

back into

b + f = 35

gives

15 + f = 35

, so

f = 20

. The value of

f - b

is

20 - 15 = 5$.

Step-by-Step Solution

1

Define variables and write the system of linear equations representing the problem.

Let

b

be the number of boxwood shrubs and

f

be the number of fern plants. The system of equations is:

\begin{aligned} b + f &= 35 \\ 18b + 12f &= 510 \end{aligned}

We need to translate the word problem into a mathematical system of two linear equations with two variables.

2

Express one variable in terms of the other from the first equation and substitute it into the second equation.

From

b + f = 35

, we get

f = 35 - b

.
Substituting this into the second equation gives:

18b + 12(35 - b) = 510

Substitution is a standard method to reduce a system of two equations to a single equation in one variable.

3

Solve the resulting single-variable equation for

b

.

18b + 420 - 12b = 510

6b + 420 = 510

6b = 90

b = 15

By distributing the coefficient and combining like terms, we isolate and solve for the variable b.

4

Substitute the value of

b

back into the first equation to solve for

f

, and then calculate

f - b

.

15 + f = 35 \Rightarrow f = 20

f - b = 20 - 15 = 5

Finding the individual values of both variables allows us to calculate the required difference.

Key Concept

Solving systems of linear equations in two variables using substitution or elimination, and evaluating a linear combination of the solution.

Alternative Method

Instead of substitution, the system can be solved using elimination. Multiply the first equation,

b + f = 35

, by

12

to get

12b + 12f = 420

. Subtract this from the second equation,

18b + 12f = 510

, to eliminate

f

:

(18b - 12b) + (12f - 12f) = 510 - 420

, which simplifies to

6b = 90

, giving

b = 15

. Then, find

f = 20

and compute

f - b = 5

.

Estimated Time:1m 30s

Question 1466Question →

Volcanic eruptions can have a profound impact on Earth's climate system by releasing gases that interact with solar radiation. During the 1991 eruption of Mount Pinatubo, the injection of sulfur dioxide into the stratosphere created a massive aerosol cloud ________ global average temperatures decreased by about 0.5 degrees Celsius over the following year.

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

cloud; consequently,

cloud, consequently,

cloud consequently

cloud, but

Show answer & explanation

Answer: cloud; consequently,

Answer

cloud; consequently,

The correct option correctly uses a semicolon to separate two independent clauses, followed by the transition 'consequently' and a comma. This creates a grammatically correct structure and logically connects the cause (the creation of the aerosol cloud) with its effect (the temperature drop).

Step-by-Step Solution

1

Identify the structure of the two clauses surrounding the blank.

The clause before the blank ('During the 1991 eruption of Mount Pinatubo, the injection of sulfur dioxide into the stratosphere created a massive aerosol cloud') and the clause after the blank ('global average temperatures decreased by about 0.5 degrees Celsius over the following year') are both independent clauses because they can each stand alone as complete sentences.

Knowing whether the clauses are independent or dependent determines the type of punctuation and conjunction needed to connect them.

2

Determine the logical relationship between the two independent clauses.

The first clause describes a cause (the creation of an aerosol cloud) and the second clause describes its direct effect (a drop in global temperatures).

This helps select the appropriate transition or conjunction that accurately reflects the meaning of the sentence.

3

Evaluate the choices to find the one that correctly punctuates the boundary between the two independent clauses while expressing the correct logical relationship.

The option containing a semicolon followed by the conjunctive adverb 'consequently' and a comma correctly joins the two independent clauses and establishes a cause-and-effect relationship. The comma splice, run-on, and contrast conjunction options are grammatically or logically incorrect.

Semicolons are standard punctuation for linking independent clauses, and using 'consequently' correctly matches the cause-and-effect logic.

Key Concept

Clause Boundaries and Linking

Estimated Time:1m 0s

Question 1467Question →

\begin{aligned} y &= x^2 - 5 \\ y &= 4x \end{aligned}

If $(x, y)$ is a solution to the system of equations above and $y > 0$ , what is the value of $y$ ?

Show answer & explanation

Answer: 20

Answer

20

Substituting

y = 4x

into the equation

y = x^2 - 5

yields

4x = x^2 - 5

. Rearranging this equation into standard form gives

x^2 - 4x - 5 = 0

. Factoring the quadratic expression gives

(x - 5)(x + 1) = 0

, which means

x = 5

or

x = -1

. Substituting these values back into

y = 4x

gives the coordinates of the two solutions:

(5, 20)

and

(-1, -4)

. Since the question specifies that

y > 0

, the correct value is the positive

y

-value, which is 20.

Step-by-Step Solution

1

Substitute the expression for

y

from the second equation into the first equation.

4x = x^2 - 5

This eliminates

y

and creates a single equation in terms of

x

.

2

Subtract

4x

from both sides to write the quadratic equation in standard form.

x^2 - 4x - 5 = 0

Standard quadratic form

ax^2 + bx + c = 0

is required to solve by factoring.

3

Factor the quadratic equation.

(x - 5)(x + 1) = 0

Factoring helps find the values of

x

that satisfy the equation.

4

Solve for

x

by setting each factor to zero.

x = 5

or

x = -1

Applying the zero product property yields the possible

x

-coordinates of the solutions.

5

Calculate the corresponding

y

-values using the equation

y = 4x

.

For

x = 5

,

y = 20

. For

x = -1

,

y = -4

.

This determines the coordinates of the intersection points, which are

(5, 20)

and

(-1, -4)

.

6

Apply the constraint

y > 0

to identify the correct value of

y

.

y = 20

Since

-4

is not greater than zero, the only valid solution is

(5, 20)

, giving

y = 20

.

Key Concept

Solving a system of linear and quadratic equations using substitution.

Question 1468Question →

For all $x > 5$ , which of the following is equivalent to the expression $\frac{3x^2 - 14x - 5}{3x + 1} - \frac{x^2 - 25}{x - 5}$ ?

-10

0

2x

2x - 10

Show answer & explanation

Answer: $-10$

Answer

-10

Factoring the numerators allows both rational expressions to be simplified. The first expression,

\frac{3x^2 - 14x - 5}{3x + 1}

, factors as

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

, which simplifies to

x - 5

since

x > 5

. The second expression,

\frac{x^2 - 25}{x - 5}

, factors as

\frac{(x - 5)(x + 5)}{x - 5}

, which simplifies to

x + 5

. Subtracting the second simplified expression from the first gives

(x - 5) - (x + 5) = x - 5 - x - 5 = -10

. Therefore, the equivalent expression is

-10

.

Step-by-Step Solution

1

Factor the numerator of the first rational expression,

3x^2 - 14x - 5

.

(3x + 1)(x - 5)

To find common factors that can be simplified with the denominator

3x + 1

.

2

Simplify the first expression,

\frac{3x^2 - 14x - 5}{3x + 1}

, for

x > 5

.

x - 5

Since

x > 5

,

3x + 1 \neq 0

, allowing us to divide out the common factor

(3x + 1)

.

3

Factor the numerator of the second rational expression,

x^2 - 25

.

(x - 5)(x + 5)

Using the difference of squares identity,

a^2 - b^2 = (a - b)(a + b)

.

4

Simplify the second expression,

\frac{x^2 - 25}{x - 5}

, for

x > 5

.

x + 5

Since

x > 5

,

x - 5 \neq 0

, allowing us to divide out the common factor

(x - 5)

.

5

Subtract the second simplified expression from the first.

(x - 5) - (x + 5) = x - 5 - x - 5 = -10

Distributing the negative sign through the parentheses and combining like terms yields the final simplified equivalent value.

Key Concept

Simplifying rational expressions by factoring and performing polynomial operations.

Question 1469Question →

The quadratic equation $x^2 - 4x + 2 = 0$ has roots $r_1$ and $r_2$ . A second quadratic equation, $x^2 - px + q = 0$ , has roots $\left(r_1 + \frac{1}{r_2}\right)$ and $\left(r_2 + \frac{1}{r_1}\right)$ , where $p$ and $q$ are constants. What is the value of $p$ ?

Show answer & explanation

Answer: 6

Answer

The value of

p

is 6.

By Vieta's formulas, the sum of the roots of

x^2 - 4x + 2 = 0

is

r_1 + r_2 = 4

and the product of the roots is

r_1 r_2 = 2

. The sum of the roots of the second equation

x^2 - px + q = 0

is

p

. Therefore,

p = \left(r_1 + \frac{1}{r_2}\right) + \left(r_2 + \frac{1}{r_1}\right) = (r_1 + r_2) + \frac{r_1 + r_2}{r_1 r_2}

. Substituting the known values yields

p = 4 + \frac{4}{2} = 6

.

Step-by-Step Solution

1

Determine the sum and product of the roots of the first equation.

r_1 + r_2 = 4

and

r_1 r_2 = 2

Vieta's formulas state that for

x^2 - Bx + C = 0

, the sum of the roots is

B

and the product is

C

.

2

Express the sum of the roots of the second equation, which is

p

.

p = (r_1 + r_2) + \left(\frac{1}{r_1} + \frac{1}{r_2}\right)

The sum of the roots of

x^2 - px + q = 0

is

p

.

3

Simplify the fractional part of the equation.

\frac{1}{r_1} + \frac{1}{r_2} = \frac{r_1 + r_2}{r_1 r_2}

Finding a common denominator allows us to write the sum of reciprocals in terms of the sum and product of the roots.

4

Substitute the values of the sum and product into the expression for

p

.

p = 4 + \frac{4}{2} = 6

This evaluates the expression to find the final value of

p

.

Key Concept

Relating the roots and coefficients of quadratic equations using Vieta's formulas.

Question 1470Question →

If $(2y - 1)^2 = 49$ and $y > 0$ , what is the value of $y$ ?

Show answer & explanation

Answer: 4

Answer

The correct answer is

4

.

By taking the square root of both sides of the equation

(2y - 1)^2 = 49

, we get

2y - 1 = 7

or

2y - 1 = -7

. Solving

2y - 1 = 7

gives

2y = 8

, or

y = 4

. Solving

2y - 1 = -7

gives

2y = -6

, or

y = -3

. Since it is given that

y > 0

, the value of

y

must be

4

.

Step-by-Step Solution

1

Take the square root of both sides of the equation.

2y - 1 = 7

or

2y - 1 = -7

Applying the square root property of equality to isolate the linear term.

2

Solve each linear equation for

y

.

y = 4

or

y = -3

Isolating

y

in each equation by adding

1

and then dividing by

2

.

3

Apply the constraint

y > 0

to identify the final value.

y = 4

The value

y = -3

is negative, so it does not satisfy the constraint

y > 0

.

Key Concept

Solving quadratic equations of the form

(ax + b)^2 = c

using the square root property.

Question 1471Question →

In the $xy$ -plane, a circle has center $(6, 8)$ and is tangent to the $y$ -axis. A line with a positive slope $m$ passes through the origin and is tangent to the circle. If $m$ is written as a fraction in simplest form, $\frac{p}{q}$ , what is the value of $p + q$ ?

Show answer & explanation

Answer: 31

Answer

The value of

p + q

is 31.

The radius of the circle is 6 since the center is

(6, 8)

and it is tangent to the

y

-axis. Substituting the line

y = mx

into the circle's equation

(x-6)^2 + (y-8)^2 = 36

yields a quadratic equation in

x

:

(1+m^2)x^2 - (12+16m)x + 64 = 0

. Since the line is tangent to the circle, there is exactly one solution, meaning the discriminant of this quadratic must equal zero:

(12+16m)^2 - 4(1+m^2)(64) = 0

. Simplifying this equation yields

24m - 7 = 0

, which gives a slope of

m = \frac{7}{24}

. Since the fraction is in simplest form,

p=7

and

q=24

, and their sum is

31

.

Step-by-Step Solution

1

Determine the radius of the circle.

r = 6

Since the center is at

(6, 8)

and the circle is tangent to the

y

-axis, the radius is the horizontal distance from the center to the

y

-axis.

2

Write the equation of the circle.

(x-6)^2 + (y-8)^2 = 36

A circle with center

(h, k)

and radius

r

has the equation

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

.

3

Substitute

y = mx

into the circle's equation.

(1+m^2)x^2 - (12+16m)x + 64 = 0

Substituting the equation of the line

y = mx

into the circle's equation and expanding gives a quadratic equation in terms of

x

.

4

Set the discriminant of the quadratic equation to zero.

\Delta = (12+16m)^2 - 256(1+m^2) = 0

For the line to be tangent to the circle, the system must have exactly one real solution, meaning the quadratic equation must have a discriminant of zero.

5

Solve for the positive slope

m

.

m = \frac{7}{24}

Expanding and simplifying the discriminant equation yields

24m - 7 = 0

, which gives

m = \frac{7}{24}

.

6

Calculate

p + q

using the simplest form fraction.

31

Since

m = \frac{7}{24}

is in simplest form,

p = 7

and

q = 24

. Therefore,

p + q = 7 + 24 = 31

.

Key Concept

Nonlinear Systems of Equations

Question 1472Question →

In the inequality $-2(3x - k) - 5(x + 1) > 17$ , $k$ is an integer constant. If the maximum integer value of $x$ that satisfies the inequality is $2$ , what is the least possible value of $k$ ?

Show answer & explanation

Answer: 23

Answer

The correct answer is 23.

By simplifying the inequality to

x < \frac{2k - 22}{11}

, we establish that the upper bound of the solution interval must be strictly greater than

2

but less than or equal to

3

for

2

to be the maximum integer solution. Solving the resulting compound inequality

2 < \frac{2k - 22}{11} \le 3

yields

22 < k \le 27.5

. The smallest integer within this range is

23

.

Step-by-Step Solution

1

Distribute the constants on the left side of the inequality.

-6x + 2k - 5x - 5 > 17

Applying the distributive property simplifies the expression and removes the parentheses.

2

Combine like terms on the left side and isolate the

x

term.

-11x + 2k - 5 > 17 \implies -11x > 22 - 2k

Grouping

x

terms together and moving the constant terms to the other side prepares the inequality for division.

3

Divide both sides by

-11

and flip the inequality sign.

x < \frac{22 - 2k}{-11} \implies x < \frac{2k - 22}{11}

Dividing by a negative number reverses the direction of the inequality sign from greater-than (

>

) to less-than (

<

).

4

Set up the inequality for the maximum integer solution to be

2

.

2 < \frac{2k - 22}{11} \le 3

For

2

to be the largest integer satisfying

x < L

(where

L

is the boundary),

2

must be strictly less than

L

, and

L

must be less than or equal to the next integer,

3

.

5

Solve the compound inequality for the parameter

k

.

22 < 2k - 22 \le 33 \implies 44 < 2k \le 55 \implies 22 < k \le 27.5

Multiplying all parts by 11, adding 22, and dividing by 2 isolates

k

.

6

Find the least integer value of

k

in the interval

(22, 27.5]

.

23

The integers that satisfy

22 < k \le 27.5

are

23, 24, 25, 26,

and

27

. The least of these values is

23

.

Key Concept

Solving linear inequalities in one variable involving parameter bounds, negative coefficients, and integer solution constraints.

Question 1473Question →

At a municipal water desalination facility, the filtration rate of a reverse osmosis membrane decreases linearly over time due to particle accumulation. The daily volume of purified water, $V$ , in thousands of gallons per day, can be modeled as a function of the number of days, $t$ , since the membrane was last serviced. The graph of this relationship in the $tV$ -plane has a $t$ -intercept of $100$ and passes through the point $(20, 64)$ . Which of the following is the best interpretation of the slope of the graph of this relationship?

The daily volume of purified water decreases by

0.8

thousand gallons each day after the membrane is serviced.

The daily volume of purified water decreases by

1.25

thousand gallons each day after the membrane is serviced.

The daily volume of purified water decreases by

80

thousand gallons each day after the membrane is serviced.

For every decrease of

1

thousand gallons in the daily volume of purified water,

0.8

days have passed since the membrane was serviced.

Show answer & explanation

Answer: The daily volume of purified water decreases by $0.8$ thousand gallons each day after the membrane is serviced.

Answer

The daily volume of purified water decreases by

0.8

thousand gallons each day after the membrane is serviced.

The correct answer describes that the daily volume of purified water decreases by

0.8

thousand gallons each day. The

t

-intercept of

100

indicates the point

(100, 0)

is on the graph, and the problem states the graph passes through

(20, 64)

. The slope of the relationship is

m = \frac{0 - 64}{100 - 20} = -0.8

. Since the vertical axis represents the daily volume of purified water (in thousands of gallons) and the horizontal axis represents the number of days, the slope of

-0.8

represents a decrease of

0.8

thousand gallons of water per day.

Step-by-Step Solution

1

Identify the coordinates of two points on the line from the given context.

The

t

-intercept of

100

corresponds to the point

(100, 0)

. The second point is given directly as

(20, 64)

.

Two points on the line are needed to calculate the slope of the linear relationship.

2

Calculate the slope (

m

) using the slope formula

m = \frac{V_2 - V_1}{t_2 - t_1}

.

m = \frac{0 - 64}{100 - 20} = \frac{-64}{80} = -0.8

.

The slope of the line represents the rate of change of the daily volume of water (

V

) with respect to the elapsed days (

t

).

3

Interpret the meaning of the slope in context.

A slope of

-0.8

indicates that for every increase of

1

day in

t

, the daily volume of water

V

decreases by

0.8

units (thousands of gallons).

The slope value of

-0.8

represents a daily reduction of

0.8

thousand gallons of purified water.

Key Concept

Interpreting the slope of a linear model in a real-world context

Alternative Method

Instead of calculating the slope directly from the formula, one can write the equation of the line in point-slope form or slope-intercept form. Since

(100,0)

is the

t

-intercept, the line can be represented as

V = m(t - 100)

. Substituting the point

(20, 64)

gives

64 = m(20 - 100)

, which simplifies to

64 = -80m

, solving to

m = -0.8

. This slope represents the daily change in the volume

V

for each unit change in day

t

.

Estimated Time:2m 30s

Question 1474Question →

The quadratic function $f$ is defined by $f(x) = (x - 4)^2 + 7$ . What are the coordinates of the vertex of the graph of $y = f(x)$ in the $xy$ -plane?

(-4, 7)

(4, -7)

(4, 7)

(7, 4)

Show answer & explanation

Answer: $(4, 7)$

Answer

The coordinates of the vertex are

(4, 7)

.

The quadratic function is given in vertex form,

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

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

xy

-plane is

(h, k)

. Comparing

f(x) = (x - 4)^2 + 7

to this form, we get

h = 4

and

k = 7

. Therefore, the coordinates of the vertex are

(4, 7)

.

Step-by-Step Solution

1

Identify the standard vertex form of a quadratic function.

The vertex form of a quadratic function is

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

, where

(h, k)

is the vertex of the parabola.

This standard form allows direct extraction of the vertex coordinates by matching the variables.

2

Compare the given function

f(x) = (x - 4)^2 + 7

to the standard vertex form.

Comparing the two equations, we identify that

a = 1

,

h = 4

, and

k = 7

.

Matching coefficients and constants determines the values of the vertex parameters.

3

Write the coordinates of the vertex

(h, k)

using the identified values.

The vertex is at

(4, 7)

.

Substituting

h = 4

and

k = 7

into

(h, k)

yields the final vertex coordinates.

Key Concept

Identifying the vertex of a quadratic function from its vertex form.

Question 1475Question →

For all $x > 0$ , which of the following is equivalent to the expression $\left(\sqrt{x} - \frac{1}{\sqrt{x}}\right)^2 \left(x + 1 + \frac{1}{x}\right)$ ?

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

\frac{(x^2+1)(x^2+x+1)}{x^2}

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

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

Show answer & explanation

Answer: $\frac{(x-1)(x^3-1)}{x^2}$

Answer

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

The correct expression is obtained by first converting

\sqrt{x} - \frac{1}{\sqrt{x}}

to

\frac{x-1}{\sqrt{x}}

, and then squaring it to get

\frac{(x-1)^2}{x}

. Next, the term

x + 1 + \frac{1}{x}

is converted to

\frac{x^2+x+1}{x}

. Multiplying these two results yields

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

. Grouping

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

and substituting it with the difference of cubes identity,

x^3-1

, gives the final equivalent expression

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

.

Step-by-Step Solution

1

Simplify the first factor by writing

\sqrt{x} - \frac{1}{\sqrt{x}}

under a common denominator and squaring it.

\left(\frac{x-1}{\sqrt{x}}\right)^2 = \frac{(x-1)^2}{x}

This allows us to combine the terms in the first factor before multiplication.

2

Write the second factor

x + 1 + \frac{1}{x}

under a common denominator.

\frac{x^2+x+1}{x}

This allows us to multiply the two fractional factors easily.

3

Multiply the two simplified factors together.

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

Multiplying the numerators and denominators of the fractions gives the product of the terms.

4

Apply the difference of cubes factorization formula:

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

.

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

This simplifies the product to its final equivalent form.

Key Concept

Simplifying rational expressions and applying algebraic identities (specifically the binomial square and difference of cubes).

Alternative Method

Alternatively, you can substitute a convenient value for

x

, such as

x = 4

, into the original expression and each of the options, then compare the results.

Estimated Time:2m 0s

Question 1476Question →

For a real solution $(x, y)$ to the system of equations below, which of the following is a possible value of $x + y$ ?

\begin{aligned} x^2 + 3y^2 &= 21 \\ x^2 - 2y &= 5 \end{aligned}

-5

-3

1

5

Show answer & explanation

Answer: 5

Answer

The correct answer is 5, which is the sum of the coordinates of the real solution (3, 2) of the system.

Substituting

x^2 = 2y + 5

into the first equation yields the quadratic equation

3y^2 + 2y - 16 = 0

. Solving this quadratic equation gives

y = 2

or

y = -8/3

. Since

x^2

must be non-negative for real solutions,

y = -8/3

yields no real solutions because

x^2 = 2(-8/3) + 5 = -1/3 < 0

. For

y = 2

, we have

x^2 = 2(2) + 5 = 9

, which gives

x = 3

or

x = -3

. The real solutions to the system are

(3, 2)

and

(-3, 2)

. The possible values for

x + y

are

3 + 2 = 5

and

-3 + 2 = -1

. Therefore, 5 is a possible value of

x + y

.

Step-by-Step Solution

1

Rearrange the second equation to express

x^2

in terms of

y

.

x^2 = 2y + 5

This allows for direct substitution into the first equation to eliminate the

x

variable.

2

Substitute the expression for

x^2

into the first equation and simplify the resulting quadratic equation in terms of

y

.

3y^2 + 2y - 16 = 0

Substituting

2y + 5

for

x^2

in

x^2 + 3y^2 = 21

gives

(2y + 5) + 3y^2 = 21

, which simplifies to

3y^2 + 2y - 16 = 0

.

3

Factor or use the quadratic formula to solve for

y

.

y = 2

or

y = -8/3

The quadratic equation factors as

(3y + 8)(y - 2) = 0

, giving the roots

y = 2

and

y = -8/3

.

4

Find the corresponding values of

x

for each

y

and determine the real solutions.

For

y = 2

,

x^2 = 2(2) + 5 = 9 \implies x = \pm 3

. For

y = -8/3

,

x^2 = 2(-8/3) + 5 = -1/3

, which has no real solutions.

A real number squared cannot be negative, so only

y = 2

yields real solutions:

(3, 2)

and

(-3, 2)

.

5

Calculate the possible values of

x + y

for the real solutions.

For

(3, 2)

,

x + y = 3 + 2 = 5

. For

(-3, 2)

,

x + y = -3 + 2 = -1

.

Evaluating the target expression for both valid real solutions gives 5 and -1. Since 5 is the only value present in the options, it is the correct answer.

Key Concept

Solving systems of nonlinear equations algebraically using substitution and analyzing real solutions.

Question 1477Question →

For a wireless sensor node in an environmental monitoring network, the battery life $B$ , in days, when transmitting $n$ data packets per day is modeled by a linear function. Under standard operating conditions, the node's battery lasts for $120$ days when transmitting $80$ packets per day, and it lasts for $90$ days when transmitting $140$ packets per day. Under a new energy-saving firmware, the battery consumption rate per packet transmitted is reduced by $40\%$ , and the static power draw is also reduced, increasing the maximum battery life (when zero packets are transmitted) by $10\%$ . Under this new firmware, what is the battery life, in days, of a sensor node that transmits $150$ data packets per day?

Show answer & explanation

Answer: 131

Answer

131

To solve this problem, we first determine the linear relationship

B = mn + B_0

under standard operating conditions. The rate of change (slope

m

) represents the change in battery life per packet transmitted per day:

m = \frac{90 - 120}{140 - 80} = -0.5

days per packet. Substituting this back into the linear equation gives

120 = -0.5(80) + B_0

, which simplifies to

120 = -40 + B_0

, yielding a y-intercept of

B_0 = 160

days. Under the energy-saving firmware, the battery consumption rate per packet is reduced by

40\%

. This means the rate at which battery life decreases per packet changes from

0.5

to

0.5 \times (1 - 0.40) = 0.3

, giving us a new slope of

-0.3

. The maximum battery life (y-intercept) increases by

10\%

, making the new y-intercept

160 \times 1.10 = 176

. The new linear model is

B_{\text{new}} = -0.3n + 176

. Evaluating this equation for

n = 150

packets per day yields

B_{\text{new}} = -0.3(150) + 176 = -45 + 176 = 131

days.

Step-by-Step Solution

1

Determine the linear relationship representing standard operating conditions by using the two given points,

(80, 120)

and

(140, 90)

.

Slope

m = -0.5

days per packet and y-intercept

B_0 = 160

days.

Establishing the initial linear equation is necessary to obtain the base battery consumption rate and maximum battery capacity.

2

Calculate the new slope and y-intercept parameters under the energy-saving firmware by applying the specified percentage changes.

New slope

m_{\text{new}} = -0.3

days per packet and new y-intercept

B_{0,\text{new}} = 176

days.

To model the linear relationship under the updated power management firmware.

3

Formulate the new linear function

B_{\text{new}} = -0.3n + 176

and substitute

150

for

n

.

Battery life

B = 131

days.

To find the expected battery life at the target daily packet transmission rate.

Key Concept

Interpreting how physical rates and initial values map to the slope and y-intercept of a linear model, and applying transformations to these parameters.

Question 1478Question →

Although a newly formed council of medieval history scholars has compiled an extensive digital database of rare fourteenth-century manuscripts, _______ has struggled to secure the long-term financial support necessary to keep the archive open to public researchers. Consequently, historians worldwide worry that access to these fragile primary sources will soon be restricted.

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

they

it

them

these

Show answer & explanation

Answer: it

Answer

The singular pronoun 'it' correctly agrees with the singular collective noun 'council' and serves as the subject of the clause.

The pronoun in the blank must refer to the singular collective noun 'council,' which is the main subject of the preceding subordinate clause. Because 'council' is singular, the pronoun must also be singular. Additionally, since the pronoun functions as the subject of the verb phrase 'has struggled,' it must be in the subjective case. The singular pronoun 'it' correctly meets both of these requirements.

Step-by-Step Solution

1

Identify the antecedent of the pronoun in the sentence.

The antecedent is 'council,' which is a singular collective noun.

Determining whether the antecedent is singular or plural is necessary to ensure pronoun-antecedent agreement.

2

Identify the grammatical role of the pronoun in the clause.

The pronoun acts as the subject of the verb phrase 'has struggled.'

Determining the pronoun's role helps identify the correct case (subjective case).

3

Select the pronoun that matches both the singular number and the subjective case.

The singular subjective pronoun 'it' is the correct choice.

This pronoun satisfies both number agreement and case requirements.

Key Concept

Pronoun-Antecedent Agreement and Case

Question 1479Question →

In the $xy$ -plane, a line has a $y$ -intercept of $(0, d)$ and passes through the point $(4, d - 3)$ , where $d$ is a constant. If the line also passes through the point $(12, k)$ , which of the following expressions represents the value of $k$ ?

d + 9

d - 16

d - 9

d - 36

Show answer & explanation

Answer: $d - 9$

Answer

d - 9

The correct answer represents the value of

k

as

d - 9

. The slope

m

of the line passing through

(0, d)

and

(4, d - 3)

is

\frac{(d - 3) - d}{4 - 0} = -\frac{3}{4}

. Using the

y

-intercept

(0, d)

, the equation of the line is

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

. Substituting the point

(12, k)

gives

k = -\frac{3}{4}(12) + d = d - 9

.

Step-by-Step Solution

1

Calculate the slope of the line using the points

(0, d)

and

(4, d - 3)

.

m = -\frac{3}{4}

The slope formula is

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

. Substituting the given coordinates yields

m = \frac{(d - 3) - d}{4 - 0} = -\frac{3}{4}

.

2

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

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

Since the line's

y

-intercept is

(0, d)

, the constant

b

in the slope-intercept equation

y = mx + b

is equal to

d

.

3

Substitute the point

(12, k)

into the equation of the line to solve for

k

.

k = d - 9

Substituting

x = 12

and

y = k

into

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

gives

k = -\frac{3}{4}(12) + d = -9 + d

, which simplifies to

d - 9

.

Key Concept

Determining the equation and coordinates of a line in the coordinate plane given its slope and intercepts.

Question 1480Question →

For a certain constant $a$ , the quadratic equation $3x^2 - 2ax + a - 4 = 0$ has real roots $x_1$ and $x_2$ . If $x_1^2 + x_2^2 = \frac{28}{9}$ , what is the sum of all possible values of $a$ ?

\frac{3}{4}

\frac{3}{2}

-\frac{3}{2}

3

Show answer & explanation

Answer: $\frac{3}{2}$

Answer

The sum of all possible values of

a

is \frac{3}{2}.

The correct answer is derived by expressing the sum of the squares of the roots,

x_1^2 + x_2^2

, as

(x_1 + x_2)^2 - 2x_1x_2

. Applying Vieta's formulas to the equation

3x^2 - 2ax + a - 4 = 0

yields

x_1 + x_2 = \frac{2a}{3}

and

x_1 x_2 = \frac{a - 4}{3}

. Substituting these values into the identity gives

\frac{4a^2}{9} - \frac{2(a - 4)}{3} = \frac{28}{9}

. Multiplying by 9 and simplifying leads to the quadratic equation

2a^2 - 3a - 2 = 0

. Since this equation has real solutions and the original equation always has real roots for any real

a

, the sum of all possible values of

a

is given by the negative linear coefficient divided by the leading coefficient, which is \frac{3}{2}.

Step-by-Step Solution

1

Express the sum of squares

x_1^2 + x_2^2

in terms of the sum and product of the roots.

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

This allows the use of Vieta's formulas to relate the roots of the quadratic equation to its coefficients.

2

Apply Vieta's formulas to the quadratic equation

3x^2 - 2ax + a - 4 = 0

to find the sum and product of the roots.

x_1 + x_2 = \frac{2a}{3}

and

x_1 x_2 = \frac{a - 4}{3}

For any quadratic equation

Ax^2 + Bx + C = 0

, the sum of the roots is

-\frac{B}{A}

and the product is

\frac{C}{A}

.

3

Substitute the expressions for the sum and product of the roots into the identity from Step 1 and set it equal to the given value.

\left(\frac{2a}{3}\right)^2 - 2\left(\frac{a - 4}{3}\right) = \frac{28}{9}

This sets up a solvable equation for the parameter

a

.

4

Simplify the algebraic equation by multiplying both sides by 9 and expanding terms.

4a^2 - 6(a - 4) = 28 \implies 4a^2 - 6a - 4 = 0 \implies 2a^2 - 3a - 2 = 0

Eliminating denominators and grouping terms yields a standard form quadratic equation in

a

.

5

Verify that the solutions for

a

are real and that they yield real roots for

x

in the original equation.

For

2a^2 - 3a - 2 = 0

, the discriminant

\Delta_a = 25 > 0

, so

a

has real solutions. The discriminant of the original equation is

\Delta_x = 4a^2 - 12a + 48

, which is positive for all real values of

a

since its own discriminant is negative.

This confirms that the solutions for

a

are valid and satisfy the condition that

x_1

and

x_2

are real numbers.

6

Use Vieta's formula on the quadratic equation

2a^2 - 3a - 2 = 0

to find the sum of all possible values of

a

.

Sum of values

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

The sum of the roots of a quadratic equation in the form

Aa^2 + Ba + C = 0

is equal to

-\frac{B}{A}

.

Key Concept

Vieta's Formulas and Algebraic Identities for Quadratic Equations

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

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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

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Key Concept

Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

Alternative Method

Answer

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes