All practice questions

Question 1681Question →

The function $f$ is defined by $f(x) = (x - 5)^2 + 3$ . If the graph of $y = f(x)$ is translated 4 units down in the $xy$ -plane to create the graph of the function $g$ , what is the vertex of the graph of $g$ ?

(1, 3)

(-5, -1)

(5, -1)

(5, 7)

Show answer & explanation

Answer: $(5, -1)$

Answer

The vertex of the graph of

g

is

(5, -1)

.

The vertex of a quadratic function of the form

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

is

(h, k)

. For the function

f(x) = (x - 5)^2 + 3

, the vertex is

(5, 3)

. Translating the graph of a function vertically downward by 4 units decreases the

y

-coordinate of every point on the graph by 4, while the

x

-coordinate remains unchanged. Applying this transformation to the vertex coordinates yields

(5, 3 - 4)

, which simplifies to

(5, -1)

.

Step-by-Step Solution

1

Identify the vertex of the original function

f(x) = (x - 5)^2 + 3

.

The vertex of the parabola

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

is

(h, k)

. For

f(x) = (x - 5)^2 + 3

, the vertex is

(5, 3)

.

Finding the original vertex is the starting point for applying the translation.

2

Apply the vertical translation of 4 units downward to the vertex coordinates.

A downward translation of 4 units subtracts 4 from the

y

-coordinate of the vertex:

(5, 3 - 4) = (5, -1)

.

Translating a graph vertically downward decreases the

y

-value of all its points by the translation distance.

Key Concept

Identifying the vertex of a quadratic function in vertex form and applying vertical translations.

Estimated Time:45s

Question 1682Question →

An online retail store charges a flat shipping fee plus a separate fee per package for bulk deliveries. The total shipping cost, $C$ , in dollars, for an order containing $x$ standard packages and $y$ deluxe packages can be modeled by the equation $C = 15x + 10y + 25$ . If the total shipping cost for a certain order is $195$ dollars and the order contains 8 deluxe packages, how many standard packages are in the order?

6

5

4

8

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6. Substituting the total cost of

195

for

C

and the 8 deluxe packages for

y

into the equation

C = 15x + 10y + 25

yields

195 = 15x + 10(8) + 25

. Simplifying the equation gives

195 = 15x + 105

. Subtracting 105 from both sides results in

90 = 15x

. Dividing both sides by 15 gives

x = 6

.

Step-by-Step Solution

1

Substitute the given values into the linear equation.

Substituting

C = 195

and

y = 8

into the equation

C = 15x + 10y + 25

yields

195 = 15x + 10(8) + 25

.

This sets up the equation with only one variable,

x

, which we need to solve for.

2

Simplify the constants on the right side of the equation.

195 = 15x + 80 + 25

, which simplifies to

195 = 15x + 105

.

Combining like terms simplifies the algebraic expression.

3

Isolate the variable term

15x

.

Subtracting 105 from both sides of the equation gives

15x = 195 - 105

, which simplifies to

15x = 90

.

To solve for

x

, we must isolate the term containing the variable on one side.

4

Solve for the variable

x

.

x = \frac{90}{15} = 6

.

Dividing by the coefficient of

x

isolates the variable completely.

Key Concept

Solving linear equations in two variables by substituting known values and isolating the variable.

Question 1683Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, has vertex $(3, 12)$ and passes through the point $(1, 0)$ . The function $g$ is defined by $g(x) = f(x + d) - 4$ , where $d$ is a constant. If the $y$ -intercept of the graph of $g$ is $(0, 5)$ and the vertex of the graph of $g$ lies in the second quadrant, what is the value of $d$ ?

Show answer & explanation

Answer: 4

Answer

The value of

d

is 4.

First, the equation of the function

f(x)

is determined in vertex form. Since the vertex of

f

is

(3, 12)

, we write

f(x) = a(x - 3)^2 + 12

. Substituting the point

(1, 0)

into this equation gives

0 = a(1 - 3)^2 + 12

, which yields

a = -3

. Thus,

f(x) = -3(x - 3)^2 + 12

. The transformation

g(x) = f(x + d) - 4

shifts the graph of

f

left by

d

units and down by 4 units, so the vertex of the graph of

g

is

(3 - d, 8)

. For this vertex to lie in the second quadrant, the

x

-coordinate must be negative, meaning

3 - d < 0

, or

d > 3

. The

y

-intercept of

g

is

(0, 5)

, so

g(0) = 5

. Since

g(0) = f(d) - 4

, we have

f(d) = 9

. Substituting

d

into

f(x)

gives

-3(d - 3)^2 + 12 = 9

, which simplifies to

(d - 3)^2 = 1

. Solving for

d

gives

d = 4

or

d = 2

. Since

d > 3

, the value of

d

must be 4.

Step-by-Step Solution

1

Write the function

f(x)

in vertex form and substitute the point

(1, 0)

to solve for

a

.

f(x) = -3(x - 3)^2 + 12

The vertex

(h, k)

is given as

(3, 12)

, and the point

(1, 0)

lies on the graph.

2

Determine the vertex of

g(x)

based on the horizontal and vertical translations of

f(x)

.

The vertex of

g

is

(3 - d, 8)

.

The transformation

g(x) = f(x + d) - 4

shifts the vertex of

f(x)

left by

d

units and down by 4 units.

3

Establish the constraint on

d

using the quadrant of the vertex of

g

.

d > 3

For the vertex

(3 - d, 8)

to lie in the second quadrant, the

x

-coordinate must be negative.

4

Set up an equation for

d

using the

y

-intercept of

g(x)

.

f(d) = 9

The

y

-intercept is

(0, 5)

, so

g(0) = 5

. Substituting this into

g(x) = f(x + d) - 4

gives

5 = f(d) - 4

.

5

Solve

f(d) = 9

for

d

.

d = 2

or

d = 4

Substituting

d

into

f(x)

gives

-3(d - 3)^2 + 12 = 9

, which simplifies to

(d - 3)^2 = 1

.

6

Select the correct value of

d

that satisfies the quadrant constraint.

d = 4

Since

d > 3

, the value

d = 2

is discarded, leaving

d = 4

as the only valid solution.

Key Concept

Vertex form and transformations of quadratic functions

Question 1684Question →

If $\sqrt{x + 2} = x$ , what is the value of $x - 1$ ?

-2

1

2

0

Show answer & explanation

Answer: 1

Answer

The value of

x - 1

is

1

.

The valid solution to the radical equation is

x = 2

. Substituting this into the expression

x - 1

yields

2 - 1 = 1

.

Step-by-Step Solution

1

Square both sides of the equation to eliminate the radical.

x + 2 = x^2

Squaring both sides removes the square root, allowing us to solve the resulting equation.

2

Rewrite the equation in standard quadratic form.

x^2 - x - 2 = 0

Moving all terms to one side sets the quadratic equation to zero, which is necessary for factoring.

3

Factor the quadratic equation.

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

, which gives the potential solutions

x = 2

and

x = -1

.

Factoring allows us to find the roots of the quadratic equation.

4

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

For

x = 2

,

\sqrt{2 + 2} = 2

is true. For

x = -1

,

\sqrt{-1 + 2} = -1 \implies 1 = -1

is false.

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

5

Calculate the value of

x - 1

using the valid solution.

2 - 1 = 1

The question asks for the value of the expression

x - 1

rather than just the variable

x

.

Key Concept

Radical Equations and Extraneous Solutions

Estimated Time:1m 0s

Question 1685Question →

For all positive real numbers $x$ and $y$ , the expression $\frac{(4x^3 y^2)^2}{\sqrt{16x^8 y^6}}$ can be written in the equivalent form $ax^b y^c$ , where $a$ , $b$ , and $c$ are positive constants. What is the value of $a + b + c$ ?

Show answer & explanation

Answer: 7

Answer

7

The expression simplifies to

4x^2 y^1

by applying the power of a product rule to the numerator to get

16x^6 y^4

, and simplifying the radical in the denominator to get

4x^4 y^3

. Dividing the terms yields

4x^2 y^1

. Identifying the coefficients and exponents gives

a = 4

,

b = 2

, and

c = 1

. The sum of these values is

4 + 2 + 1 = 7

.

Step-by-Step Solution

1

Simplify the numerator of the expression.

(4x^3 y^2)^2 = 16x^6 y^4

Apply the power of a product rule

(ab)^n = a^n b^n

and the power of a power rule

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

.

2

Simplify the denominator of the expression.

\sqrt{16x^8 y^6} = 4x^4 y^3

Take the square root of the coefficient and apply fractional exponent rules for positive variables:

\sqrt{16} = 4

,

\sqrt{x^8} = x^{8/2} = x^4

, and

\sqrt{y^6} = y^{6/2} = y^3

.

3

Divide the numerator by the denominator.

\frac{16x^6 y^4}{4x^4 y^3} = 4x^2 y^1

Divide the coefficients and subtract the exponents of the corresponding variables:

\frac{16}{4} = 4

,

x^{6-4} = x^2

, and

y^{4-3} = y^1

.

4

Identify the values of

a

,

b

, and

c

, and sum them.

a = 4

,

b = 2

,

c = 1

, and

a + b + c = 7

Match the simplified form

4x^2 y^1

to

ax^b y^c

to find

a

,

b

, and

c

, then calculate their sum.

Key Concept

Simplifying equivalent algebraic expressions using exponent rules and radical properties.

Estimated Time:1m 30s

Question 1686Question →

A commercial hydroponic farm monitors the volume of nutrient solution in a reservoir. The table below shows the volume of solution remaining, $V$ , in liters, after $t$ hours of operation.

Time ( $t$ , hours)	Volume ( $V$ , liters)
0	450
10	432
20	414
30	396

The relationship between the time and the volume of remaining solution can be modeled by a linear equation. Which of the following is the best interpretation of the slope of the graph of this equation in the $tV$ -plane?

The nutrient solution is being depleted at a rate of 1.8 liters per hour.

The initial volume of the nutrient solution in the reservoir is 450 liters.

The reservoir will be completely empty after 250 hours.

The volume of the nutrient solution decreases by 1 liter every 1.8 hours.

Show answer & explanation

Answer: The nutrient solution is being depleted at a rate of 1.8 liters per hour.

Answer

The nutrient solution is being depleted at a rate of 1.8 liters per hour.

The slope of a linear function represents the constant rate of change of the dependent variable with respect to the independent variable. By selecting two data points from the table, such as

(0, 450)

and

(10, 432)

, the slope is calculated as

\frac{432 - 450}{10 - 0} = -1.8

. Since the volume

V

is measured in liters and the time

t

is in hours, this slope represents a decrease of

1.8

liters of nutrient solution per hour of operation. Therefore, the nutrient solution is being depleted at a rate of

1.8

liters per hour.

Step-by-Step Solution

1

Calculate the slope using two points from the table.

Using the points

(0, 450)

and

(10, 432)

, the slope is

\frac{432 - 450}{10 - 0} = \frac{-18}{10} = -1.8

.

The slope of a linear relationship is given by the change in the dependent variable (Volume) divided by the change in the independent variable (Time).

2

Interpret the unit of the slope in context.

The unit of the slope is liters per hour (

\text{L/h}

), representing a change of

-1.8

liters for every hour of operation.

Determining the units of the rate helps verify what physical quantity the slope represents.

3

Relate the negative sign of the slope to the context of depletion.

A negative slope of

-1.8

means the volume of nutrient solution decreases by

1.8

liters each hour, which translates to a depletion rate of

1.8

liters per hour.

Connecting the mathematical sign to the real-world action clarifies whether the quantity is increasing or decreasing.

Key Concept

Interpreting the slope of a linear equation as a constant rate of change in a real-world context.

Estimated Time:1m 30s

Question 1687Question →

In a certain video game, players earn points by completing daily quests and weekly challenges. Last month, a player completed a total of $24$ activities, which consisted of daily quests and weekly challenges, and earned a total of $190$ points. Each daily quest completed was worth $5$ points, and each weekly challenge completed was worth $15$ points. How many weekly challenges did the player complete?

5

7

9

17

Show answer & explanation

Answer: $7$

Answer

The player completed

7

weekly challenges.

To find the number of weekly challenges, we set up a system of linear equations. Let

d

be the number of daily quests and

w

be the number of weekly challenges. We have

d + w = 24

and

5d + 15w = 190

. Expressing

d

as

24 - w

and substituting it into the second equation gives

5(24 - w) + 15w = 190

, which simplifies to

120 - 5w + 15w = 190

. Combining like terms yields

120 + 10w = 190

, so

10w = 70

, which gives

w = 7

. This matches the correct value.

Step-by-Step Solution

1

Represent the situation with a system of linear equations.

Let

d

represent the number of daily quests completed and

w

represent the number of weekly challenges completed. The system is:

\begin{aligned} d + w &= 24 \\ 5d + 15w &= 190 \end{aligned}

This sets up the mathematical model using variables for the two unknown quantities.

2

Express one variable in terms of the other using the first equation.

d = 24 - w

This allows for substitution into the second equation to reduce the system to a single variable.

3

Substitute the expression into the second equation and expand.

5(24 - w) + 15w = 190 \implies 120 - 5w + 15w = 190

This substitutes the representation of

d

to solve for

w

directly.

4

Simplify the equation and solve for

w

.

120 + 10w = 190 \implies 10w = 70 \implies w = 7

Combining like terms and isolating the variable yields the number of weekly challenges.

Key Concept

Solving systems of linear equations in two variables using substitution or elimination.

Alternative Method

Instead of substitution, the elimination method can be used. Multiply the first equation,

d + w = 24

, by

-5

to get

-5d - 5w = -120

. Add this equation to the second equation,

5d + 15w = 190

, to eliminate

d

, resulting in

10w = 70

, which simplifies to

w = 7

.

Estimated Time:1m 30s

Question 1688Question →

If $x$ is a solution to the equation $\frac{2x - 3}{x - 1} - \frac{2}{x - 2} = \frac{1}{x^2 - 3x + 2}$ , what is the value of $2x - 1$ ?

1

5

6

9

Show answer & explanation

Answer: 6

Answer

6

The correct answer is the value obtained by solving the rational equation for

x

and then calculating the expression

2x-1

. Multiplying the equation by the least common denominator

(x-1)(x-2)

yields the quadratic equation

2x^2 - 9x + 7 = 0

, which factors into

(2x - 7)(x - 1) = 0

. The potential solution

x = 1

is extraneous because it makes the denominators of the original equation equal to zero. Thus, the only valid solution is

x = 7/2

. Substituting this value into the expression gives

2(7/2) - 1 = 6

.

Step-by-Step Solution

1

Identify the least common denominator (LCD) of the rational expressions in the equation.

The LCD is

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

. Since these denominators cannot be zero, we must have

x \neq 1

and

x \neq 2

.

Finding the LCD allows us to eliminate the denominators by multiplying both sides of the equation.

2

Multiply both sides of the equation by the LCD,

(x - 1)(x - 2)

, to clear the fractions.

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

This simplifies the rational equation into a polynomial equation.

3

Expand the products and simplify the equation.

(2x^2 - 7x + 6) - (2x - 2) = 1 \implies 2x^2 - 9x + 8 = 1

Expanding the terms is necessary to group like terms and form a quadratic equation.

4

Subtract 1 from both sides to write the quadratic equation in standard form.

2x^2 - 9x + 7 = 0

Standard form (

ax^2 + bx + c = 0

) is required to factor or apply the quadratic formula.

5

Factor the quadratic equation.

(2x - 7)(x - 1) = 0

, which gives potential solutions

x = \frac{7}{2}

and

x = 1

.

Factoring helps find the values of

x

that satisfy the polynomial equation.

6

Check the potential solutions against the original denominators to identify any extraneous solutions.

For

x = 1

, the denominators

x - 1

and

x^2 - 3x + 2

become zero, so

x = 1

is extraneous. For

x = \frac{7}{2}

, the denominators are non-zero, so

x = \frac{7}{2}

is the only valid solution.

Multiplying by variables can introduce extraneous solutions that make the original rational expressions undefined.

7

Calculate the value of the requested expression,

2x - 1

, using the valid solution

x = \frac{7}{2}

.

2\left(\frac{7}{2}\right) - 1 = 7 - 1 = 6

The question asks for the value of the expression

2x - 1

, not just the value of

x

.

Key Concept

Solving rational equations by clearing denominators, factoring the resulting quadratic equation, and checking for extraneous solutions.

Question 1689Question →

If $\frac{x}{8} - \frac{2}{x} = 0$ and $x > 0$ , what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

4

The correct answer is 4. Adding the term

\frac{2}{x}

to both sides of the equation yields

\frac{x}{8} = \frac{2}{x}

. Cross-multiplying the terms gives

x^2 = 16

. Solving for

x

by taking the square root of both sides gives

x = 4

or

x = -4

. Since the question specifies that

x > 0

, the negative value is discarded, leaving 4 as the only valid solution.

Step-by-Step Solution

1

Add

\frac{2}{x}

to both sides of the equation.

\frac{x}{8} = \frac{2}{x}

To isolate the rational terms on opposite sides of the equation.

2

Cross-multiply the terms.

x^2 = 16

To eliminate the denominators and form a quadratic equation.

3

Solve for

x

by taking the square root of both sides.

x = 4

or

x = -4

Taking the square root of 16 yields both positive and negative solutions.

4

Apply the constraint

x > 0

.

x = 4

The question specifies that

x

must be greater than 0, which excludes the negative solution.

Key Concept

Solving rational equations by isolating terms, cross-multiplying, and applying given constraints.

Question 1690Question →

A parabola passes through the point $(0, 12)$ on the $y$ -axis and intersects the $x$ -axis at two distinct points, $P$ and $Q$ . The line connecting $P$ to the $y$ -intercept has a slope of $2$ , while the line connecting $Q$ to the $y$ -intercept has a slope of $-6$ . What is the maximum $y$ -value achieved by this parabola?

Show answer & explanation

Answer: 16

Answer

The maximum

y

-value achieved by the parabola is 16.

The slopes of the lines connecting the

x

-intercepts

P(x_1, 0)

and

Q(x_2, 0)

to the

y

-intercept

C(0, 12)

determine their coordinates. The slope of the line through

P

and

C

is

\frac{12 - 0}{0 - x_1} = 2

, which gives

x_1 = -6

. The slope of the line through

Q

and

C

is

\frac{12 - 0}{0 - x_2} = -6

, which gives

x_2 = 2

. The factored form of the parabola is

y = a(x + 6)(x - 2)

. Using the

y

-intercept

C(0, 12)

, we find

12 = a(6)(-2)

, meaning

a = -1

. The

x

-coordinate of the vertex is the midpoint of the intercepts, which is

\frac{-6 + 2}{2} = -2

. Substituting

x = -2

into the equation gives

y = -(-2 + 6)(-2 - 2) = 16

. Since the leading coefficient is negative, this represents the maximum

y

-value.

Step-by-Step Solution

1

Find the coordinates of

P(x_1, 0)

and

Q(x_2, 0)

using the slopes of the lines from these points to the

y

-intercept

C(0, 12)

.

P = (-6, 0)

and

Q = (2, 0)

The slope formula between

(x_1, 0)

and

(0, 12)

is

\frac{12 - 0}{0 - x_1} = 2

, which gives

x_1 = -6

. Similarly, for the second line,

\frac{12 - 0}{0 - x_2} = -6

, which gives

x_2 = 2

.

2

Write the quadratic equation in factored form using the

x

-intercepts.

y = a(x + 6)(x - 2)

Since the parabola intersects the

x

-axis at

(-6, 0)

and

(2, 0)

, these points are the roots of the quadratic function.

3

Solve for the leading coefficient

a

by substituting the

y

-intercept

(0, 12)

into the equation.

a = -1

Substituting

x = 0

and

y = 12

gives

12 = a(0 + 6)(0 - 2)

, which simplifies to

12 = -12a

, so

a = -1

.

4

Determine the

x

-coordinate of the vertex.

x = -2

The axis of symmetry of a parabola lies exactly midway between its

x

-intercepts:

x = \frac{-6 + 2}{2} = -2

.

5

Calculate the maximum

y

-value by evaluating the function at the

x

-coordinate of the vertex.

y = 16

Substituting

x = -2

into

y = -(x + 6)(x - 2)

gives

y = -(-2 + 6)(-2 - 2) = -(4)(-4) = 16

.

Key Concept

Using

x

-intercepts and coordinate geometry to find the vertex of a parabola.

Question 1691Question →

A researcher models the population of a certain species of bacteria in a culture. The population $N(t)$ of bacteria $t$ hours after the culture is established is modeled by the function

N(t) = N_0 \cdot b^{\frac{t}{3}}

where $N_0$ is the initial population of the bacteria and $b$ is a constant. If the population of the bacteria increases by $44\%$ every 2 hours, what is the value of $b$ ?

1.200

1.275

1.660

1.728

Show answer & explanation

Answer: 1.728

Answer

1.728

The correct answer is

1.728

. Since the population increases by

44\%

every 2 hours, the population at

t = 2

is

1.44

times the initial population, meaning

N(2) = 1.44 \cdot N_0

. Substituting

t = 2

into the model

N(t) = N_0 \cdot b^{\frac{t}{3}}

yields

N_0 \cdot b^{\frac{2}{3}} = 1.44 \cdot N_0

. Dividing both sides by

N_0

gives

b^{\frac{2}{3}} = 1.44

. Raising both sides to the power of

\frac{3}{2}

isolates

b

as

b = (1.44)^{\frac{3}{2}}

. Since

1.44 = (1.2)^2

, we can simplify this expression using exponent rules:

b = (1.2^2)^{\frac{3}{2}} = 1.2^3 = 1.728

.

Step-by-Step Solution

1

Write the equation relating the population at time

t = 2

hours to the initial population at

t = 0

hours using the given percentage increase.

N(2) = 1.44 \cdot N_0

An increase of

44\%

means the population becomes

100\% + 44\% = 144\%

of its initial value, which corresponds to multiplying by a factor of

1.44

.

2

Substitute the function definition

N(t) = N_0 \cdot b^{\frac{t}{3}}

into the equation for

t = 2

.

N_0 \cdot b^{\frac{2}{3}} = 1.44 \cdot N_0

This allows us to set up an equation to solve for the constant

b

.

3

Divide both sides of the equation by

N_0

and isolate

b

.

b^{\frac{2}{3}} = 1.44

Since the initial population

N_0

is positive, we can divide both sides by

N_0

to isolate the exponential base term.

4

Solve for

b

by raising both sides of the equation to the power of

\frac{3}{2}

.

b = (1.44)^{\frac{3}{2}}

To solve for

b

, we multiply the exponent

\frac{2}{3}

by its reciprocal

\frac{3}{2}

.

5

Evaluate the expression

(1.44)^{\frac{3}{2}}

using exponent properties.

b = (1.2^2)^{\frac{3}{2}} = 1.2^3 = 1.728

Expressing

1.44

as

1.2^2

allows us to simplify the fractional exponent using the power of a power rule,

(x^a)^b = x^{ab}

.

Key Concept

Solving exponential equations by applying exponent rules and interpreting exponential growth factors in context.

Question 1692Question →

In 1948, physicist Richard Feynman introduced a pictorial representation for the mathematical expressions governing the behavior of subatomic particles. These diagrams served a vital _______ complex quantum field theory calculations that had previously taken weeks to complete.

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

purpose: they simplified

purpose, they simplified

purpose; which simplified

purpose they simplified

Show answer & explanation

Answer: purpose: they simplified

Answer

purpose: they simplified

The option that uses a colon to connect the independent clauses is correct. The clause before the colon ('These diagrams served a vital purpose') is grammatically independent. The clause after the colon ('they simplified complex quantum field theory calculations...') is also independent and directly explains the purpose. Standard English conventions allow a colon to link two independent clauses when the second clause explains or illustrates the first.

Step-by-Step Solution

1

Identify the structure of the clauses surrounding the blank.

The first part of the sentence ('These diagrams served a vital purpose') is an independent clause. The second part ('they simplified complex quantum field theory calculations...') is also an independent clause that explains the first.

Determining clause independence is necessary to choose the correct punctuation.

2

Evaluate the punctuation rules for joining these clauses.

A colon can join two independent clauses when the second clause explains or expands upon the first. A semicolon cannot be followed by a dependent clause (such as 'which simplified...'), a comma alone cannot join two independent clauses, and omitting punctuation entirely creates a run-on sentence.

Applying conventions of Standard English to select the correct punctuation mark.

Key Concept

Using colons to connect independent clauses where the second explains the first.

Question 1693Question →

In the 1930s, agricultural chemist George Washington Carver demonstrated that crop rotation could restore nitrogen to depleted soil. Carver encouraged local farmers to plant peanuts and sweet potatoes, arguing that these ________ cultivation would not only improve soil health but also provide a reliable source of income.

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

crops

crop's

crops'

crop

Show answer & explanation

Answer: crops'

Answer

crops'

The plural possessive noun 'crops'' is correct because it agrees with the plural demonstrative pronoun 'these' (referring to 'peanuts and sweet potatoes') and properly shows possession over the noun 'cultivation'.

Step-by-Step Solution

1

Determine if the noun should be singular or plural.

Plural

The demonstrative pronoun 'these' refers to the plural nouns 'peanuts and sweet potatoes', requiring a plural noun ('crops') to follow it.

2

Determine if the noun should be possessive.

Possessive

The noun must modify 'cultivation' to show that the cultivation belongs to the crops, requiring the possessive form ('crops'').

Key Concept

Plural and Possessive Nouns and Pronouns

Estimated Time:1m 0s

Question 1694Question →

African elephants communicate over vast distances using infrasound, which consists of low-frequency sounds below the limit of human hearing. These low-frequency vibrations travel through the ground and can be detected by other elephants through their sensitive ______ this seismic messaging allows distant herds to coordinate their movements, locate water sources, and avoid potential threats.

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

feet; this

feet, this

feet this

feet, but this

Show answer & explanation

Answer: feet; this

Answer

feet; this

The correct answer successfully links two independent clauses. The first clause ends with the word 'feet' and the second begins with 'this'. In Standard Written English, two independent clauses can be joined by a semicolon without a coordinating conjunction.

Step-by-Step Solution

1

Identify the clause boundaries and types in the sentence.

The text contains two independent clauses: 'These low-frequency vibrations travel through the ground and can be detected by other elephants through their sensitive feet' and 'this seismic messaging allows distant herds to coordinate...'

Determining whether the clauses are independent or dependent is necessary to apply the correct punctuation rules.

2

Determine the appropriate punctuation or coordinating conjunction to link two independent clauses.

Two independent clauses must be joined by a period, a semicolon, or a comma paired with a coordinating conjunction (such as 'and', 'but', or 'so').

Using only a comma or no punctuation at all creates a grammatical error.

3

Evaluate the choices based on grammatical correctness and logical relationship.

The choice using a semicolon to link the independent clauses is grammatically correct. The choice with only a comma creates a comma splice, the choice with no punctuation creates a run-on, and the choice with 'but' is logically incorrect because there is no contrast between the clauses.

Eliminating incorrect grammatical structures and illogical transition words yields the only correct answer.

Key Concept

Clause Boundaries and Linking

Question 1695Question →

The population of a certain species of plankton in a lake is modeled by the function

P(t) = P_0 \cdot (1.69)^{\frac{t}{d}}

where

P_0

is the initial population,

t

is the time, in days, since the population was first measured, and

d

is a positive constant. The population of the plankton increases by

30\%

every

h

hours, where

h

is a positive constant. If the model is accurate, what is the value of the ratio

\frac{h}{d}

?

0.5

2

12

48

Show answer & explanation

Answer: 12

Answer

The correct value of the ratio is 12.

The correct answer shows the value of 12. Since

t

is measured in days, the time interval of

h

hours must be converted to

\frac{h}{24}

days. A

30\%

increase corresponds to a multiplier of

1.3

. Substituting

t = \frac{h}{24}

into the function gives

(1.69)^{\frac{h}{24d}} = 1.3

. Rewriting

1.69

as

1.3^2

yields

\left(1.3^2\right)^{\frac{h}{24d}} = 1.3

, which simplifies to

1.3^{\frac{h}{12d}} = 1.3^1

. Equating the exponents gives

\frac{h}{12d} = 1

, which solves to

\frac{h}{d} = 12

.

Step-by-Step Solution

1

Express the growth interval in days to match the unit of

t

in the model.

Since there are 24 hours in a day, an interval of

h

hours is equivalent to

\frac{h}{24}

days.

The variable

t

in the model

P(t) = P_0 \cdot (1.69)^{\frac{t}{d}}

represents time in days, so all time variables must be in days.

2

Relate the percentage increase to the population multiplier.

An increase of

30\%

corresponds to a multiplier of

1 + 0.30 = 1.3

.

A

30\%

increase means the new population is

130\%

of the previous population.

3

Set up the equation using the given model at

t = \frac{h}{24}

.

P_0 \cdot (1.69)^{\frac{h}{24d}} = 1.3 \cdot P_0

, which simplifies to

(1.69)^{\frac{h}{24d}} = 1.3

.

At

t = \frac{h}{24}

days, the population must be

1.3

times the initial population

P_0

.

4

Rewrite the base of the exponential term and solve for the ratio.

Since

1.69 = 1.3^2

, the equation becomes

\left(1.3^2\right)^{\frac{h}{24d}} = 1.3^1

, which simplifies to

1.3^{\frac{2h}{24d}} = 1.3^1

, leading to

\frac{h}{12d} = 1

.

Expressing both sides with the same base allows for equating their exponents to solve for the target ratio.

5

Solve for the final ratio

\frac{h}{d}

.

\frac{h}{d} = 12

.

Multiplying both sides of the equation

\frac{h}{12d} = 1

by 12 isolates the ratio

\frac{h}{d}

.

Key Concept

Exponential Equations with Base Manipulation and Unit Conversion

Alternative Method

Instead of substituting a specific value for

t

, we can equate the daily growth factors. The model's daily growth factor is

(1.69)^{1/d}

. A

30\%

increase every

h

hours corresponds to a factor of

1.3

every

h

hours. Since there are

24

hours in a day, there are

\frac{24}{h}

such intervals in a day, yielding a daily growth factor of

(1.3)^{24/h}

. Setting the factors equal:

(1.69)^{1/d} = (1.3)^{24/h} \implies (1.3^2)^{1/d} = (1.3)^{24/h} \implies \frac{2}{d} = \frac{24}{h} \implies \frac{h}{d} = 12

.

Estimated Time:3m 0s

Question 1696Question →

The value of a certain investment is modeled by the function $f(t) = 500(1.21)^{\frac{t}{2}}$ , where $f(t)$ represents the value of the investment, in dollars, $t$ years after the initial investment was made. Which of the following is the annual growth rate, as a percentage, of the investment?

21\%

10\%

10.5\%

46.41\%

Show answer & explanation

Answer: $10\%$

Answer

The annual growth rate of the investment is

10\%

.

To find the annual growth rate, we need to rewrite the function

f(t) = 500(1.21)^{\frac{t}{2}}

in the form

f(t) = P(1 + r)^t

, where

P

is the initial value,

1 + r

is the annual growth factor, and

r

is the annual growth rate. Applying the power of a power rule for exponents, we can rewrite the expression as

f(t) = 500\left(1.21^{\frac{1}{2}}\right)^t

. Since raising a number to the power of

\frac{1}{2}

is equivalent to taking the square root, this simplifies to

f(t) = 500(\sqrt{1.21})^t = 500(1.1)^t

. The annual growth factor is

1.1

, which can be written as

1 + 0.1

. Therefore, the annual growth rate

r

is

0.1

, which corresponds to

10\%

.

Step-by-Step Solution

1

Identify the given exponential function and rewrite the expression to isolate the annual growth factor.

The function is

f(t) = 500(1.21)^{\frac{t}{2}}

. Using exponent rules, we can rewrite the term

(1.21)^{\frac{t}{2}}

as

\left(1.21^{\frac{1}{2}}\right)^t

.

Isolating the variable

t

in the exponent allows us to identify the annual growth factor.

2

Calculate the base of the exponential term.

1.21^{\frac{1}{2}} = \sqrt{1.21} = 1.1

. Thus, the rewritten function is

f(t) = 500(1.1)^t

.

Simplifying the base gives us the actual growth factor per unit of time (per year).

3

Determine the percentage growth rate from the growth factor.

The growth factor is

1.1

, which is equal to

1 + 0.1

. The growth rate

r

is

0.1

. Converting

0.1

to a percentage gives

0.1 \times 100\% = 10\%

.

The growth rate is the decimal increase per year, which we convert to a percentage.

Key Concept

Manipulating exponential expressions to identify rates of growth or decay.

Question 1697Question →

A landscaping company sells two types of soil mixtures: a basic mixture and a premium mixture. Each bag of basic mixture contains $4$ pounds of compost and $8$ pounds of sand. Each bag of premium mixture contains $6$ pounds of compost and $5$ pounds of sand. A landscaper purchases a combination of bags containing a total of $46$ pounds of compost and $57$ pounds of sand. How many bags of premium mixture did the landscaper purchase?

Show answer & explanation

Answer: 5

Answer

5

The system of equations representing the scenario is

4b + 6p = 46

for compost and

8b + 5p = 57

for sand, where

b

represents the number of bags of basic mixture and

p

represents the number of bags of premium mixture. Multiplying the first equation by

2

gives

8b + 12p = 92

. Subtracting the second equation from this yields

(8b + 12p) - (8b + 5p) = 92 - 57

, which simplifies to

7p = 35

. Dividing by

7

gives

p = 5

. Thus, the landscaper purchased

5

bags of premium mixture.

Step-by-Step Solution

1

Define variables and write the system of equations based on the given context.

Let

b

represent the number of basic mixture bags and

p

represent the number of premium mixture bags. The system is:

\begin{aligned} 4b + 6p &= 46 \\ 8b + 5p &= 57 \end{aligned}

Translating the word problem into a system of linear equations is necessary to solve for the unknowns.

2

Multiply the first equation by

2

to facilitate the elimination method.

8b + 12p = 92

Aligning the coefficients of

b

allows us to eliminate

b

by subtracting the two equations.

3

Subtract the second equation from the new equation.

(8b + 12p) - (8b + 5p) = 92 - 57 \implies 7p = 35

This isolates the variable

p

by eliminating the variable

b

.

4

Solve for

p

.

p = 5

Dividing both sides of the equation by

7

gives the final number of premium mixture bags.

Key Concept

Solving systems of linear equations in context

Question 1698Question →

For all positive values of $x$ , which of the following is equivalent to the expression $3x^{\frac{1}{2}} (2x^{\frac{3}{2}} - x^{-\frac{1}{2}})$ ?

6x^{\frac{3}{4}} - 3

6x^2 - 3x

6x^2 - 3

5x^2 - 3

Show answer & explanation

Answer: $6x^2 - 3$

Answer

6x^2 - 3

Distributing the term

3x^{\frac{1}{2}}

to both terms inside the parentheses yields

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

. Multiplying the coefficients and adding the exponents according to the rule

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

gives

6x^{\frac{1}{2} + \frac{3}{2}} - 3x^{\frac{1}{2} - \frac{1}{2}} = 6x^2 - 3x^0

. Since

x^0 = 1

for any positive

x

, the simplified equivalent expression is

6x^2 - 3

.

Step-by-Step Solution

1

Distribute the term

3x^{\frac{1}{2}}

to both terms inside the parentheses.

3x^{\frac{1}{2}} \cdot 2x^{\frac{3}{2}} - 3x^{\frac{1}{2}} \cdot x^{-\frac{1}{2}}

Apply the distributive property

a(b - c) = ab - ac

to expand the expression.

2

Multiply the coefficients and apply the product rule for exponents,

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

, to each product.

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

When multiplying terms with the same base, keep the base and add the exponents.

3

Simplify the arithmetic in the exponents and evaluate the resulting terms.

6x^2 - 3

Since

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

and

\frac{1}{2} - \frac{1}{2} = 0

, the expression simplifies to

6x^2 - 3x^0

. Because

x > 0

,

x^0 = 1

, making the final expression

6x^2 - 3

.

Key Concept

Equivalent Algebraic Expressions

Alternative Method

Substitute a simple value for

x

, such as

x = 4

. The original expression evaluates to

3(4)^{1/2}(2(4)^{3/2} - 4^{-\frac{1}{2}}) = 3(2)(2(8) - \frac{1}{2}) = 6(16 - 0.5) = 6(15.5) = 93

. Evaluating the correct expression

6x^2 - 3

at

x = 4

yields

6(16) - 3 = 96 - 3 = 93

. Evaluating the other options at

x = 4

yields different values.

Estimated Time:1m 30s

Question 1699Question →

A commercial bakery uses a mixing bowl that initially contains some flour. A machine adds flour to the bowl at a constant rate. After the machine has been running for $8$ minutes, the total mass of the flour in the bowl is $14$ kilograms. After the machine has been running for $20$ minutes, the total mass of the flour in the bowl is $29$ kilograms. The relationship between the total mass of the flour in the bowl, $M$ , in kilograms, and the time the machine has been running, $t$ , in minutes, is linear. What is the mass of the flour, in kilograms, in the bowl before the machine starts running?

Show answer & explanation

Answer: 4

Answer

4

The relationship between the total mass of the flour,

M

, and the time,

t

, is linear and can be represented by the equation

M = mt + b

, where

m

is the rate at which flour is added and

b

is the initial mass of the flour in the bowl. Using the two given points,

(8, 14)

and

(20, 29)

, the slope

m

is calculated as

m = \frac{29 - 14}{20 - 8} = \frac{15}{12} = 1.25

kilograms per minute. Substituting

m = 1.25

and the point

(8, 14)

into the linear equation gives

14 = 1.25(8) + b

, which simplifies to

14 = 10 + b

. Solving for

b

yields

b = 4

. Therefore, the mass of the flour in the bowl before the machine starts running is

4

kilograms.

Step-by-Step Solution

1

Find the rate of change (slope) of the linear relationship.

The rate is

1.25

kilograms per minute.

The slope of a linear relationship represents the constant rate at which flour is added to the bowl.

2

Set up the linear equation and solve for the y-intercept.

The initial mass is

4

kilograms.

The y-intercept represents the initial mass of the flour in the bowl at

t = 0

minutes.

Key Concept

Interpreting the y-intercept of a linear relationship in context

Question 1700Question →

In the $xy$ -plane, the graph of a linear function $f$ passes through the points $(k, 2k + 3)$ and $(2k, 5k - 1)$ , where $k$ is a constant. If the $y$ -intercept of the graph of $f$ is $-9$ , what is the value of $k$ ?

Show answer & explanation

Answer: 16

Answer

The value of

k

is 16.

The value of

k

is 16 because when

k = 16

, the points on the graph are

(16, 35)

and

(32, 79)

. The slope of the line is

\frac{79 - 35}{32 - 16} = 2.75

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

y = 2.75x + b

. Using the point

(16, 35)

, we get

35 = 2.75(16) + b \implies 35 = 44 + b \implies b = -9

, which matches the given

y

-intercept of

-9

.

Step-by-Step Solution

1

Calculate the slope of the line in terms of

k

using the two given points

(k, 2k + 3)

and

(2k, 5k - 1)

.

The slope

m

is

\frac{3k - 4}{k}

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

.

2

Use the

y

-intercept of

-9

, which corresponds to the point

(0, -9)

, along with the point

(k, 2k + 3)

to write another expression for the slope.

The slope

m

is

\frac{2k + 12}{k}

.

The slope must be constant for all points on the line, so the slope between the

y

-intercept and one of the points must equal the slope between the two points.

3

Equate the two slope expressions and solve for

k

.

k = 16

Setting the two expressions for the slope equal to each other gives

\frac{3k - 4}{k} = \frac{2k + 12}{k}

. Multiplying by

k

on both sides yields

3k - 4 = 2k + 12

, which simplifies to

k = 16

.

Key Concept

Linear Functions and Graphs

Answer

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Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

Alternative Method

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

Alternative Method

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Common Mistakes

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Common Mistakes

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Common Mistakes

Alternative Method

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Key Concept

Common Mistakes