All practice questions

Question 1521Question →

The following passage is adapted from a study on cooperative breeding in birds. Which form of the noun 'helper' correctly completes the blank in the sentence to ensure it conforms to the conventions of Standard English?

Fill in the blanks below

In many avian species, cooperative breeding relies on helper birds who assist a breeding pair in raising offspring rather than establishing their own nests. These contributions—which typically include foraging for food, defending the territory from predators, and maintaining the nest structure—significantly increase the overall survival rate of the hatchlings while reducing the workload of the biological parents.

Show answer & explanation

Answer

helpers'

The correct answer is the plural possessive form 'helpers''. The context indicates that the contributions belong to multiple 'helper' birds mentioned in the preceding sentence, as signaled by the plural demonstrative adjective 'These'. To form the possessive of a plural noun ending in -s, an apostrophe is placed after the final -s.

Step-by-Step Solution

1

Determine the number (singular or plural) of the noun required by the context.

The noun must be plural because it is preceded by the plural demonstrative adjective 'These'. Therefore, the base noun is 'helpers'.

The modifier 'These' must agree in number with the noun it modifies.

2

Determine the case (possessive or non-possessive) of the noun required by the syntax.

The noun must be possessive to modify the noun 'contributions', showing that the contributions belong to the helpers. For a plural noun ending in 's', the possessive is formed by adding an apostrophe after the 's'.

A noun modifying another noun to show ownership or association must be in the possessive case.

Key Concept

Plural and Possessive Nouns

Estimated Time:1m 0s

Question 1522Question →

The graph of the quadratic function $g$ in the $xy$ -plane is a parabola with vertex $(2, -5)$ . Which of the following equations could define the function $g$ ?

g(x) = (x + 2)^2 - 5

g(x) = (x - 2)^2 - 5

g(x) = (x - 2)^2 + 5

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

Show answer & explanation

Answer: $g(x) = (x - 2)^2 - 5$

Answer

The equation

g(x) = (x - 2)^2 - 5

could define the function

g

.

The vertex form of a quadratic function is given by

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

, where the point

(h, k)

is the vertex of the parabola. We are given that the vertex is

(2, -5)

, which means

h = 2

and

k = -5

. Substituting these values into the vertex form equation gives

g(x) = a(x - 2)^2 - 5

. Letting

a = 1

yields the equation

g(x) = (x - 2)^2 - 5

.

Step-by-Step Solution

1

Identify the vertex form of a quadratic function.

The vertex form of a quadratic function is written as

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

, where

(h, k)

is the vertex of the parabola and

a

is a non-zero constant.

Recognizing the vertex form allows us to directly substitute the coordinates of the given vertex.

2

Substitute the given vertex coordinates into the vertex form equation.

Given the vertex is

(2, -5)

, we substitute

h = 2

and

k = -5

into the vertex form:

g(x) = a(x - 2)^2 - 5

. If we assume

a = 1

, this simplifies to

g(x) = (x - 2)^2 - 5

.

This matches one of the given choices to find the equation that could define the function.

Key Concept

Vertex form of a quadratic function

Estimated Time:45s

Question 1523Question →

For a constant $k$ , the circle $(x - 5)^2 + (y - 5)^2 = 18$ and the line $y = kx$ are graphed in the $xy$ -plane. For how many integer values of $k$ will the circle and the line intersect at exactly two points?

5

6

7

8

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6. By substituting the line equation

y = kx

into the circle equation, we obtain the quadratic equation

(k^2 + 1)x^2 - 10(k + 1)x + 32 = 0

. For the circle and line to intersect at exactly two points, this equation must have two distinct real roots, meaning its discriminant must be positive. This leads to the inequality

7k^2 - 50k + 7 < 0

, which factors as

(7k - 1)(k - 7) < 0

. The solution is the interval

\frac{1}{7} < k < 7

. The integer values of

k

in this range are

1, 2, 3, 4, 5,

and

6

, giving a total of 6 integers.

Step-by-Step Solution

1

Substitute the line equation

y = kx

into the circle equation

(x - 5)^2 + (y - 5)^2 = 18

.

(x - 5)^2 + (kx - 5)^2 = 18

Substitution reduces the system of equations to a single quadratic equation in

x

, representing the

x

-coordinates of the intersection points.

2

Expand the terms and write the equation in standard quadratic form

Ax^2 + Bx + C = 0

.

(k^2 + 1)x^2 - 10(k + 1)x + 32 = 0

Putting the equation in standard form is necessary to analyze its discriminant.

3

Set the discriminant

\Delta = b^2 - 4ac

strictly greater than zero to ensure exactly two distinct real solutions.

[-10(k+1)]^2 - 4(k^2 + 1)(32) > 0 \Rightarrow 7k^2 - 50k + 7 < 0

A quadratic equation has two distinct real roots if and only if its discriminant is positive.

4

Solve the quadratic inequality

7k^2 - 50k + 7 < 0

by factoring the quadratic expression.

\frac{1}{7} < k < 7

The roots of

(7k - 1)(k - 7) = 0

are

k = \frac{1}{7}

and

k = 7

. Since the leading coefficient is positive, the expression is negative between these roots.

5

Identify and count the integer values of

k

that lie within the interval

(\frac{1}{7}, 7)

.

The integers are

1, 2, 3, 4, 5, 6

, which gives a total of 6 values.

The integers that are strictly greater than

\frac{1}{7}

and strictly less than

7

are the whole numbers from

1

to

6

.

Key Concept

Solving systems consisting of a circle and a line algebraically by utilizing the quadratic discriminant to find the condition for two real intersection points.

Question 1524Question →

In the $xy$ -plane, the graph of the linear equation $y - 4 = m(x - 6)$ , where $m$ is a constant, has a $y$ -intercept of $(0, 7)$ . What is the $x$ -coordinate of the $x$ -intercept of the line?

-4

-2

14

8

Show answer & explanation

Answer: 14

Answer

The correct answer is 14. This is the x-coordinate of the point where the line crosses the x-axis, which is (14, 0).

To find the value of

m

, substitute the y-intercept

(0, 7)

into the equation:

7 - 4 = m(0 - 6)

, which simplifies to

3 = -6m

, so

m = -\frac{1}{2}

. The equation of the line is

y - 4 = -\frac{1}{2}(x - 6)

. The x-intercept occurs when

y = 0

. Substituting

y = 0

gives

-4 = -\frac{1}{2}(x - 6)

, which simplifies to

8 = x - 6

, so

x = 14

.

Step-by-Step Solution

1

Substitute the y-intercept

(0, 7)

into the given equation to solve for the constant

m

.

7 - 4 = m(0 - 6) \implies 3 = -6m \implies m = -\frac{1}{2}

The y-intercept point

(0, 7)

must satisfy the equation of the line, allowing us to find the slope

m

.

2

Write the linear equation using the calculated value of

m

.

y - 4 = -\frac{1}{2}(x - 6)

Knowing the slope allows us to write the complete linear equation in point-slope form.

3

Find the x-intercept of the line by setting

y = 0

and solving for

x

.

0 - 4 = -\frac{1}{2}(x - 6) \implies -4 = -\frac{1}{2}(x - 6) \implies 8 = x - 6 \implies x = 14

The x-intercept is the point on the line where the y-coordinate is equal to 0.

Key Concept

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

Alternative Method

Alternatively, write the equation in slope-intercept form

y = mx + b

. Since the y-intercept is

(0, 7)

, we have

b = 7

, so the equation is

y = mx + 7

. Since the line passes through the point

(6, 4)

(determined by the point-slope form

y - 4 = m(x - 6)

), substitute this point into the equation:

4 = m(6) + 7 \implies -3 = 6m \implies m = -\frac{1}{2}

. Thus, the equation is

y = -\frac{1}{2}x + 7

. Setting

y = 0

for the x-intercept gives

0 = -\frac{1}{2}x + 7 \implies \frac{1}{2}x = 7 \implies x = 14

.

Estimated Time:1m 30s

Question 1525Question →

Peatlands cover only three percent of the Earth's land surface, yet they store more carbon than all of the world's forests combined. However, when these ecosystems are drained or burned, their stored carbon is released back into the _______ conservationists are increasingly prioritizing peatland restoration as a critical strategy to mitigate global warming. Which choice completes the text so that it conforms to the conventions of Standard English?

atmosphere, therefore,

atmosphere therefore

atmosphere, so

atmosphere, because

Show answer & explanation

Answer: atmosphere, so

Answer

atmosphere, so

The option containing 'atmosphere, so' is correct because it uses the coordinating conjunction 'so' preceded by a comma to appropriately link two independent clauses. The first clause describes the release of carbon when peatlands are damaged, and the second clause describes the resulting conservation efforts.

Step-by-Step Solution

1

Identify the boundary between the two independent clauses in the sentence.

The first independent clause ends with 'atmosphere', and the second independent clause begins with 'conservationists'.

Recognizing where the independent clauses meet helps determine the correct punctuation and linking mechanism required.

2

Evaluate the grammar of the coordinating conjunction and punctuation options.

A coordinating conjunction like 'so' preceded by a comma is grammatically correct and logically connects the cause and effect.

Two independent clauses must be joined by a semicolon, a period, or a comma plus a coordinating conjunction.

Key Concept

Clause Boundaries and Linking

Question 1526Question →

The system of equations below consists of a linear equation and a quadratic equation:

\begin{aligned} y - 2x &= 5 \\ y &= x^2 - 3x - 1 \end{aligned}

Let

(x_1, y_1)

and

(x_2, y_2)

represent the two distinct real coordinate points where the graphs of these equations intersect. If

y_1 > y_2

, what is the value of

x_1 - x_2

?

Show answer & explanation

Answer: 7

Answer

7

Solving the system algebraically by expressing

y

as

2x + 5

and substituting it into the quadratic equation leads to

x^2 - 5x - 6 = 0

. Factoring this equation yields

x = 6

and

x = -1

. Evaluating these values in the linear equation gives the coordinates

(6, 17)

and

(-1, 3)

. Since

17 > 3

, the point

(6, 17)

corresponds to

(x_1, y_1)

and

(-1, 3)

corresponds to

(x_2, y_2)

. The difference

x_1 - x_2

is equal to

6 - (-1) = 7

.

Step-by-Step Solution

1

Solve for

y

in the linear equation to express it in terms of

x

.

y = 2x + 5

This allows for substitution into the quadratic equation to eliminate one variable.

2

Substitute

2x + 5

for

y

in the quadratic equation.

2x + 5 = x^2 - 3x - 1

To create a single-variable quadratic equation in terms of

x

.

3

Rearrange the equation by subtracting

2x

and

5

from both sides to set it equal to zero.

x^2 - 5x - 6 = 0

Setting the quadratic equation to zero is the standard first step to find its roots.

4

Factor the quadratic trinomial.

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

, which gives

x = 6

and

x = -1

.

Factoring finds the

x

-coordinates of the points of intersection.

5

Substitute the

x

-values back into the linear equation

y = 2x + 5

to determine the corresponding

y

-values.

For

x = 6

,

y = 2(6) + 5 = 17

. For

x = -1

,

y = 2(-1) + 5 = 3

. The intersection points are

(6, 17)

and

(-1, 3)

.

This defines the full coordinate pairs of the system's solutions.

6

Apply the condition

y_1 > y_2

to identify

(x_1, y_1)

and

(x_2, y_2)

.

Since

17 > 3

, the first solution is

(x_1, y_1) = (6, 17)

and the second solution is

(x_2, y_2) = (-1, 3)

.

This aligns the coordinate values with the variables defined in the problem constraint.

7

Calculate the value of

x_1 - x_2

.

6 - (-1) = 7

Subtracting a negative value is equivalent to adding its absolute value, yielding the final required answer.

Key Concept

Nonlinear Systems of Equations

Question 1527Question →

A manufacturer plans to produce two models of smartphones: Model X and Model Y. The assembly of each Model X smartphone requires $2.5$ hours, and the assembly of each Model Y smartphone requires $4$ hours. A total of $800$ hours of assembly time is allocated for producing these two models. If the manufacturer produces $120$ Model X smartphones, what is the maximum number of Model Y smartphones that can be produced using the remaining allocated assembly time?

125

128

150

275

Show answer & explanation

Answer: 125

Answer

The maximum number of Model Y smartphones that can be produced is 125.

The correct answer is the value that satisfies the linear relation under the given constraints. Setting up the equation

2.5x + 4y = 800

and substituting

120

for

x

results in

300 + 4y = 800

. Subtracting

300

from both sides yields

4y = 500

, and dividing by

4

gives

125

Model Y smartphones.

Step-by-Step Solution

1

Set up a linear equation representing the total assembly time.

2.5x + 4y = 800

, where

x

is the number of Model X smartphones and

y

is the number of Model Y smartphones.

The total assembly time is the sum of the time spent on Model X (

2.5

hours per phone) and Model Y (

4

hours per phone), which must equal the allocated

800

hours.

2

Substitute the given number of Model X smartphones into the equation.

2.5(120) + 4y = 800

, which simplifies to

300 + 4y = 800

.

The problem states that the manufacturer produces

120

Model X smartphones.

3

Isolate the variable representing Model Y smartphones to find its value.

4y = 500

, which simplifies to

y = 125

.

Subtracting

300

from both sides of the equation and then dividing by

4

isolates the variable

y

.

Key Concept

Linear Equations in Two Variables

Estimated Time:1m 30s

Question 1528Question →

In the system of equations below,

k

is a constant.

\begin{aligned} x^2 + y^2 + 8y &= 9 \\ x^2 + y &= k \end{aligned}

If the system has exactly 3 distinct real solutions

(x, y)

, what is the value of

k

?

Show answer & explanation

Answer: 1

Answer

The value of

k

is 1.

Substituting

x^2 = k - y

into

x^2 + y^2 + 8y = 9

gives

y^2 + 7y + k - 9 = 0

. For the system to have exactly 3 distinct real solutions, the vertex of the parabola

y = -x^2 + k

must lie on the circle, which corresponds to the root

y = k

. Substituting

y = k

into the quadratic equation yields

k^2 + 8k - 9 = 0

, which gives

k = 1

or

k = -9

. For

k = 1

, the roots of the quadratic are

y = 1

and

y = -8

. The root

y = 1

yields 1 real solution,

(0, 1)

, and the root

y = -8

yields 2 real solutions,

(3, -8)

and

(-3, -8)

, for a total of 3 real solutions. For

k = -9

, the roots are

y = -9

and

y = 2

. The root

y = 2

does not yield any real solutions for

x

because

2 > -9

, so the system has only 1 real solution. Thus,

k = 1

.

Step-by-Step Solution

1

Express

x^2

in terms of

y

and

k

from the second equation.

x^2 = k - y

(with constraint

y \le k

for real

x

)

To prepare for substitution into the first equation and establish the boundary condition for real solutions.

2

Substitute

x^2 = k - y

into the first equation.

y^2 + 7y + k - 9 = 0

To create a single quadratic equation in terms of

y

.

3

Analyze the conditions on the roots of the quadratic equation to get exactly 3 distinct real solutions.

One root must equal

k

and the other must be less than

k

.

A root

y = k

yields 1 real solution for

x

(

x = 0

), while a root

y < k

yields 2 real solutions (

x = \pm\sqrt{k-y}

).

4

Find the candidate values of

k

by setting

y = k

in the quadratic equation.

k^2 + 8k - 9 = 0 \implies k = 1

or

k = -9

To find the values of

k

where the parabola's vertex lies on the circle.

5

Verify which candidate value of

k

satisfies all conditions.

For

k = 1

, the roots are

y = 1

and

y = -8 < 1

(3 solutions). For

k = -9

, the roots are

y = -9

and

y = 2 > -9

(1 solution). Therefore,

k = 1

.

To ensure the second root is strictly less than

k

, guaranteeing exactly 3 solutions.

Key Concept

Analyzing the number of solutions in a nonlinear system of equations using algebraic substitution and boundary constraints.

Estimated Time:3m 0s

Question 1529Question →

An online retailer uses a linear relationship to determine the shipping cost of an order based on the total weight of the items in the order. The table below shows the shipping cost, $y$ , in dollars, for an order with a total weight of $x$ pounds.

Weight ( $x$ pounds)	Shipping Cost ( $y$ dollars)
$2$	$9.50$
$6$	$17.50$

What is the shipping cost, in dollars, for an order with a total weight of $11$ pounds?

Show answer & explanation

Answer: 27.5

Answer

The correct shipping cost for an order weighing 11 pounds is 27.5 dollars.

The linear relationship has a constant rate of change (slope). Using the coordinates from the table,

(2, 9.50)

and

(6, 17.50)

, the slope

m

is calculated as

\frac{17.50 - 9.50}{6 - 2} = \frac{8}{4} = 2

. Using the point-slope form, the equation of the line is

y - 9.50 = 2(x - 2)

, which simplifies to

y = 2x + 5.50

. Substituting

x = 11

gives

y = 2(11) + 5.50 = 27.5

.

Step-by-Step Solution

1

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

m = 2

Determines the rate at which the shipping cost increases per additional pound of weight.

2

Formulate the linear equation representing the relationship.

y = 2x + 5.50

Establishes a functional relationship to calculate costs for any given weight.

3

Evaluate the function at the target weight of 11 pounds.

y = 27.5

Determines the final cost of shipping for the specified weight of 11 pounds.

Key Concept

Determining and evaluating linear functions represented in tabular form

Estimated Time:1m 30s

Question 1530Question →

In the quadratic equation $3x^2 - kx + 24 = 0$ , $k$ is a positive constant. If one of the roots of the equation is twice the other root, what is the value of $k$ ?

6

12

18

36

Show answer & explanation

Answer: 18

Answer

18

The correct answer is 18. Let the roots of the equation be

r

and

2r

. The product of the roots is given by

\frac{c}{a} = \frac{24}{3} = 8

. Therefore,

2r^2 = 8

, which means

r^2 = 4

. Since

k > 0

, the roots must be positive, so

r = 2

and the two roots are 2 and 4. The sum of the roots is

2 + 4 = 6

. According to Vieta's formulas, the sum of the roots is also equal to

-\frac{b}{a} = \frac{k}{3}

. Setting the two expressions for the sum of the roots equal to each other gives

\frac{k}{3} = 6

, which simplifies to

k = 18

.

Step-by-Step Solution

1

Define the roots and set up the product of the roots using Vieta's formulas.

Let the roots be

r

and

2r

. The product of the roots is

r \times 2r = 2r^2

. From the equation

3x^2 - kx + 24 = 0

, the product of the roots is also

\frac{c}{a} = \frac{24}{3} = 8

.

This establishes a relationship between the given ratio of the roots and the coefficients of the equation.

2

Solve for the root variable

r

.

Setting

2r^2 = 8

gives

r^2 = 4

. Since

k

is a positive constant, the sum of the roots must be positive, which means the roots themselves must be positive. Thus,

r = 2

, and the roots are

2

and

4

.

Finding the actual values of the roots is necessary to calculate their sum.

3

Use Vieta's formula for the sum of the roots to find

k

.

The sum of the roots is

2 + 4 = 6

. According to Vieta's formulas, the sum of the roots is

-\frac{b}{a} = \frac{k}{3}

. Setting them equal gives

\frac{k}{3} = 6

, which simplifies to

k = 18

.

This directly isolates and solves for the unknown constant

k

.

Key Concept

Using Vieta's formulas to relate the roots of a quadratic equation

ax^2 + bx + c = 0

to its coefficients: the sum of the roots is

-\frac{b}{a}

and the product of the roots is

\frac{c}{a}

.

Question 1531Question →

For all $x > 2$ , which of the following is equivalent to the expression $\frac{3x^2 - 12}{x^2 - 2x} - \frac{2x^2 + 2x - 12}{x^2 + 3x}$ ?

\frac{x+2}{x}

\frac{x+10}{x}

\frac{x-12}{x}

\frac{x+14}{x}

Show answer & explanation

Answer: $\frac{x+10}{x}$

Answer

\frac{x+10}{x}

The correct answer is obtained by factoring both rational expressions and simplifying them before subtracting. The first term factors into

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

, which simplifies to

\frac{3(x+2)}{x}

. The second term factors into

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

, which simplifies to

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

. Subtracting these two expressions yields

\frac{3(x+2) - 2(x-2)}{x} = \frac{3x+6-2x+4}{x} = \frac{x+10}{x}

.

Step-by-Step Solution

1

Factor the numerators and denominators of both rational expressions.

The expression becomes

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

.

Factoring helps identify common binomial factors in the numerator and denominator.

2

Simplify both terms by canceling common factors.

For

x > 2

, the expression simplifies to

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

.

Since

x > 2

, the terms

x-2

and

x+3

are non-zero and can be canceled.

3

Combine the simplified terms over the common denominator

x

.

\frac{3(x+2) - 2(x-2)}{x} = \frac{3x + 6 - (2x - 4)}{x}

Both terms share the common denominator

x

.

4

Distribute the negative sign in the numerator and combine like terms.

\frac{3x + 6 - 2x + 4}{x} = \frac{x+10}{x}

Distributing subtraction to the terms inside

(2x-4)

yields

-2x + 4

.

Key Concept

Simplifying and subtracting rational expressions by factoring and finding a common denominator

Question 1532Question →

Company A and Company B both rent moving trucks. The total cost $A(d)$ , in dollars, for renting a truck from Company A for $d$ days is given by $A(d) = 35d + 75$ . The table below shows some values of the total cost $B(d)$ , in dollars, for renting a truck from Company B for $d$ days.

Days ( $d$ )	Total Cost ( $B(d)$ )
2	150
5	300

If the relationship between $d$ and $B(d)$ is linear, for how many days of rental will the total cost at Company B be exactly $95$ dollars more than the total cost at Company A?

2

3

5

8

Show answer & explanation

Answer: 8

Answer

The total cost at Company B will be exactly 95 dollars more than the total cost at Company A for a rental of 8 days.

To find the number of days for which Company B's cost is exactly 95 dollars more than Company A's cost, we first find the linear function for Company B. The rate of change is

\frac{300 - 150}{5 - 2} = 50

dollars per day. The initial cost is

150 - 50(2) = 50

dollars, giving

B(d) = 50d + 50

. We then set up the equation

B(d) - A(d) = 95

, which is

(50d + 50) - (35d + 75) = 95

. Simplifying this equation gives

15d - 25 = 95

. Adding 25 to both sides gives

15d = 120

, and dividing by 15 gives

d = 8

.

Step-by-Step Solution

1

Determine the linear cost function for Company B,

B(d) = md + b

, using the points

(2, 150)

and

(5, 300)

from the table.

The slope is

m = \frac{300 - 150}{5 - 2} = 50

. Using

(2, 150)

to find the y-intercept:

150 = 50(2) + b \Rightarrow b = 50

. Thus,

B(d) = 50d + 50

.

We need to find the equation representing the total cost of Company B to set up the comparison.

2

Set up an equation where the total cost at Company B is equal to the total cost at Company A plus 95 dollars.

B(d) = A(d) + 95 \Rightarrow 50d + 50 = (35d + 75) + 95

.

This represents the condition that Company B's cost is 95 dollars more than Company A's cost.

3

Simplify the equation and solve for the number of days,

d

.

50d + 50 = 35d + 170 \Rightarrow 15d = 120 \Rightarrow d = 8

.

Isolating the variable

d

gives the exact number of days required.

Key Concept

Comparing linear functions and solving linear equations representing real-world contexts.

Question 1533Question →

The function $f$ is defined by $f(x) = a \cdot 3^x$ , where $a$ is a constant. If $f(2) = 45$ , what is the value of $f(1)$ ?

Show answer & explanation

Answer: 15

Answer

15

To find the value of

f(1)

, we first determine the value of the constant

a

. We are given that

f(2) = 45

, so substituting

x = 2

into the function definition

f(x) = a \cdot 3^x

gives

45 = a \cdot 3^2

. Simplifying

3^2

to

9

yields

45 = 9a

, which means

a = 5

. Now that we know

a = 5

, we can write the function as

f(x) = 5 \cdot 3^x

. To find

f(1)

, we substitute

x = 1

into this equation, yielding

f(1) = 5 \cdot 3^1 = 15

.

Step-by-Step Solution

1

Substitute the given point

(2, 45)

into the function equation to solve for

a

.

a = 5

Since

f(2) = 45

, we have

45 = a \cdot 3^2 = 9a

, which gives

a = 5

.

2

Evaluate the function at

x = 1

using the value of

a = 5

.

f(1) = 15

Substituting

a = 5

and

x = 1

into

f(x) = a \cdot 3^x

gives

f(1) = 5 \cdot 3^1 = 15

.

Key Concept

Evaluating and solving exponential functions given initial conditions or points.

Question 1534Question →

A deep-sea research submersible's internal cabin pressure, $P$ , in atmospheres ( $\text{atm}$ ), is modeled by a linear function of its depth below the ocean surface, $d$ , in meters. At the surface ( $d = 0$ ), the internal pressure is $1.0\text{ atm}$ . For every increase in depth of $100\text{ meters}$ , the internal pressure increases by $0.05\text{ atm}$ . The submersible descends from the surface at a constant rate of $2.5\text{ meters per second}$ . What is the rate of increase of the internal cabin pressure, in $\text{atm}$ per hour, as the submersible descends?

Show answer & explanation

Answer: 4.5

Answer

The rate of increase of the internal cabin pressure is

4.5

atmospheres per hour.

The rate of change of internal pressure with depth is

\frac{0.05\text{ atm}}{100\text{ m}} = 0.0005\text{ atm/m}

. Since the submersible descends at a rate of

2.5\text{ m/s}

, we convert this rate of depth change to hours:

2.5\text{ m/s} \times 3600\text{ s/hr} = 9000\text{ m/hr}

. Multiplying the pressure change per meter by the depth change per hour gives the rate of change of the internal pressure per hour:

0.0005\text{ atm/m} \times 9000\text{ m/hr} = 4.5\text{ atm/hr}

.

Step-by-Step Solution

1

Calculate the rate of pressure increase per meter of depth.

0.0005\text{ atm/m}

The pressure increases by

0.05\text{ atm}

for every

100\text{ meters}

of depth, giving a rate of change of

\frac{0.05\text{ atm}}{100\text{ m}} = 0.0005\text{ atm/m}

.

2

Determine the distance descended by the submersible in one hour.

9000\text{ meters}

With

3600\text{ seconds}

in one hour and a descent speed of

2.5\text{ m/s}

, the submersible descends a total of

2.5\text{ m/s} \times 3600\text{ s} = 9000\text{ meters}

in one hour.

3

Calculate the rate of internal pressure increase per hour.

4.5\text{ atm/hr}

Multiply the rate of change of pressure per meter (

0.0005\text{ atm/m}

) by the hourly descent distance (

9000\text{ m}

) to obtain

0.0005 \times 9000 = 4.5\text{ atm/hr}

.

Key Concept

Interpreting Linear Relationships in Context

Alternative Method

Write the cabin pressure as a function of time

t

, in seconds:

P(t) = 1.0 + 0.0005(2.5t) = 1.0 + 0.00125t

. The slope of this line,

0.00125\text{ atm/s}

, represents the rate of increase per second. To convert this rate to hours, multiply by

3600\text{ seconds/hour}

:

0.00125 \times 3600 = 4.5\text{ atm/hr}

.

Estimated Time:2m 0s

Question 1535Question →

In temperate forests, mycorrhizal fungi form vast underground networks that connect the roots of distant trees. Through these subterranean pathways, mature trees can share vital nutrients with shaded seedlings that would otherwise struggle to _______ this cooperative relationship helps maintain the overall health and resilience of the entire forest ecosystem.

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

survive, this

survive this

survive; this

survive, but this

Show answer & explanation

Answer: survive; this

Answer

survive; this

The correct option is the choice that uses a semicolon to separate the two independent clauses. A semicolon is a standard punctuation mark used to link two closely related independent clauses when a coordinating conjunction is not present.

Step-by-Step Solution

1

Identify the boundaries of the clauses in the sentence.

The sentence contains two independent clauses: the first ends at the word 'survive' and the second starts with 'this'.

Both clauses possess their own subjects and verbs and can stand alone as complete thoughts.

2

Determine the logical relationship between the two independent clauses.

The second clause explains and expands upon the significance of the tree cooperation mentioned in the first clause.

Understanding this connection helps rule out contrastive conjunctions.

3

Select the punctuation that correctly links two independent clauses without introducing grammatical errors.

A semicolon is the appropriate punctuation mark to link these two closely related independent clauses without any coordinating conjunction.

Using a semicolon avoids both run-on sentences and comma splices.

Key Concept

Clause Boundaries and Linking

Question 1536Question →

In the $xy$ -plane, the graph of the linear equation $3x + 4y = k$ , where $k$ is a positive constant, is tangent to the circle with equation $x^2 + y^2 - 2x - 4y = 20$ . What is the value of $k$ ?

Show answer & explanation

Answer: 36

Answer

The value of the constant

k

is

36

.

To find the value of

k

where the line

3x + 4y = k

is tangent to the circle, we first rewrite the circle's equation

x^2 + y^2 - 2x - 4y = 20

in standard form by completing the square:

(x^2 - 2x + 1) + (y^2 - 4y + 4) = 20 + 1 + 4

, which simplifies to

(x - 1)^2 + (y - 2)^2 = 25

. This is a circle centered at

(1, 2)

with a radius of

5

. A line is tangent to a circle if the perpendicular distance from the center of the circle to the line is equal to the radius of the circle. Using the distance formula

\frac{|Ax_0 + By_0 - C|}{\sqrt{A^2 + B^2}}

for the point

(1, 2)

and the line

3x + 4y - k = 0

, we set up the equation:

\frac{|3(1) + 4(2) - k|}{\sqrt{3^2 + 4^2}} = 5

. This simplifies to

\frac{|11 - k|}{5} = 5

, or

|11 - k| = 25

. Solving the absolute value equation gives

11 - k = 25 \Rightarrow k = -14

and

11 - k = -25 \Rightarrow k = 36

. Since the problem specifies that

k

is a positive constant, the correct answer is

36

.

Step-by-Step Solution

1

Rewrite the circle's equation by completing the square for both variables.

(x - 1)^2 + (y - 2)^2 = 25

To identify the center and radius of the circle.

2

Identify the center and radius from the standard form of the circle's equation.

Center is

(1, 2)

and radius is

5

.

The standard form of a circle is

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

.

3

Apply the condition for tangency: the perpendicular distance from the center

(1, 2)

to the line

3x + 4y - k = 0

must equal the radius

5

.

\frac{|3(1) + 4(2) - k|}{\sqrt{3^2 + 4^2}} = 5

A line is tangent to a circle if and only if the distance from the center to the line equals the radius.

4

Simplify the distance equation and solve the resulting absolute value equation for

k

.

|11 - k| = 25

, yielding

k = -14

or

k = 36

.

To find all mathematically possible values of the constant

k

.

5

Select the positive value for

k

as specified by the problem constraints.

k = 36

The problem states that

k

is a positive constant.

Key Concept

The relationship between a line and a circle in a nonlinear system, specifically using the distance from the center to a tangent line to solve for an unknown constant.

Question 1537Question →

During a chemistry experiment, the temperature of a liquid increases at a constant rate. The temperature of the liquid is $24^\circ\text{C}$ at $5$ minutes after the experiment starts, and $48^\circ\text{C}$ at $15$ minutes after the experiment starts. The temperature $T$ , in degrees Celsius, of the liquid $t$ minutes after the experiment starts can be modeled by the equation $T = mt + b$ , where $m$ and $b$ are constants. What is the value of $b$ ?

Show answer & explanation

Answer: 12

Answer

12

The correct answer is 12. By translating the given information into two points on the line,

(5, 24)

and

(15, 48)

, we can find the slope

m = \frac{48 - 24}{15 - 5} = 2.4

. Substituting this slope and the point

(5, 24)

into

T = mt + b

yields

24 = 2.4(5) + b

, which simplifies to

24 = 12 + b

. Solving for

b

gives

b = 12

.

Step-by-Step Solution

1

Identify the data points representing the relationship between time and temperature.

The two data points are

(t_1, T_1) = (5, 24)

and

(t_2, T_2) = (15, 48)

.

These points represent coordinates

(t, T)

on the line representing the temperature over time.

2

Calculate the slope

m

of the linear equation.

m = \frac{48 - 24}{15 - 5} = \frac{24}{10} = 2.4

The slope represents the constant rate of temperature increase per minute.

3

Substitute the slope

m

and the coordinates of one point into the equation

T = mt + b

to solve for the y-intercept

b

.

24 = 2.4(5) + b \implies 24 = 12 + b \implies b = 12

This determines the value of the constant

b

, which corresponds to the initial temperature of the liquid.

Key Concept

Determining the y-intercept of a linear model in two variables given two points.

Question 1538Question →

If $3^{x+1} = k$ , where $k > 0$ , which of the following is equivalent to $27^{x-1}$ ?

\frac{k^3}{729}

\frac{k^3}{81}

\frac{k^3}{27}

\frac{k^3}{9}

Show answer & explanation

Answer: $\frac{k^3}{729}$

Answer

The expression \frac{k^3}{729}

To express

27^{x-1}

in terms of

k

, we can rewrite both the given equation and the target expression using base

3

. The given equation

3^{x+1} = k

can be adjusted to find the value of

3^{x-1}

by dividing both sides by

3^2

, which yields

3^{x-1} = \frac{k}{9}

. The target expression

27^{x-1}

can be rewritten as

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

. Substituting

3^{x-1} = \frac{k}{9}

into this expression gives

\left(\frac{k}{9}\right)^3 = \frac{k^3}{729}

.

Step-by-Step Solution

1

Express the given equation in terms of a simpler base

3

exponent.

3^{x-1} = \frac{k}{9}

We start with the given equation

3^{x+1} = k

. To relate this to the exponent

x-1

, we divide both sides of the equation by

3^2 = 9

:

3^{x-1} = \frac{3^{x+1}}{3^2} = \frac{k}{9}

.

2

Rewrite the target expression

27^{x-1}

with base

3

.

27^{x-1} = (3^{x-1})^3

Since

27 = 3^3

, we can rewrite the target expression as

27^{x-1} = (3^3)^{x-1} = 3^{3(x-1)} = (3^{x-1})^3

.

3

Substitute the expression for

3^{x-1}

from Step 1 into the expression from Step 2.

\frac{k^3}{729}

Substituting

3^{x-1} = \frac{k}{9}

into

(3^{x-1})^3

gives

\left(\frac{k}{9}\right)^3 = \frac{k^3}{9^3} = \frac{k^3}{729}

.

Key Concept

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

Question 1539Question →

An online store sells two types of monthly subscription plans: a basic plan for $12 per month and a premium plan for$ 20 per month. In April, the store had a total of 150 active subscribers for these two plans and collected a total of $2,280 in subscription fees. How many of the active subscribers in April were on the premium plan?

15

50

60

90

Show answer & explanation

Answer: 60

Answer

60

By representing the number of basic subscribers as

x

and premium subscribers as

y

, we set up the system of equations

x + y = 150

and

12x + 20y = 2280

. Solving for

x

in the first equation gives

x = 150 - y

. Substituting this into the second equation yields

12(150 - y) + 20y = 2280

. Distributing the 12 and combining like terms gives

1800 + 8y = 2280

. Subtracting 1800 from both sides gives

8y = 480

, and dividing by 8 results in

y = 60

. Therefore, there were 60 premium subscribers.

Step-by-Step Solution

1

Define variables for the unknown quantities: let

x

represent the number of basic subscribers and let

y

represent the number of premium subscribers.

The system of equations is:

x + y = 150

and

12x + 20y = 2280

.

Translate the given information about total subscribers and total revenue into algebraic equations.

2

Express

x

in terms of

y

using the first equation (

x = 150 - y

) and substitute it into the second equation.

The substituted equation is:

12(150 - y) + 20y = 2280

, which expands to

1800 - 12y + 20y = 2280

.

Eliminate one variable to create a single equation with only one variable,

y

.

3

Simplify the equation and isolate

y

.

1800 + 8y = 2280 \implies 8y = 480 \implies y = 60

.

Combine like terms and use basic operations to solve for the number of premium subscribers.

Key Concept

Solving a system of two linear equations using substitution or elimination.

Estimated Time:1m 30s

Question 1540Question →

To survive in the arid environment of the Sahara Desert, the Saharan silver ant (*Cataglyphis bombycina*) utilizes unique physiological adaptations to forage during the hottest parts of the day. While most other desert creatures must seek shade to avoid overheating, this _____ dense, triangular-shaped silvery hairs reflect both visible and near-infrared sunlight, keeping the insect remarkably cool.

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

ant

ants

ant's

ants'

Show answer & explanation

Answer: ant's

Answer

ant's

The correct choice is the singular possessive form 'ant's'. The singular demonstrative adjective 'this' indicates that the noun must be singular. Additionally, because the noun modifies 'dense, triangular-shaped silvery hairs' to show ownership or association, it must be in the possessive form.

Step-by-Step Solution

1

Determine whether the noun needs to be singular or plural.

Singular

The noun is preceded by the singular demonstrative adjective 'this', and the subsequent sentence uses the singular pronoun 'it' ('allowing it to forage'), indicating a singular subject.

2

Determine whether the noun needs to be possessive or non-possessive.

Possessive

The noun modifies the noun phrase 'dense, triangular-shaped silvery hairs', indicating that the hairs belong to the ant.

3

Combine the singular and possessive requirements to select the correct grammatical form.

ant's

The spelling 'ant's' represents the singular possessive form of the noun.

Key Concept

Plural and Possessive Nouns and Pronouns

Estimated Time:1m 0s

Answer

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