All practice questions

If $x$ and $y$ are positive numbers with $x > 1$ that satisfy the system of equations below, what is the value of $x$ ?

x^y = 2^{12}

(2x)^{y-2} = 2^8

Show answer & explanation

Answer: 8

Answer

The correct answer is 8.

The correct answer is 8. By manipulating the system of equations, we can express the second equation in terms of

x^y

. Substituting the value

2^{12}

for

x^y

allows us to express

x

in terms of

y

as

x = 2^{(y+2)/2}

. Substituting this back into the first equation results in the quadratic equation

y^2 + 2y - 24 = 0

. Since

y

must be positive, we find

y = 4

. Using this value of

y

, we solve for

x

and obtain

x = 8

, which is greater than 1, satisfying all given conditions.

Step-by-Step Solution

1

Distribute the exponent in the second equation:

(2x)^{y-2} = 2^8

.

2^{y-2} \cdot x^{y-2} = 2^8

Applying the power of a product rule:

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

.

2

Multiply both sides of the equation by

x^2

to align with the term

x^y

.

2^{y-2} \cdot x^y = 2^8 \cdot x^2

Since

x^{y-2} \cdot x^2 = x^{y-2+2} = x^y

, this allows us to substitute the value of

x^y

from the first equation.

3

Substitute

x^y = 2^{12}

into the equation.

2^{y-2} \cdot 2^{12} = 2^8 \cdot x^2

This uses the given system's first equation to eliminate the

x^y

term.

4

Simplify the left side and divide both sides by

2^8

to isolate

x^2

.

x^2 = 2^{y+2}

2^{y-2} \cdot 2^{12} = 2^{y+10}

. Dividing both sides by

2^8

yields

2^{y+10-8} = 2^{y+2}

.

5

Take the square root of both sides to find an expression for

x

.

x = 2^{\frac{y+2}{2}}

Since

x > 1

, we take the positive root:

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

.

6

Substitute the expression for

x

back into the first equation

x^y = 2^{12}

.

\left(2^{\frac{y+2}{2}}\right)^y = 2^{12}

This yields a single equation in terms of

y

.

7

Equate the exponents and solve the resulting quadratic equation for

y

.

y = 4

(since

y > 0

)

The equation is

2^{\frac{y^2+2y}{2}} = 2^{12} \implies \frac{y^2+2y}{2} = 12 \implies y^2+2y-24=0 \implies (y+6)(y-4)=0

. Since

y

must be positive,

y = 4

.

8

Substitute

y = 4

back into the expression for

x

.

x = 8

x = 2^{\frac{4+2}{2}} = 2^3 = 8

.

Key Concept

Solving systems of exponential equations by equating bases, applying exponent rules, and solving quadratic equations.

Question 1662Question →

A biologist is studying a population of bacteria that triples in size every 4 hours. The population of the bacteria $t$ hours after the start of the study can be modeled by the function $P(t) = P_0 \cdot 3^{kt}$ , where $P_0$ is the initial population and $k$ is a constant. What is the value of $k$ ?

Show answer & explanation

Answer: 0.25

Answer

The correct answer is 0.25 (or 1/4).

The population triples every 4 hours, meaning that after

t

hours, the population has tripled

\frac{t}{4}

times. Thus, the growth model is

P(t) = P_0 \cdot 3^{\frac{t}{4}}

. Comparing this to the given expression

P(t) = P_0 \cdot 3^{kt}

, we get

kt = \frac{t}{4}

. Dividing both sides by

t

yields

k = \frac{1}{4}

, or

0.25

.

Step-by-Step Solution

1

Write the general formula for exponential growth where the population triples every 4 hours.

P(t) = P_0 \cdot 3^{\frac{t}{4}}

If a population triples every 4 hours, it will be multiplied by 3 a total of

\frac{t}{4}

times after

t

hours.

2

Set the exponent of the general model equal to the exponent of the given model.

kt = \frac{t}{4}

Both models represent the same population growth function, so their exponents must be equal.

3

Solve for the constant

k

.

k = \frac{1}{4}

(or

0.25

)

Divide both sides of the equation by

t

.

Key Concept

Exponential growth functions and representing time intervals in the exponent.

Question 1663Question →

If $16^{x-1} - 10 \cdot 4^{x-2} + 1 = 0$ , what is the sum of all values of $x$ that satisfy the equation?

Show answer & explanation

Answer: 2

Answer

The sum of all values of

x

that satisfy the equation is 2.

By writing

16^{x-1}

as

\frac{(4^x)^2}{16}

and

10 \cdot 4^{x-2}

as

\frac{10 \cdot 4^x}{16}

, multiplying the entire equation by 16 yields

(4^x)^2 - 10(4^x) + 16 = 0

. Factoring this quadratic equation gives

(4^x - 2)(4^x - 8) = 0

, which yields

4^x = 2

and

4^x = 8

. Solving these exponential equations results in

x = 0.5

and

x = 1.5

. Adding these values together gives

0.5 + 1.5 = 2

.

Step-by-Step Solution

1

Express each exponential term in the equation using a base of 4.

\frac{4^{2x}}{16} - \frac{10 \cdot 4^x}{16} + 1 = 0

Since

16 = 4^2

, we can write

16^{x-1} = (4^2)^{x-1} = 4^{2x-2} = \frac{4^{2x}}{16}

. Similarly,

4^{x-2} = \frac{4^x}{16}

.

2

Multiply every term in the equation by 16 to eliminate the fractions.

4^{2x} - 10 \cdot 4^x + 16 = 0

This simplifies the expression into a standard polynomial form that is quadratic in terms of

4^x

.

3

Use the substitution

u = 4^x

(where

u > 0

) to transform the equation into a quadratic equation in terms of

u

, and solve it.

u^2 - 10u + 16 = 0 \implies (u - 2)(u - 8) = 0 \implies u = 2

or

u = 8

Substituting

u

allows us to factor the quadratic expression using standard factoring techniques.

4

Substitute

4^x

back for

u

and solve for

x

using base 2 conversion.

For

u = 2

:

4^x = 2 \implies 2^{2x} = 2^1 \implies x = 0.5

. For

u = 8

:

4^x = 8 \implies 2^{2x} = 2^3 \implies x = 1.5

.

We must convert the solutions for the intermediate variable

u

back to the original variable

x

.

5

Calculate the sum of all found values of

x

.

0.5 + 1.5 = 2

The question asks for the sum of all real values of

x

that satisfy the equation.

Key Concept

Solving exponential equations of quadratic form by using variable substitution and exponent laws.

Question 1664Question →

Within the field of behavioral economics, the concept of 'present bias' describes the cognitive tendency of individuals to overvalue immediate rewards at the expense of their long-term financial well-being. Visual reminders of future goals can often mitigate this _______ they do not completely eliminate the impulsive decision-making that leads to sub-optimal choices.

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

bias they

bias, they

bias; they

bias, because they

Show answer & explanation

Answer: bias; they

Answer

bias; they

The correct option properly separates two independent clauses with a semicolon. Semicolons are standard punctuation marks used to link two closely related independent clauses without the need for a coordinating conjunction.

Step-by-Step Solution

1

Identify the clause boundaries in the sentence.

The sentence contains two independent clauses: 'Visual reminders of future goals can often mitigate this bias' and 'they do not completely eliminate the impulsive decision-making that leads to sub-optimal choices.'

Determining where one clause ends and another begins is necessary to select the correct punctuation or linking word.

2

Evaluate the grammatical relationship between the two independent clauses.

Since both clauses are independent and express related but distinct ideas, they must be separated by a period, a semicolon, or a comma paired with a coordinating conjunction (like 'but').

This helps narrow down the acceptable punctuation methods.

3

Test the choices against Standard English conventions and the logical flow of the sentence.

Using a semicolon properly separates the two independent clauses without creating a run-on or comma splice, and it maintains a logical connection.

This confirms the correct choice.

Key Concept

Semicolons are used to link two independent clauses that are closely related in thought.

Question 1665Question →

If $\frac{20}{x - 3} = 4$ , what is the value of $x + 2$ ?

Show answer & explanation

Answer: 10

Answer

10

To solve the equation

\frac{20}{x - 3} = 4

, multiply both sides by

x - 3

to get

20 = 4(x - 3)

. Distributing the 4 gives

20 = 4x - 12

. Adding 12 to both sides yields

32 = 4x

, and dividing by 4 gives

x = 8

. Substituting

x = 8

into the expression

x + 2

gives

8 + 2 = 10

.

Step-by-Step Solution

1

Multiply both sides of the equation by the denominator

x - 3

.

20 = 4(x - 3)

To eliminate the fraction and rewrite the rational equation in linear form.

2

Distribute the constant on the right side of the equation.

20 = 4x - 12

To remove the parentheses by multiplying 4 by both

x

and

-3

.

3

Add 12 to both sides of the equation.

32 = 4x

To isolate the term with the variable

x

on one side of the equation.

4

Divide both sides of the equation by 4.

x = 8

To solve for the variable

x

.

5

Substitute the value of

x

into the expression

x + 2

.

10

The question asks for the value of the expression

x + 2

rather than just the variable

x

.

Key Concept

Solving rational equations by clearing the denominator and isolating the variable to evaluate algebraic expressions.

Question 1666Question →

An agricultural drone is spraying liquid fertilizer on a field at a constant rate. The total amount of fertilizer remaining in the drone's tank, in liters, can be modeled by a linear function of the time, in minutes, since the drone began spraying. After $3$ minutes of spraying, $84$ liters of fertilizer remain in the tank. After $8$ minutes of spraying, $54$ liters of fertilizer remain in the tank. How many minutes of spraying will it take for the tank to become completely empty?

Show answer & explanation

Answer: 17

Answer

17

The correct answer is 17. The volume of fertilizer in the tank decreases at a constant rate of 6 liters per minute. Since 84 liters remain after 3 minutes, the initial volume of fertilizer in the tank was 102 liters. Dividing the initial volume of 102 liters by the rate of 6 liters per minute yields 17 minutes for the tank to be completely empty.

Step-by-Step Solution

1

Calculate the constant rate of change (slope) of the remaining fertilizer volume.

-6 liters per minute

The slope formula is used with the two data points representing time and volume:

(3, 84)

and

(8, 54)

.

2

Determine the initial volume of fertilizer in the tank (y-intercept).

102 liters

Using the slope-intercept form

F(t) = mt + b

, we substitute the slope

m = -6

and the point

(3, 84)

to solve for

b

.

3

Find the time when the volume of remaining fertilizer reaches 0.

17 minutes

Setting the linear function equal to 0 and solving for time gives the total duration until the tank is empty.

Key Concept

Linear Functions and Graphs

Question 1667Question →

Architect Zaha Hadid was renowned for her dynamically curved designs that challenged traditional geometry. In her design of the Heydar Aliyev Center in Azerbaijan, the ______ exterior flows seamlessly into the surrounding plaza, blurring the boundary between the structure and the city.

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

building

buildings

building's

buildings'

Show answer & explanation

Answer: building's

Answer

building's

The correct answer is the singular possessive noun 'building's' because it correctly modifies the noun 'exterior' to show that the exterior belongs to the single structure mentioned in the sentence, the Heydar Aliyev Center.

Step-by-Step Solution

1

Determine whether the target noun should be singular or plural based on the context.

The passage discusses Zaha Hadid's design of a single structure, the Heydar Aliyev Center. Therefore, the singular form of 'building' is required.

Identifying the number of the noun prevents the incorrect use of a plural form.

2

Determine if the noun needs to be possessive.

The noun directly modifies 'exterior', indicating that the exterior belongs to the building. This indicates a possessive form is required.

A possessive form is needed to show ownership or association with the noun that follows.

Key Concept

Plural and Possessive Nouns

Estimated Time:1m 0s

Question 1668Question →

The population of a colony of bacteria is modeled by the function $P(t) = 250 \cdot 8^{\frac{t}{6} + 1}$ , where $P(t)$ is the estimated number of bacteria $t$ hours after the start of an experiment. The function can be rewritten in the equivalent form $P(t) = A \cdot 2^{\frac{m}{k}}$ , where $m$ is the number of 20-minute intervals that have elapsed since the start of the experiment, and $A$ and $k$ are constants. What is the value of $A + k$ ?

506

2,002

2,006

2,018

Show answer & explanation

Answer: 2,006

Answer

2,006

The correct answer is 2,006. By writing the base 8 as 2^3 and applying the exponent rules, the function becomes 250 * 2^(t/2 + 3) = 2000 * 2^(t/2). Since m represents 20-minute intervals and there are 3 intervals in an hour, we have t = m/3. Substituting this into the function yields 2000 * 2^(m/6), which gives A = 2000 and k = 6. The sum is 2000 + 6 = 2006.

Step-by-Step Solution

1

Express the base 8 in terms of base 2 to match the target form.

P(t) = 250 \cdot (2^3)^{\frac{t}{6} + 1}

Converting to base 2 allows simplification of the exponent to match the required base of 2 in the target equation.

2

Apply the exponent power rule

(b^y)^z = b^{yz}

to simplify the expression.

P(t) = 250 \cdot 2^{3\left(\frac{t}{6} + 1\right)} = 250 \cdot 2^{\frac{t}{2} + 3}

Multiplying the exponents simplifies the expression into a single power of base 2.

3

Isolate the constant coefficient by separating the terms in the exponent using the rule

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

.

P(t) = 250 \cdot 2^3 \cdot 2^{\frac{t}{2}} = 250 \cdot 8 \cdot 2^{\frac{t}{2}} = 2000 \cdot 2^{\frac{t}{2}}

This separates the initial coefficient multiplier from the variable growth term.

4

Convert the time variable from hours (

t

) to 20-minute intervals (

m

).

Since there are 3 twenty-minute intervals in one hour,

m = 3t

, which means

t = \frac{m}{3}

. Substituting this into the equation yields

P(t) = 2000 \cdot 2^{\frac{m/3}{2}} = 2000 \cdot 2^{\frac{m}{6}}

.

This expresses the population model in terms of the new time unit variable,

m

, as required.

5

Identify the constants

A

and

k

and calculate their sum.

A = 2000

,

k = 6

, so

A + k = 2000 + 6 = 2006

.

Comparing the final model to

A \cdot 2^{\frac{m}{k}}

directly gives the values of the constants.

Key Concept

Rewriting and interpreting exponential functions by manipulating bases and changing variables.

Estimated Time:2m 0s

Question 1669Question →

In the system of quadratic equations below, $c$ is an integer constant:

2x^2 - 11x + c = 0

x^2 - 15x + 4c = 0

If the first equation has two distinct real solutions and the second equation has no real solutions, what is the value of $c$ ?

Show answer & explanation

Answer: 15

Answer

15

For the first equation

2x^2 - 11x + c = 0

to have two distinct real solutions, its discriminant must be positive:

\Delta_1 = (-11)^2 - 4(2)(c) = 121 - 8c > 0

, which gives

c < 15.125

. For the second equation

x^2 - 15x + 4c = 0

to have no real solutions, its discriminant must be negative:

\Delta_2 = (-15)^2 - 4(1)(4c) = 225 - 16c < 0

, which gives

c > 14.0625

. The only integer

c

that satisfies the inequality

14.0625 < c < 15.125

is

15

.

Step-by-Step Solution

1

Set up the inequality for the first equation's discriminant to have two distinct real solutions.

c < 15.125

A quadratic equation

ax^2 + bx + c = 0

has two distinct real solutions if its discriminant,

b^2 - 4ac

, is greater than zero. For

2x^2 - 11x + c = 0

, this means

(-11)^2 - 4(2)(c) > 0

, simplifying to

121 - 8c > 0

, or

c < 15.125

.

2

Set up the inequality for the second equation's discriminant to have no real solutions.

c > 14.0625

A quadratic equation has no real solutions if its discriminant is less than zero. For

x^2 - 15x + 4c = 0

, the discriminant is

(-15)^2 - 4(1)(4c) < 0

, simplifying to

225 - 16c < 0

, or

c > 14.0625

.

3

Find the integer value of

c

that satisfies both inequalities.

c = 15

Combining the results from the previous steps, we get

14.0625 < c < 15.125

. Since

c

is specified to be an integer, the only integer that lies in this open interval is

15

.

Key Concept

Using the discriminant to determine the number of real solutions of quadratic equations

Question 1670Question →

At a local farmer's market, a vendor sells two types of fruit baskets: a standard basket and a deluxe basket. The standard basket contains $3$ apples and $2$ oranges. The deluxe basket contains $5$ apples and $4$ oranges. If a customer bought a combination of these baskets containing a total of $31$ apples and $22$ oranges, what is the total number of baskets the customer bought?

Show answer & explanation

Answer: 9

Answer

The total number of baskets the customer bought is 9.

The correct answer is obtained by setting up a system of equations where

s

is the number of standard baskets and

d

is the number of deluxe baskets. The equations

3s + 5d = 31

and

2s + 4d = 22

represent the totals of apples and oranges respectively. Dividing the second equation by 2 gives

s + 2d = 11

, which can be rearranged to

s = 11 - 2d

. Substituting this expression into the first equation yields

3(11 - 2d) + 5d = 31

, which simplifies to

33 - d = 31

, meaning

d = 2

. Substituting

d = 2

back gives

s = 7

. The total number of baskets is the sum of both types,

7 + 2 = 9

.

Step-by-Step Solution

1

Define variables and set up the system of linear equations.

Let

s

be the number of standard baskets and

d

be the number of deluxe baskets. The equations are:

3s + 5d = 31

(for apples) and

2s + 4d = 22

(for oranges).

To represent the relationships between the number of baskets and the total quantities of fruits mathematically.

2

Solve the system of equations using substitution.

From the second equation, dividing by 2 yields

s + 2d = 11

, so

s = 11 - 2d

. Substituting this into the first equation gives

3(11 - 2d) + 5d = 31

, which simplifies to

33 - d = 31

, leading to

d = 2

.

To find the number of deluxe baskets purchased.

3

Calculate the number of standard baskets and sum the two counts to find the total number of baskets.

Substitute

d = 2

into

s = 11 - 2d

to get

s = 7

. The total number of baskets is

s + d = 7 + 2 = 9

.

To answer the specific question asking for the total number of baskets bought.

Key Concept

Systems of Linear Equations

Question 1671Question →

If $\sqrt{2x + 15} - 6 = x$ , what is the value of $x + 5$ ?

-3

-2

2

8

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

The correct answer is 2. To solve the equation

\sqrt{2x + 15} - 6 = x

, we first isolate the radical by adding

6

to both sides, obtaining

\sqrt{2x + 15} = x + 6

. Squaring both sides yields

2x + 15 = x^2 + 12x + 36

. Rearranging the terms to form a quadratic equation gives

x^2 + 10x + 21 = 0

, which factors as

(x + 7)(x + 3) = 0

. This gives two potential solutions:

x = -7

and

x = -3

. We must check both potential solutions in the original equation. For

x = -3

, the equation holds true:

\sqrt{2(-3) + 15} - 6 = 3 - 6 = -3

. For

x = -7

, it does not:

\sqrt{2(-7) + 15} - 6 = 1 - 6 = -5 \neq -7

. Thus,

x = -3

is the only valid solution. Finally, substituting this into the expression

x + 5

gives

-3 + 5 = 2

.

Step-by-Step Solution

1

Isolate the radical term on one side of the equation.

\sqrt{2x + 15} = x + 6

Before squaring both sides, the radical term must be isolated to avoid cross-terms on that side.

2

Square both sides of the equation to eliminate the radical.

2x + 15 = (x + 6)^2 \implies 2x + 15 = x^2 + 12x + 36

Squaring a square root cancels the radical, allowing us to solve the resulting algebraic equation.

3

Rearrange the equation into standard quadratic form:

ax^2 + bx + c = 0

.

x^2 + 10x + 21 = 0

Subtracting

2x

and

15

from both sides collects all terms on one side so the quadratic can be factored.

4

Factor the quadratic equation.

(x + 7)(x + 3) = 0 \implies x = -7 \text{ or } x = -3

Finding two numbers that multiply to 21 and add to 10 allows us to factor the quadratic and find potential solutions.

5

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

For

x = -3

,

\sqrt{2(-3) + 15} - 6 = -3

is true. For

x = -7

,

\sqrt{2(-7) + 15} - 6 = -7

is false (gives

-5 \neq -7

). Thus,

x = -3

is the only valid solution.

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

6

Evaluate the expression

x + 5

using the valid solution.

-3 + 5 = 2

The question asks for the value of

x + 5

, not just the value of

x

.

Key Concept

Solving radical equations by isolating the radical, squaring both sides, checking for extraneous solutions, and evaluating algebraic expressions.

Question 1672Question →

Although obsidian is commonly associated with dark, glass-like formations, its color varies significantly depending on the chemical composition of the source lava. For instance, microscopic inclusions of iron oxide can impart a deep reddish-brown ______ clusters of microscopic gas bubbles trapped during the cooling process can produce a golden or iridescent sheen.

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

hue, meanwhile,

hue; meanwhile,

hue meanwhile

hue, because

Show answer & explanation

Answer: hue; meanwhile,

Answer

The correct option is the one that uses a semicolon followed by the conjunctive adverb and a comma to link the two independent clauses.

The text presents two independent clauses: the first describes how iron oxide affects obsidian's color, and the second describes how gas bubbles do so. To connect these two independent clauses, standard punctuation must be used. Using a semicolon followed by a conjunctive adverb and a comma is a grammatically correct way to link independent clauses while showing a transition between them.

Step-by-Step Solution

1

Identify the clause boundaries in the second sentence.

The sentence contains two independent clauses: the first ends with the word describing the color ('hue'), and the second begins with the subject describing the gas bubbles ('clusters of microscopic gas bubbles...').

Locating where one complete thought ends and the next begins helps determine the required punctuation.

2

Evaluate the grammatical link needed between two independent clauses.

Two independent clauses must be joined by a period, a semicolon, or a coordinating conjunction with a comma. A conjunctive adverb like 'meanwhile' requires a semicolon before it when joining two clauses.

Joining two independent clauses with only a comma creates a comma splice, and joining them with no punctuation creates a run-on sentence.

3

Assess the logical connection between the clauses.

The clauses present two different, parallel examples of how inclusions affect obsidian's color. A transitional word like 'meanwhile' is logically appropriate to show this relationship, while a causal subordinator like 'because' is incorrect.

Ensuring the connector fits both grammatical rules and logical meaning determines the correct choice.

Key Concept

Clause Boundaries and Linking

Question 1673Question →

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

Show answer & explanation

Answer: 4

Answer

4

To solve the rational equation, multiply both sides by the least common denominator,

x^2 - 4

. This results in the equation

x(x + 2) - 8(x - 2) = 8

. Expanding and combining like terms yields the quadratic equation

x^2 - 6x + 8 = 0

. Factoring this equation gives

(x - 2)(x - 4) = 0

, which yields the potential solutions

x = 2

and

x = 4

. However,

x = 2

must be excluded because it makes the denominators of the original equation equal to zero. Therefore, the only valid solution is

x = 4

.

Step-by-Step Solution

1

Determine the least common denominator (LCD) for the terms in the equation.

The LCD is

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

.

Multiplying the entire equation by the LCD clears all rational expressions.

2

Multiply the entire equation by the LCD to eliminate the denominators.

x(x + 2) - 8(x - 2) = 8

This simplifies the rational equation into a polynomial equation.

3

Expand, simplify, and solve the resulting quadratic equation.

x^2 - 6x + 8 = 0 \Rightarrow (x - 2)(x - 4) = 0 \Rightarrow x = 2 \text{ or } x = 4

Factoring the quadratic equation gives the potential solutions.

4

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

Substituting

x = 2

leads to division by zero, making the expression undefined. Substituting

x = 4

yields a valid identity:

\frac{4}{2} - \frac{8}{6} = \frac{2}{3}

. Therefore,

x = 4

is the only valid solution.

Values that make any denominator in the original equation equal to zero must be excluded.

Key Concept

Solving rational equations and checking for extraneous solutions.

Question 1674Question →

The passage below contains a blank. Based on the grammatical context of the passage, what form of the noun 'planet' correctly completes the blank?

Fill in the blanks below

In the early seventeenth century, German astronomer Johannes Kepler analyzed detailed observational data of Mars collected by his predecessor, Tycho Brahe. Through rigorous calculations, Kepler discovered that planetary orbits are elliptical rather than perfectly circular. This breakthrough revolutionized astronomy by challenging the long-held assumption that a path through space must conform to ancient ideals of geometric perfection.

Show answer & explanation

Answer

planet's

The correct answer is the singular possessive noun 'planet's'. The singular indefinite article 'a' preceding the blank indicates that the noun must be singular, while the presence of the noun 'path' immediately following the blank indicates that the possessive form is required to show ownership or association.

Step-by-Step Solution

1

Analyze the noun modified by the blank to determine if possession is required.

The blank precedes the noun 'path', indicating that the noun in the blank must be in the possessive case to show that the path belongs to the noun ('the path of a planet').

Possessive nouns are used to show ownership or association with a following noun.

2

Determine the grammatical number (singular or plural) of the noun in the blank.

The blank is preceded by the singular indefinite article 'a', which requires a singular noun. Therefore, the singular possessive form is needed.

The singular indefinite article 'a' cannot modify a plural noun.

3

Form the singular possessive of the noun 'planet'.

Adding an apostrophe and an 's' to the singular noun 'planet' yields 'planet's'.

Singular nouns are made possessive by adding an apostrophe and an 's'.

Key Concept

Plural and Possessive Nouns and Pronouns

Question 1675Question →

In the quadratic equation $2x^2 - 11x + c = 0$ , $c$ is a constant. If the sum of the squares of the two real solutions to the equation is $\frac{85}{4}$ , what is the value of $c$ ?

9

18

-9

\frac{9}{2}

Show answer & explanation

Answer: 9

Answer

9

The sum of the solutions of the quadratic equation is

11/2

and the product of the solutions is

c/2

. The sum of the squares of the solutions is

(11/2)^2 - 2(c/2) = 121/4 - c

. Setting this equal to the given value of

85/4

and solving for

c

yields

c = 9

.

Step-by-Step Solution

1

Express the sum and product of the solutions using Vieta's formulas.

For the quadratic equation

2x^2 - 11x + c = 0

, the sum of the solutions is

x_1 + x_2 = -\frac{b}{a} = \frac{11}{2}

, and the product of the solutions is

x_1 x_2 = \frac{c}{a} = \frac{c}{2}

.

Vieta's formulas relate the coefficients of a polynomial to the sum and product of its roots.

2

Use the algebraic identity for the sum of squares of two variables.

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

This identity allows us to express the sum of squares in terms of the sum and product of the roots.

3

Substitute the expressions from Step 1 into the identity and set it equal to the given sum of squares.

\left(\frac{11}{2}\right)^2 - 2\left(\frac{c}{2}\right) = \frac{85}{4} \Rightarrow \frac{121}{4} - c = \frac{85}{4}

To solve for

c

using the given value of the sum of squares, which is

\frac{85}{4}

.

4

Isolate the constant

c

.

c = \frac{121}{4} - \frac{85}{4} = \frac{36}{4} = 9

To find the numerical value of

c

.

Key Concept

Vieta's Formulas and Root Identities

Question 1676Question →

In the $xy$ -plane, the line with equation $ax + 4y = 36$ , where $a$ is a constant, has a $y$ -intercept of $(0, p)$ and an $x$ -intercept of $(q, 0)$ , where $p$ and $q$ are positive integers. If $p + q = 15$ , what is the value of $a$ ?

Show answer & explanation

Answer: 6

Answer

6

To find the value of

a

, we first determine the

y

-intercept of the line by setting

x = 0

in the equation

ax + 4y = 36

. This gives

4y = 36

, so

y = 9

. Thus, the

y

-intercept is

(0, 9)

, which means

p = 9

. Using the given relationship

p + q = 15

, we substitute

p = 9

to find

q = 6

. The

x

-intercept is therefore

(6, 0)

. Substituting these coordinates back into the line's equation gives

a(6) + 4(0) = 36

, which simplifies to

6a = 36

. Solving for

a

yields

a = 6

.

Step-by-Step Solution

1

Set

x = 0

in the equation

ax + 4y = 36

to find the

y

-intercept.

4y = 36 \implies y = 9

, so

p = 9

.

The

y

-intercept of a graph is the point where

x = 0

.

2

Substitute

p = 9

into the equation

p + q = 15

to solve for

q

.

9 + q = 15 \implies q = 6

.

We are given that the sum of the

y

-coordinate of the

y

-intercept and the

x

-coordinate of the

x

-intercept is

15

.

3

Substitute the

x

-intercept

(6, 0)

into the equation

ax + 4y = 36

to solve for

a

.

a(6) + 4(0) = 36 \implies 6a = 36 \implies a = 6

.

Since the

x

-intercept is

(q, 0)

and

q = 6

, the point

(6, 0)

must satisfy the equation of the line.

Key Concept

Finding and using intercepts of a linear equation in two variables.

Question 1677Question →

If the polynomial $p(x) = x^2 - kx + 12$ is divisible by $x - 3$ , where $k$ is a constant, what is the value of $k$ ?

Show answer & explanation

Answer: 7

Answer

7

According to the Factor Theorem, if a polynomial

p(x)

is divisible by

x - c

, then

p(c) = 0

. For the polynomial

p(x) = x^2 - kx + 12

to be divisible by

x - 3

, we must have

p(3) = 0

. Substituting

x = 3

into the polynomial gives

3^2 - k(3) + 12 = 0

. Simplifying this equation yields

9 - 3k + 12 = 0

, which combines to

21 - 3k = 0

. Solving for

k

gives

3k = 21

, which simplifies to

k = 7

.

Step-by-Step Solution

1

Apply the Factor Theorem to find the root of the polynomial.

Since

x - 3

is a factor of

p(x)

, evaluating the polynomial at

x = 3

must equal 0, so

p(3) = 0

.

The Factor Theorem states that

x - c

is a factor of a polynomial

p(x)

if and only if

p(c) = 0

.

2

Substitute

x = 3

into

p(x) = x^2 - kx + 12

and set it equal to 0.

3^2 - k(3) + 12 = 0

This sets up an algebraic equation to solve for the unknown constant

k

.

3

Simplify the equation and solve for

k

.

9 - 3k + 12 = 0 \implies 21 - 3k = 0 \implies 3k = 21 \implies k = 7

Combine like terms and isolate

k

using basic algebraic operations.

Key Concept

The Factor Theorem states that a polynomial

p(x)

has a factor

(x - c)

if and only if

p(c) = 0

.

Question 1678Question →

The graph of the quadratic function $f$ in the $xy$ -plane is a parabola with vertex $(3, 12)$ . If the graph passes through the point $(5, 8)$ , what is the $y$ -value of the point on the graph where $x = 1$ ?

Show answer & explanation

Answer: 8

Answer

The correct answer is

8

.

The vertex of the parabola is given as

(3, 12)

, which means the axis of symmetry is the line

x = 3

. Since a parabola is symmetric with respect to its axis of symmetry, any two points on the parabola that are equidistant from this line must share the same

y

-coordinate. The given point has an

x

-coordinate of

5

, which is

5 - 3 = 2

units to the right of the axis of symmetry. The target point has an

x

-coordinate of

1

, which is

3 - 1 = 2

units to the left of the axis of symmetry. Because both points are exactly

2

units away from the axis of symmetry, their

y

-coordinates are equal. Therefore, the

y

-value of the point where

x = 1

is

8

.

Step-by-Step Solution

1

Determine the axis of symmetry of the parabola.

The axis of symmetry is the vertical line

x = 3

.

The vertex of a parabola

(h, k)

always lies on its axis of symmetry, which is

x = h

.

2

Find the horizontal distance from the axis of symmetry to the given point.

The distance from

x = 3

to

x = 5

is

|5 - 3| = 2

units.

This measures how far the point is horizontally from the line of symmetry.

3

Find the horizontal distance from the axis of symmetry to the target point.

The distance from

x = 3

to

x = 1

is

|1 - 3| = 2

units.

This determines if the target point is symmetric to the given point.

4

Equate the y-values using symmetry.

The

y

-value at

x = 1

is

8

.

Because both

x = 5

and

x = 1

are

2

units away from the axis of symmetry, their corresponding

y

-values must be identical.

Key Concept

Symmetry of Quadratic Graphs

Question 1679Question →

A botanist models the rate of water transpiration of a plant species under various temperature conditions. The table below shows the estimated transpiration rate, $R$ , in milligrams of water per square decimeter of leaf area per hour ( $\text{mg}/(\text{dm}^2\cdot\text{h})$ ), at various ambient temperatures, $t$ , in degrees Celsius ( $^\circ\text{C}$ ).

Temperature ( $t$ )	Transpiration rate ( $R$ )
$15$	$84.5$
$20$	$108.5$
$25$	$132.5$
$30$	$156.5$

The relationship between the ambient temperature and the transpiration rate is linear. Based on the model, what is the estimated increase in the transpiration rate, in milligrams of water per square meter of leaf area per minute, for each increase of $1$ degree Celsius in the ambient temperature? (Note: $1\text{ square meter} = 100\text{ square decimeters}$ )

Show answer & explanation

Answer: 8

Answer

8

To find the estimated increase in transpiration rate per degree Celsius in the new units, we first determine the rate of change in the original units from the table. The change in temperature is

20 - 15 = 5^\circ\text{C}

, and the corresponding change in transpiration rate is

108.5 - 84.5 = 24\text{ mg}/(\text{dm}^2\cdot\text{h})

. The rate of change is

\frac{24}{5} = 4.8\text{ mg}/(\text{dm}^2\cdot\text{h})

per

1^\circ\text{C}

temperature increase. Converting this rate to square meters, we multiply by

100

because

1\text{ square meter} = 100\text{ square decimeters}

, giving

480\text{ mg}/(\text{m}^2\cdot\text{h})

. Finally, to convert to minutes, we divide by

60

because

1\text{ hour} = 60\text{ minutes}

, which yields

\frac{480}{60} = 8\text{ mg}/(\text{m}^2\cdot\text{min})

per

1^\circ\text{C}

temperature increase.

Step-by-Step Solution

1

Find the rate of change of the transpiration rate with respect to temperature from the given data table.

Slope =

4.8\text{ mg}/(\text{dm}^2\cdot\text{h})

per

1^\circ\text{C}

The rate of change represents the increase in transpiration rate for each

1^\circ\text{C}

increase in temperature in the original units.

2

Convert the unit of area in the rate of change from square decimeters to square meters.

Rate =

480\text{ mg}/(\text{m}^2\cdot\text{h})

per

1^\circ\text{C}

Since

1\text{ square meter} = 100\text{ square decimeters}

, the rate per square meter is

100

times the rate per square decimeter.

3

Convert the unit of time in the rate of change from hours to minutes.

Rate =

8\text{ mg}/(\text{m}^2\cdot\text{min})

per

1^\circ\text{C}

Since there are

60

minutes in an hour, dividing the hourly rate by

60

gives the rate per minute.

Key Concept

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

Estimated Time:2m 30s

Question 1680Question →

Which punctuation mark must be placed in the blank to properly connect the clauses and conform to the conventions of Standard English?

Fill in the blanks below

Many astronomical objects emit radiation at wavelengths invisible to the human eye therefore, astronomers must use specialized space telescopes to capture infrared, ultraviolet, and X-ray data.

Show answer & explanation

Answer

A semicolon (;) must be used to separate the two independent clauses before the conjunctive adverb 'therefore'.

A semicolon is the correct punctuation mark to place in the blank. It properly links two independent clauses ('Many astronomical objects emit radiation...' and 'therefore, astronomers must use...') without a coordinating conjunction. When a conjunctive adverb like 'therefore' transitions between two independent clauses, a semicolon is required before it.

Step-by-Step Solution

1

Identify the grammatical structure of the clauses surrounding the blank.

The clause before the blank ('Many astronomical objects emit radiation at wavelengths invisible to the human eye') and the clause after the blank ('therefore, astronomers must use specialized space telescopes to capture infrared, ultraviolet, and X-ray data') are both independent clauses.

Knowing whether the clauses are independent or dependent helps determine the correct punctuation mark needed to connect them.

2

Determine the correct punctuation mark to join these two independent clauses when the second begins with the conjunctive adverb 'therefore'.

A semicolon is required to link the two independent clauses, preventing a comma splice or run-on sentence.

Standard English conventions dictate that two independent clauses not joined by a coordinating conjunction (like 'and' or 'but') must be separated by a semicolon or a period, especially when a conjunctive adverb introduces the second clause.

Key Concept

Using a semicolon to link independent clauses separated by a conjunctive adverb.

Estimated Time:1m 0s

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