All practice questions

Question 1881Question →

The daily profit, in dollars, of a small company is modeled by the quadratic function $P(x) = -2x^2 + 120x - 1000$ , where $x$ represents the number of items the company produces and sells each day. For what number of items produced and sold, greater than $20$ , will the company break even (meaning its daily profit is $\$0$ )?

10

30

40

50

Show answer & explanation

Answer: 50

Answer

The company breaks even when it produces and sells 50 items.

To find the break-even points, set the profit function to zero:

-2x^2 + 120x - 1000 = 0

. Dividing both sides by

-2

yields

x^2 - 60x + 500 = 0

. Factoring this equation gives

(x - 10)(x - 50) = 0

, which results in the solutions

x = 10

and

x = 50

. Since the question specifies that the number of items must be greater than

20

, the correct answer is 50.

Step-by-Step Solution

1

Set the profit function equal to zero to find the break-even points.

-2x^2 + 120x - 1000 = 0

Breaking even means the profit,

P(x)

, is equal to 0.

2

Divide the entire equation by the common factor of

-2

to simplify the quadratic expression.

x^2 - 60x + 500 = 0

Simplifying the quadratic expression makes it easier to factor.

3

Factor the quadratic equation by finding two numbers that multiply to

500

and add to

-60

.

(x - 10)(x - 50) = 0

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

4

Solve for

x

by setting each factor equal to zero.

x = 10

or

x = 50

The zero product property states that if a product of factors is zero, at least one factor must be zero.

5

Apply the constraint that the number of items must be greater than

20

.

x = 50

Out of the two roots, only 50 is greater than 20.

Key Concept

Solving quadratic equations in real-world contexts by finding roots and applying constraints.

Question 1882Question →

A local veterinarian clinic recorded the primary diets of 120 animals (cats and dogs). The results are summarized in the table below.

Animal	Dry food	Wet food	Total
Cats	35	25	60
Dogs	45	15	60
Total	80	40	120

If one of these animals is selected at random, what is the probability that the animal is a dog, given that the animal's primary diet is dry food?

\frac{9}{16}

\frac{3}{8}

\frac{3}{4}

\frac{5}{8}

Show answer & explanation

Answer: $\frac{9}{16}$

Answer

The correct answer is the option representing

\frac{9}{16}

.

The correct answer is

\frac{9}{16}

. To find the probability that the selected animal is a dog given that its primary diet is dry food, we restrict our focus to the animals that eat dry food. According to the table, there are 80 animals in total whose primary diet is dry food. Among these 80 animals, 45 are dogs. Therefore, the probability is

\frac{45}{80}

, which simplifies to

\frac{9}{16}

.

Step-by-Step Solution

1

Identify the given condition in the question.

The condition is that the selected animal's primary diet is dry food.

This restricts the sample space to only the animals in the 'Dry food' column.

2

Find the total number of animals that meet the given condition from the table.

The total number of animals whose primary diet is dry food is 80.

This value serves as the denominator for the conditional probability fraction.

3

Find the number of dogs within the conditional space identified in the previous step.

The number of dogs that eat dry food is 45.

This value serves as the numerator representing the favorable outcomes.

4

Calculate the conditional probability by dividing the favorable outcomes by the total outcomes in the restricted sample space.

\frac{45}{80} = \frac{9}{16}

Dividing the numerator by the denominator and simplifying by dividing both by 5 yields the final probability.

Key Concept

Conditional Probability from Two-Way Tables

Alternative Method

Instead of working directly with the counts in the table, you can write the conditional probability formula:

P(\text{Dog} \mid \text{Dry}) = \frac{P(\text{Dog} \cap \text{Dry})}{P(\text{Dry})} = \frac{45/120}{80/120} = \frac{45}{80} = \frac{9}{16}

.

Estimated Time:50s

Question 1883Question →

In the quadratic equation $2x^2 - 12x + k = 0$ , $k$ is a constant. If the sum of the squares of the solutions to the equation is $26$ , what is the value of $k$ ?

Show answer & explanation

Answer: 10

Answer

The value of

k

is

10

.

The correct answer is

10

. By expressing the sum of the squares of the solutions as

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

, we can substitute the sum of the solutions (

-\frac{-12}{2} = 6

) and the product of the solutions (

\frac{k}{2}

) directly into the expression. This gives

26 = 36 - k

. Solving for

k

yields

k = 10

. Alternatively, solving the quadratic equation using the quadratic formula yields solutions

3 + \frac{\sqrt{144-8k}}{4}

and

3 - \frac{\sqrt{144-8k}}{4}

. Squaring these solutions and setting their sum equal to

26

simplifies to

18 + 2\left(\frac{144-8k}{16}\right) = 26

, which also solves to

k = 10

.

Step-by-Step Solution

1

Find the sum and product of the solutions using the coefficients of the quadratic equation.

The sum of the solutions is

6

and the product of the solutions is

\frac{k}{2}

.

By Vieta's formulas, for any quadratic equation

ax^2 + bx + c = 0

with solutions

x_1

and

x_2

, the sum of the solutions is

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

and the product of the solutions is

x_1 x_2 = \frac{c}{a}

.

2

Apply the algebraic identity to express the sum of the squares of the solutions in terms of their sum and product.

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

This identity allows us to substitute the sum and product of the solutions directly without solving for the individual solutions first.

3

Substitute the values and solve for

k

.

26 = 6^2 - 2\left(\frac{k}{2}\right) \implies 26 = 36 - k \implies k = 10

Substituting the given sum of squares (

26

), the sum of solutions (

6

), and the product of solutions (

\frac{k}{2}

) allows us to solve for the unknown constant

k

directly.

Key Concept

Using the relationship between the roots and coefficients of a quadratic equation (Vieta's formulas) in combination with algebraic identities to solve for unknown constants.

Question 1884Question →

A technology company has two regional offices, Office North and Office South. In 2024, the number of employees at Office North was $25\%$ greater than the number of employees at Office South. In 2025, the number of employees at Office North increased by $20\%$ , and the number of employees at Office South decreased by $10\%$ . If the total number of employees at both offices combined in 2025 was $432$ , what was the total number of employees at both offices combined in 2024?

360

405

225

450

Show answer & explanation

Answer: 405

Answer

405

Let

S

represent the number of employees at Office South in 2024. Because Office North has

25\%

more employees than Office South in 2024, the count at Office North is

1.25S

. The combined total in 2024 is

S + 1.25S = 2.25S

. In 2025, Office North's staff increases by

20\%

, making it

1.20 \times 1.25S = 1.50S

. Office South's staff decreases by

10\%

, making it

0.90S

. The combined total in 2025 is

1.50S + 0.90S = 2.40S

. Given that the total in 2025 is

432

, we solve the equation

2.40S = 432

, which gives

S = 180

. Substituting

S = 180

into the expression for the 2024 total yields

2.25 \times 180 = 405

.

Step-by-Step Solution

1

Define variables for the number of employees in 2024. Let

S

be the number of employees at Office South in 2024. Since Office North had

25\%

more employees, represent Office North as

1.25S

. Express the total number of employees in 2024 as

S + 1.25S = 2.25S

.

Total employees in 2024 is represented by

2.25S

.

To establish a baseline relationship between the employee counts at the two offices.

2

Express the number of employees in 2025 in terms of

S

by applying the percentage changes. Office North's employees increased by

20\%

, which is

1.20 \times 1.25S = 1.50S

. Office South's employees decreased by

10\%

, which is

0.90 \times S = 0.90S

. Find the total in 2025 by adding these values:

1.50S + 0.90S = 2.40S

.

Total employees in 2025 is represented by

2.40S

.

To determine the algebraic expression representing the combined count after the percent changes.

3

Set the algebraic expression for the 2025 total equal to the given value of

432

and solve for

S

.

2.40S = 432 \implies S = 180

To find the baseline number of employees at Office South in 2024.

4

Substitute

S = 180

back into the expression for the 2024 total (

2.25S

) to calculate the final answer.

2.25 \times 180 = 405

To calculate the combined total number of employees in 2024.

Key Concept

Percents and Percent Change

Estimated Time:1m 30s

Question 1885Question →

In modern agricultural science, biochar is increasingly added to depleted soils to improve crop yields and retain vital nutrients. Derived from the thermochemical conversion of biomass in an oxygen-depleted environment, this highly porous substance acts as a physical sponge in the _______ it binds to essential minerals and prevents them from leaching away during heavy rainfall.

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

ground,

ground

ground, though

ground;

Show answer & explanation

Answer: ground;

Answer

ground;

The correct option is the one ending with a semicolon. The passage contains two independent clauses that are closely related in meaning. A semicolon is a grammatically correct way to join these two clauses without a coordinating conjunction.

Step-by-Step Solution

1

Determine the grammatical structure of the clauses on either side of the blank.

The text before the blank ('this highly porous substance acts as a physical sponge in the ground') and the text after the blank ('it binds to essential minerals and prevents them from leaching away during heavy rainfall') are both complete, independent clauses.

Identifying the clause boundaries helps determine which punctuation or conjunction is grammatically required.

2

Analyze the logical relationship between the two independent clauses.

The second clause explains the mechanism of the physical sponge described in the first clause. There is no contrast or contradiction between them.

This helps eliminate transitions that suggest an incorrect relationship, such as contrast.

3

Select the option that correctly links the two independent clauses.

A semicolon is the standard way to connect two independent clauses when no coordinating conjunction is used, making 'ground;' the correct choice.

To ensure standard English conventions are followed without creating a comma splice or run-on sentence.

Key Concept

Clause Boundaries and Linking

Question 1886Question →

If

x

satisfies the equation below, what is the value of

x - 1

?

\sqrt{30 - 2x} = x - 3

-4

2

6

8

Show answer & explanation

Answer: 6

Answer

6

The correct answer is

6

. Squaring both sides of the equation

\sqrt{30 - 2x} = x - 3

yields

30 - 2x = x^2 - 6x + 9

. Rearranging terms into standard quadratic form gives

x^2 - 4x - 21 = 0

, which factors as

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

. This gives potential solutions of

x = 7

and

x = -3

. Checking these in the original equation shows that

x = 7

is valid since

\sqrt{30 - 2(7)} = 7 - 3 \implies 4 = 4

, while

x = -3

is extraneous since

\sqrt{30 - 2(-3)} = -3 - 3 \implies 6 \neq -6

. Therefore, the value of the requested expression

x - 1

is

7 - 1 = 6

.

Step-by-Step Solution

1

Square both sides of the equation to eliminate the radical.

30 - 2x = (x - 3)^2

To solve a radical equation, squaring both sides isolates the terms under the square root.

2

Expand the right side and move all terms to one side to set the quadratic equation to zero.

x^2 - 4x - 21 = 0

Expanding

(x - 3)^2

yields

x^2 - 6x + 9

. Subtracting

30

and adding

2x

to both sides results in a standard quadratic form.

3

Factor the quadratic equation.

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

Factoring the quadratic helps find the potential solutions for

x

.

4

Identify potential solutions and substitute them back into the original equation to check for extraneous solutions.

x = 7

is the only valid solution;

x = -3

is extraneous.

Substituting

x = 7

gives

\sqrt{30 - 14} = 7 - 3

, which simplifies to

4 = 4

(true). Substituting

x = -3

gives

\sqrt{30 - 2(-3)} = -3 - 3

, which simplifies to

6 = -6

(false).

5

Calculate the value of the expression

x - 1

using the valid solution

x = 7

.

7 - 1 = 6

The question asks for the value of

x - 1

, so we substitute

7

for

x

.

Key Concept

Solving radical equations and identifying extraneous solutions.

Estimated Time:1m 35s

Question 1887Question →

A commercial bakery uses an automated flour silo. The mass of the flour in the silo, $F$ , in kilograms, is modeled as a linear function of the time $t$ , in hours, after the bakery opens. The table below shows the mass of the flour remaining in the silo at two different times during the day:

Time (hours), $t$	Mass of flour (kilograms), $F$
$3$	$1,850$
$7$	$1,490$

If the mass of the flour in the silo decreases at a constant rate of $r$ kilograms per hour, what is the value of $r$ ?

Show answer & explanation

Answer: 90

Answer

90

The rate of decrease of the flour is represented by the magnitude of the slope of the linear relationship. Using the points

(3, 1850)

and

(7, 1490)

from the table, the slope is calculated as

\frac{1490 - 1850}{7 - 3} = \frac{-360}{4} = -90

. This indicates that the mass of the flour decreases by

90

kilograms per hour. Therefore, the value of

r

is

90

.

Step-by-Step Solution

1

Identify the data points representing time and mass from the table.

(t_1, F_1) = (3, 1850)

and

(t_2, F_2) = (7, 1490)

We need two coordinates to find the slope of the linear relationship.

2

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

Slope =

\frac{1490 - 1850}{7 - 3} = -90

The slope formula

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

gives the rate of change of the mass of the flour per hour.

3

Determine the value of

r

based on the rate of decrease.

r = 90

The rate of decrease is the positive magnitude of the rate of change.

Key Concept

Interpreting rate of change (slope) from tabular data in a linear context

Estimated Time:1m 30s

Question 1888Question →

If $4^{a} \cdot 8^{b} = 32^{5}$ and $a + b = 9$ , what is the value of $b$ ?

2

5

7

11

Show answer & explanation

Answer: 7

Answer

7

By converting all bases to 2, the equation

4^a \cdot 8^b = 32^5

becomes

(2^2)^a \cdot (2^3)^b = (2^5)^5

, which simplifies to

2^{2a+3b} = 2^{25}

. Equating the exponents gives the linear equation

2a + 3b = 25

. Since we are given that

a + b = 9

, we can multiply this equation by 2 to get

2a + 2b = 18

. Subtracting the two equations yields

b = 7

.

Step-by-Step Solution

1

Express each base in the equation

4^a \cdot 8^b = 32^5

as a power of 2.

Since

4 = 2^2

,

8 = 2^3

, and

32 = 2^5

, the equation becomes

(2^2)^a \cdot (2^3)^b = (2^5)^5

.

To solve exponential equations with different bases, it is helpful to express all terms using a common base.

2

Apply the power-of-a-power property

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

and the product-of-powers property

x^m \cdot x^n = x^{m+n}

to simplify both sides.

The left side simplifies to

2^{2a} \cdot 2^{3b} = 2^{2a+3b}

, and the right side simplifies to

2^{25}

. The equation is now

2^{2a+3b} = 2^{25}

.

Simplifying the expressions allows us to equate the exponents.

3

Set the exponents equal to each other to form a linear equation.

2a + 3b = 25

.

If two exponential expressions with the same base are equal, their exponents must be equal.

4

Solve the system of equations consisting of

2a + 3b = 25

and

a + b = 9

.

Multiplying the second equation by 2 gives

2a + 2b = 18

. Subtracting this from

2a + 3b = 25

yields

(2a + 3b) - (2a + 2b) = 25 - 18

, which simplifies to

b = 7

.

Eliminating one variable allows us to solve for the other variable directly.

Key Concept

Solving exponential equations by converting terms to a common base and solving the resulting system of linear equations.

Alternative Method

Substitute the answer options for

b

back into the equations. If

b = 7

, then

a = 9 - 7 = 2

. Plugging these values into the left side of the exponential equation gives

4^2 \cdot 8^7 = 16 \cdot (2^3)^7 = 2^4 \cdot 2^{21} = 2^{25}

. The right side is

32^5 = (2^5)^5 = 2^{25}

. Since both sides are equal, 7 is the correct answer.

Estimated Time:1m 30s

Question 1889Question →

To make a specific shade of green paint, a painter mixes yellow paint and blue paint in a ratio of $3$ to $5$ . If the painter wants to make a total of $40$ gallons of this green paint, how many gallons of yellow paint are needed?

12

15

24

25

Show answer & explanation

Answer: $15$

Answer

The correct amount of yellow paint needed is

15

gallons.

The correct answer is

15

gallons. Since the ratio of yellow to blue paint is

3:5

, the total mixture consists of

3 + 5 = 8

equal parts. Yellow paint represents

3

of these

8

parts, or

\frac{3}{8}

of the total mixture. To find the amount of yellow paint needed for

40

gallons of green paint, multiply the fraction by the total volume:

\frac{3}{8} \times 40 = 15

gallons.

Step-by-Step Solution

1

Determine the fraction of the total mixture that is yellow paint.

The ratio of yellow to blue paint is

3:5

, which gives a total of

3 + 5 = 8

parts. Therefore, yellow paint represents

\frac{3}{8}

of the total mixture.

To find the amount of one component in a given total volume, we must find the part-to-whole ratio of that component to the total mixture.

2

Multiply the fraction representing yellow paint by the total desired volume of green paint.

\frac{3}{8} \times 40 = 3 \times 5 = 15

gallons.

Multiplying the part-to-whole ratio by the total volume gives the specific volume of yellow paint needed.

Key Concept

Solving part-to-whole ratio problems by summing the ratio parts to find the total parts, then multiplying the corresponding fraction by the total quantity.

Question 1890Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = -x^2 + 6x + 7$ has a vertex at $(h, k)$ . If the graph of $f$ is translated $4$ units to the right and $3$ units up to produce the graph of the function $g$ , what is the maximum value of $g$ ?

Show answer & explanation

Answer: 19

Answer

The maximum value of the function

g

is

19

.

The vertex of the original quadratic function

f(x) = -x^2 + 6x + 7

is

(3, 16)

. Since the coefficient of

x^2

is negative, the graph opens downward, making

16

the maximum value of the function. Translating the graph

3

units up shifts all

y

-values up by

3

, which increases the maximum value to

16 + 3 = 19

. The horizontal translation of

4

units to the right shifts the graph horizontally but does not affect the maximum output value.

Step-by-Step Solution

1

Find the vertex of the original quadratic function

f(x) = -x^2 + 6x + 7

.

The vertex of

f(x)

is

(3, 16)

.

Rewrite the function in vertex form,

f(x) = -(x-3)^2 + 16

, to identify the vertex

(h, k)

as

(3, 16)

.

2

Determine the maximum value of the original function

f(x)

.

The maximum value of

f(x)

is

16

.

Since the leading coefficient is negative, the parabola opens downward, and the maximum value occurs at the vertex's

y

-coordinate.

3

Apply the vertical translation to find the maximum value of the new function

g(x)

.

The maximum value of

g

is

19

.

Translating the graph

3

units up increases all

y

-values, including the maximum value, by

3

, so

16 + 3 = 19

. The horizontal translation of

4

units to the right does not affect the maximum value.

Key Concept

Quadratic Functions and Graphs

Question 1891Question →

An online retail company determines that the relationship between the selling price of a product, $x$ dollars, and the daily number of units sold, $y$ , can be modeled by a linear equation. When the selling price is $12$ dollars, the company sells $80$ units per day. For every $3$ dollars increase in the selling price, the number of units sold daily decreases by $15$ . Which of the following equations represents this relationship?

y = -5x + 140

y = -5x + 80

y = -0.2x + 82.4

y = -5x + 20

Show answer & explanation

Answer: $y = -5x + 140$

Answer

The equation

y = -5x + 140

The relationship between price

x

and units sold

y

is linear. The slope

m

represents the rate of change:

m = \frac{\Delta y}{\Delta x} = \frac{-15}{3} = -5

. Using the point-slope form with the known point

(12, 80)

, we get

y - 80 = -5(x - 12)

. Distributing the

-5

gives

y - 80 = -5x + 60

. Adding

80

to both sides results in the equation

y = -5x + 140

.

Step-by-Step Solution

1

Calculate the slope (

m

) of the linear relation.

m = \frac{\text{change in } y}{\text{change in } x} = \frac{-15}{3} = -5

The slope is the rate of change, which represents the decrease of

15

units for every

3

dollars increase in price.

2

Set up the equation using point-slope form with the point

(12, 80)

and the slope

m = -5

.

y - 80 = -5(x - 12)

Point-slope form allows us to write the equation of a line given its slope and a point it passes through.

3

Simplify the equation into slope-intercept form.

y - 80 = -5x + 60 \Rightarrow y = -5x + 140

Distributing the slope

-5

to

-12

yields

+60

, and adding

80

to both sides isolates the variable

y

.

Key Concept

Linear Equations in Two Variables

Alternative Method

Substitute the point

(12, 80)

and slope

m = -5

into the slope-intercept equation

y = mx + b

to find

b

:

80 = -5(12) + b \Rightarrow 80 = -60 + b \Rightarrow b = 140

. Thus,

y = -5x + 140

.

Estimated Time:1m 30s

Question 1892Question →

The equation $\sqrt{2x + 12} - x = -6$ has one real solution. What is this solution?

Show answer & explanation

Answer: 12

Answer

12

Substituting 12 into the original equation

\sqrt{2x + 12} - x = -6

yields

\sqrt{2(12) + 12} - 12 = \sqrt{36} - 12 = 6 - 12 = -6

. Since this creates a true statement, 12 is the unique real solution to the equation.

Step-by-Step Solution

1

Isolate the radical term by adding

x

to both sides of the equation.

\sqrt{2x + 12} = x - 6

Isolating the radical on one side allows us to eliminate it by squaring both sides in the next step.

2

Square both sides of the equation to clear the square root.

2x + 12 = (x - 6)^2

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

3

Expand the squared binomial on the right-hand side.

2x + 12 = x^2 - 12x + 36

Expanding

(x - 6)^2

to

x^2 - 12x + 36

helps set up a standard quadratic form.

4

Subtract

2x

and

12

from both sides to form a quadratic equation equal to zero.

x^2 - 14x + 24 = 0

A quadratic equation must be in the form

ax^2 + bx + c = 0

to solve by factoring or the quadratic formula.

5

Factor the quadratic equation.

(x - 12)(x - 2) = 0

Factoring finds two numbers that multiply to 24 and add to -14, which are -12 and -2.

6

Test the potential solutions

x = 12

and

x = 2

in the original equation to identify any extraneous solutions.

For

x = 2

:

\sqrt{2(2) + 12} - 2 = \sqrt{16} - 2 = 4 - 2 = 2 \neq -6

. For

x = 12

:

\sqrt{2(12) + 12} - 12 = \sqrt{36} - 12 = 6 - 12 = -6

. Thus,

x = 12

is the only valid solution.

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

Key Concept

Solving radical equations by isolating the radical, squaring both sides, solving the resulting quadratic equation, and verifying all potential solutions to check for extraneous values.

Estimated Time:1m 30s

Question 1893Question →

During a 2021 expedition to the Canadian Arctic, marine biologist Dr. Marcus Patel focused on tracking the migration patterns of bowhead whales using satellite telemetry. By analyzing the collected telemetry data, Patel aimed to understand how shifting sea ice affects the whales' feeding paths. Ultimately, the ______ research yielded crucial insights into how rapidly warming ocean temperatures are altering traditional migratory corridors in the region.

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

biologist

biologists

biologist's

biologists'

Show answer & explanation

Answer: biologist's

Answer

The singular possessive noun 'biologist's' correctly completes the sentence by showing that the research belongs to the single biologist, Dr. Marcus Patel.

The correct answer is the singular possessive noun 'biologist's'. The passage establishes that Dr. Marcus Patel is a singular marine biologist. Because the noun following the blank ('research') belongs to him, we need the singular possessive form, which is created by adding an apostrophe followed by 's'.

Step-by-Step Solution

1

Identify the noun being modified and determine if a possessive relationship is required.

The blank precedes 'research'. Since the research belongs to the scientist mentioned, a possessive form of the noun is required.

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

2

Determine the number (singular or plural) of the noun in the context of the passage.

The passage mentions 'marine biologist Dr. Marcus Patel', which refers to a single individual.

The noun must be singular to match the single biologist introduced in the first sentence.

3

Select the form that is both singular and possessive.

The singular possessive form is 'biologist's' (formed by adding -'s to the singular noun 'biologist').

This form correctly conveys that the research belongs to one biologist.

Key Concept

Plural and Possessive Nouns

Estimated Time:45s

Question 1894Question →

A coffee roasting company uses a commercial roasting machine. The temperature of the roasting drum, $T$ , in degrees Fahrenheit ( $^\circ\text{F}$ ), is modeled as a linear function of the roasting time, $t$ , in minutes, where $0 \leq t \leq 10$ . After $2$ minutes of roasting, the temperature of the drum is $280^\circ\text{F}$ . After $5$ minutes of roasting, the temperature of the drum is $385^\circ\text{F}$ . What is the rate of temperature increase, in degrees Fahrenheit per minute, of the roasting drum?

Show answer & explanation

Answer: 35

Answer

The rate of temperature increase of the roasting drum is

35

degrees Fahrenheit per minute.

The temperature,

T

, is modeled as a linear function of time,

t

. The constant rate of temperature increase corresponds to the slope of this linear function. Given two points,

(2, 280)

and

(5, 385)

, the slope is calculated as the change in temperature divided by the change in time:

\frac{385 - 280}{5 - 2} = \frac{105}{3} = 35

degrees Fahrenheit per minute.

Step-by-Step Solution

1

Identify the coordinates representing the relationship between time and temperature.

The two points are

(2, 280)

and

(5, 385)

, where the first coordinate is the time,

t

, in minutes, and the second coordinate is the temperature,

T

, in degrees Fahrenheit.

To find the rate of change of a linear relationship, we first need to determine two points

(t_1, T_1)

and

(t_2, T_2)

from the given context.

2

Calculate the rate of temperature increase as the slope of the line passing through these two points.

The slope

m

is given by

m = \frac{385 - 280}{5 - 2} = \frac{105}{3} = 35

.

The rate of temperature increase per minute is the constant slope of the linear relationship.

Key Concept

Slope of a linear relationship in context

Question 1895Question →

The function $f$ is defined by $f(x) = x^3 - 3x^2 - 10x + k$ , where $k$ is a constant. In the $xy$ -plane, the graph of $y = f(x)$ has $x$ -intercepts at $(c, 0)$ and $(2c, 0)$ , where $c$ is a positive constant. What is the value of $k$ ?

Show answer & explanation

Answer: 24

Answer

24

By applying the factor theorem to the two

x

-intercepts

(c,0)

and

(2c,0)

, we establish the system

c^3 - 3c^2 - 10c + k = 0

and

8c^3 - 12c^2 - 20c + k = 0

. Subtracting these equations yields

7c^3 - 9c^2 - 10c = 0

. Since

c > 0

, dividing by

c

gives

7c^2 - 9c - 10 = 0

, which factors as

(7c+5)(c-2)=0

, yielding the positive solution

c=2

. Substituting

c=2

back into the first equation gives

8 - 12 - 20 + k = 0

, which solves to

k = 24

.

Step-by-Step Solution

1

Set up equations for the roots

c

and

2c

using the factor theorem.

f(c) = c^3 - 3c^2 - 10c + k = 0

and

f(2c) = 8c^3 - 12c^2 - 20c + k = 0

An

x

-intercept at

(r, 0)

means that

r

is a root of the polynomial, so

f(r) = 0

.

2

Subtract the first equation from the second to eliminate the constant

k

.

7c^3 - 9c^2 - 10c = 0

Eliminating

k

allows us to solve for the root

c

directly.

3

Solve the polynomial equation for the positive constant

c

.

c = 2

Dividing the equation by

c

(since

c > 0

) yields

7c^2 - 9c - 10 = 0

, which factors as

(7c+5)(c-2) = 0

. Since

c

must be positive,

c = 2

.

4

Substitute

c = 2

back into the equation for

f(c) = 0

to solve for

k

.

k = 24

Substituting the known root value allows us to find the value of the constant coefficient

k

.

Key Concept

The relationship between a polynomial's algebraic factors, its roots, and its

x

-intercepts in the coordinate plane.

Estimated Time:2m 0s

Question 1896Question →

A deep space satellite transmits data to a ground station at a constant rate of $1.6$ megabits per second. The transmitted data consists of actual scientific data and protocol overhead. The protocol overhead accounts for $25\%$ of the total transmitted bits, meaning the actual scientific data constitutes only $75\%$ of the total transmitted bits. The ground station needs to receive $5$ scientific data files, each with a size of $1.08$ gigabytes. How many hours will it take to transmit all $5$ files? (Given that $1\\text{ byte} = 8\\text{ bits}$ , $1\\text{ megabit} = 10^6\\text{ bits}$ , and $1\\text{ gigabyte} = 10^9\\text{ bytes}$ .)

Show answer & explanation

Answer: 10

Answer

It will take 10 hours to transmit all 5 files.

The correct answer is 10. To find this, first calculate the total size of the scientific files: 5 files * 1.08 gigabytes per file = 5.4 gigabytes. Convert this to bytes: 5.4 * 10^9 bytes. Convert bytes to bits: 5.4 * 10^9 * 8 = 43.2 * 10^9 bits. Since protocol overhead is 25%, the scientific data is 75% of the total transmitted bits, so the total bits transmitted is 43.2 * 10^9 / 0.75 = 57.6 * 10^9 bits. The transmission rate is 1.6 megabits per second, or 1.6 * 10^6 bits per second. The transmission time in seconds is 57.6 * 10^9 / (1.6 * 10^6) = 36,000 seconds. Converting seconds to hours: 36,000 / 3,600 = 10 hours.

Step-by-Step Solution

1

Calculate the total size of the scientific files in gigabytes.

5.4

gigabytes

To find the total amount of scientific data that must be received.

2

Convert the total scientific data size from gigabytes to bytes and then to bits.

43.2 \\times 10^9

bits

The transmission rate is given in megabits per second, so the data size must be converted to bits for unit consistency. Since

1\\text{ GB} = 10^9\\text{ bytes}

and

1\\text{ byte} = 8\\text{ bits}

, we have

5.4 \\times 10^9 \\times 8 = 43.2 \\times 10^9

bits.

3

Calculate the total number of bits transmitted, including protocol overhead.

57.6 \\times 10^9

bits

Since protocol overhead is

25\%

, the scientific data is only

75\%

of the total bits transmitted. Thus, we divide the scientific bits by

0.75

to find the total bits transmitted:

\\frac{43.2 \\times 10^9}{0.75} = 57.6 \\times 10^9

bits.

4

Calculate the transmission time in seconds.

36,000

seconds

Divide the total bits by the transmission rate in bits per second (

1.6\\text{ Mbps} = 1.6 \\times 10^6

bits per second):

\\frac{57.6 \\times 10^9}{1.6 \\times 10^6} = 36,000

seconds.

5

Convert the transmission time from seconds to hours.

10

hours

Since there are

3,600

seconds in one hour, divide the total seconds by

3,600

:

\\frac{36,000}{3,600} = 10

.

Key Concept

Multi-step dimensional analysis and compound rate conversions incorporating percentage overhead

Alternative Method

Instead of converting units step-by-step, dimensional analysis can be set up as a single product of conversion factors: 5 files * (1.08 GB / 1 file) * (10^9 bytes / 1 GB) * (8 bits / 1 byte) * (1 total bit / 0.75 scientific bits) * (1 second / 1.6 * 10^6 bits) * (1 hour / 3600 seconds) = 10 hours.

Estimated Time:3m 0s

Question 1897Question →

In the $xy$ -plane, the graph of the function $g$ is obtained by applying a sequence of transformations to the graph of the function $f(x) = |x + 2| - 3$ . Specifically, the graph of $g$ is a vertical stretch and translation of the graph of $f$ , such that $g(x) = a f(x - h) + k$ for some constants $a$ , $h$ , and $k$ . The vertex of the graph of $g$ is located at $(1, 5)$ , and the graph of $g$ passes through the point $(0, -1)$ . What is the value of $g(3)$ ?

-5

-7

17

-43

Show answer & explanation

Answer: -7

Answer

The value of

g(3)

is

-7

.

The correct value of

-7

is obtained by first identifying the vertex of

f(x) = |x + 2| - 3

at

(-2, -3)

. Comparing this to the vertex of

g(x)

at

(1, 5)

yields the horizontal shift parameter

h = 3

and the equation

-3a + k = 5

. Using the point

(0, -1)

yields the second equation

-2a + k = -1

. Solving this system gives

a = -6

and

k = -13

. Substituting these into the formula for

g(3)

yields

-7

.

Step-by-Step Solution

1

Identify the vertex of the function

f(x) = |x + 2| - 3

.

The vertex of

f(x)

is at

(-2, -3)

.

The vertex of an absolute value function of the form

y = |x - x_0| + y_0

is located at

(x_0, y_0)

.

2

Relate the vertex of

f(x)

to the vertex of

g(x) = a f(x - h) + k

.

h = 3

and

-3a + k = 5

.

The horizontal shift

h

moves the vertex from

x = -2

to

x = 1

, so

-2 + h = 1 \implies h = 3

. The vertical stretch and translation transform the

y

-coordinate of the vertex from

-3

to

5

, so

a(-3) + k = 5

.

3

Use the given point

(0, -1)

to set up a second equation.

-2a + k = -1

.

Since the graph of

g(x)

passes through

(0, -1)

, we evaluate

g(0) = a f(0 - 3) + k = -1

. Evaluating

f(-3) = |-3 + 2| - 3 = -2

yields the equation

-2a + k = -1

.

4

Solve the system of equations for

a

and

k

.

a = -6

and

k = -13

.

Subtracting

-3a + k = 5

from

-2a + k = -1

gives

a = -6

. Substituting

a = -6

back into either equation yields

k = -13

.

5

Evaluate

g(3)

using the completed function formula

g(x) = -6 f(x - 3) - 13

.

g(3) = -7

.

We substitute

x = 3

into the equation to get

g(3) = -6 f(0) - 13

. Evaluating

f(0) = |0 + 2| - 3 = -1

gives

g(3) = -6(-1) - 13 = 6 - 13 = -7

.

Key Concept

Function transformations including horizontal translations, vertical translations, and vertical scaling.

Question 1898Question →

The graph of the quadratic function $f(x) = -2(x - 3)^2 + a$ in the $xy$ -plane has a $y$ -intercept at $(0, -10)$ , where $a$ is a constant. What is the maximum value of $f(x)$ ?

-28

-10

3

8

Show answer & explanation

Answer: $8$

Answer

The maximum value of the function is

8

.

The correct answer is

8

. Since the

y

-intercept of the graph is

(0, -10)

, we substitute

x = 0

into the function:

f(0) = -2(0 - 3)^2 + a = -10

. Simplifying this yields

-2(9) + a = -10

, which becomes

-18 + a = -10

. Adding

18

to both sides gives

a = 8

. The equation of the function is therefore

f(x) = -2(x - 3)^2 + 8

. Since this is in vertex form, the vertex is

(3, 8)

. Because the coefficient of the squared term is negative, the parabola opens downward, and the maximum value of the function is the

y

-coordinate of the vertex, which is

8

.

Step-by-Step Solution

1

Use the given

y

-intercept to set up an equation for the constant

a

.

f(0) = -10 \implies -2(0 - 3)^2 + a = -10

The

y

-intercept occurs where the input

x

is equal to

0

.

2

Simplify the expression and solve for

a

.

-2(-3)^2 + a = -10 \implies -2(9) + a = -10 \implies -18 + a = -10 \implies a = 8

Squaring

-3

yields

9

, and multiplying by

-2

gives

-18

. Adding

18

to both sides isolates

a

.

3

Identify the vertex of the quadratic function and determine the maximum value.

The function is

f(x) = -2(x - 3)^2 + 8

. The vertex of this parabola is

(3, 8)

. Since the leading coefficient

-2

is negative, the parabola opens downward, meaning the

y

-coordinate of the vertex,

8

, is the maximum value of

f(x)

.

A quadratic function in vertex form

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

has its vertex at

(h, k)

. If

a < 0

, the maximum value is

k

.

Key Concept

Identifying the vertex and maximum value of a quadratic function from its vertex form and

y

-intercept.

Question 1899Question →

If $\frac{4^{x+3}}{8^{x-1}} = 16^{3-x}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 1

Answer

1

The correct answer is 1. By writing all terms with a base of 2, the equation is expressed as

\frac{(2^2)^{x+3}}{(2^3)^{x-1}} = (2^4)^{3-x}

. Applying the power rule to simplify the exponents yields

\frac{2^{2x+6}}{2^{3x-3}} = 2^{12-4x}

. Using the quotient rule, we subtract the exponents on the left-hand side to get

2^{(2x+6)-(3x-3)} = 2^{-x+9}

. Equating the exponents gives the linear equation

-x+9 = 12-4x

. Adding

4x

and subtracting

9

from both sides results in

3x = 3

, which simplifies to

x=1

.

Step-by-Step Solution

1

Express all exponential terms using a common base of 2

\frac{(2^2)^{x+3}}{(2^3)^{x-1}} = (2^4)^{3-x}

Rewriting each base as a power of 2 allows the application of standard exponent rules.

2

Apply the power of a power rule

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

to distribute the exponents

\frac{2^{2x+6}}{2^{3x-3}} = 2^{12-4x}

Multiplying the inner exponent by each term of the outer exponent simplifies the expression.

3

Apply the quotient of powers rule

\frac{a^m}{a^n} = a^{m-n}

to combine the fraction

2^{-x+9} = 2^{12-4x}

Subtracting the exponent in the denominator from the exponent in the numerator simplifies the left-hand side.

4

Equate the exponents since the bases are equal

-x + 9 = 12 - 4x

If two exponential expressions with the same positive base are equal, their exponents must be equal.

5

Solve the linear equation for

x

x = 1

Isolating

x

by algebraic manipulation yields the final answer.

Key Concept

Solving exponential equations by finding a common base and applying the laws of exponents.

Question 1900Question →

A study by a wildlife biologist shows that the population of a certain species of songbird in a state park increases by $8\%$ of its previous year's population each year. The initial population of the songbirds in the park was $150$ . If $t$ represents the number of years since the study began, which of the following functions best models the population of songbirds, $P(t)$ , after $t$ years?

P(t) = 150 + 12t

P(t) = 162^t

P(t) = 150(1.08)^t

P(t) = 8 + 150t

Show answer & explanation

Answer: $P(t) = 150(1.08)^t$

Answer

The correct function is

P(t) = 150(1.08)^t

.

The correct answer is the function showing the initial value of

150

multiplied by the growth factor

1.08

raised to the power of

t

. Since the population increases by a constant percentage (

8\%

) each year, the growth is exponential. The general form of an exponential growth model is

P(t) = P_0(1 + r)^t

, where

P_0 = 150

is the initial value and

r = 0.08

is the growth rate. Substituting these values gives

P(t) = 150(1.08)^t

.

Step-by-Step Solution

1

Identify the initial value of the population.

The initial population of songbirds is

150

.

The initial value represents the population when

t = 0

, which corresponds to the coefficient

P_0

in the exponential growth model.

2

Determine the growth rate and calculate the growth factor.

The population increases by

8\%

each year, so the growth rate

r = 0.08

, and the growth factor is

1 + r = 1 + 0.08 = 1.08

.

An increase by a constant percentage each year represents exponential growth, where the growth factor is

1 + r

.

3

Formulate the exponential growth function.

P(t) = 150(1.08)^t

An exponential growth model is written in the form

P(t) = P_0(b)^t

, where

P_0

is the initial value and

b

is the growth factor.

Key Concept

Distinguishing between linear and exponential growth, and constructing exponential models from a percent growth rate.

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