All practice questions

Question 261Question →

During a dry season, the volume of water $V$ , in millions of gallons, in a municipal reservoir $d$ days after the start of a monitoring period is modeled by the equation $V = 248 - 1.8d$ , where $0 \leq d \leq 60$ . According to the model, at what rate, in gallons per minute, is water leaving the reservoir?

Show answer & explanation

Answer: 1250

Answer

The reservoir is losing water at a rate of

1,250

gallons per minute.

The slope of the linear equation

V = 248 - 1.8d

is

-1.8

, indicating a decrease of

1.8

million gallons of water per day. To express this rate in gallons per minute, we perform the unit conversion. First, convert

1.8

million gallons to gallons:

1.8 \times 1,000,000 = 1,800,000

gallons. Second, convert

1

day to minutes:

24 \text{ hours/day} \times 60 \text{ minutes/hour} = 1,440

minutes. Finally, divide the total gallons by the total minutes:

\frac{1,800,000 \text{ gallons}}{1,440 \text{ minutes}} = 1,250

gallons per minute.

Step-by-Step Solution

1

Identify the daily rate of water leaving the reservoir from the given linear equation.

The daily rate is

1.8

million gallons per day.

The slope of the linear equation

V = 248 - 1.8d

is

-1.8

, which represents the rate of change of the volume with respect to time in days. The negative sign indicates a decrease, meaning water is leaving at a rate of

1.8

million gallons per day.

2

Convert the volume from million gallons to gallons.

1,800,000

gallons.

To convert from millions of gallons to gallons, multiply the value by

1,000,000

, resulting in

1.8 \times 1,000,000 = 1,800,000

.

3

Convert the time unit from days to minutes.

1,440

minutes.

There are

24

hours in a day and

60

minutes in an hour, so

1 \text{ day} = 24 \times 60 = 1,440 \text{ minutes}

.

4

Calculate the rate in gallons per minute.

1,250

gallons per minute.

Divide the volume in gallons by the time in minutes:

\frac{1,800,000 \text{ gallons}}{1,440 \text{ minutes}} = 1,250

.

Key Concept

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

Question 262Question →

A solar power system stores electricity in a battery at a constant rate during the day. The total amount of energy in the battery, in kilowatt-hours (kWh), is a linear function of the number of hours after sunrise. Exactly $3$ hours after sunrise, the battery contains $14$ kWh of energy, and exactly $7$ hours after sunrise, it contains $26$ kWh of energy. If the battery's maximum capacity is $35$ kWh, how many hours after sunrise will it take for the battery to be fully charged?

Show answer & explanation

Answer: 10

Answer

10

To find the number of hours after sunrise it takes to fully charge the battery to 35 kWh, we first determine the constant rate of energy storage. The rate of change is given by the difference in energy divided by the difference in hours:

\frac{26 - 14}{7 - 3} = \frac{12}{4} = 3

kWh per hour. The relationship between energy

E

and hours

h

can be written as

E = 3h + b

. Substituting the known point

(3, 14)

gives

14 = 3(3) + b

, which simplifies to

b = 5

. Thus, the linear equation representing this relationship is

E = 3h + 5

. To find the time when the battery is fully charged to

35

kWh, set

E = 35

in the equation:

35 = 3h + 5

. Subtracting 5 from both sides yields

30 = 3h

, and dividing by 3 gives

h = 10

hours.

Step-by-Step Solution

1

Calculate the rate of energy storage per hour.

3 kWh per hour

Since the energy increases at a constant rate, the rate is the change in energy divided by the change in time:

\frac{26 - 14}{7 - 3} = 3

.

2

Determine the initial energy stored at sunrise.

5 kWh

Using the rate of 3 kWh per hour and the fact that the battery had 14 kWh at 3 hours, we subtract the energy gained in those 3 hours:

14 - 3(3) = 5

.

3

Set up the linear equation and solve for the time when the battery reaches 35 kWh.

10 hours

The linear relationship is

E = 3h + 5

. Setting

E = 35

gives

35 = 3h + 5

. Subtracting 5 from both sides gives

30 = 3h

, which simplifies to

h = 10

.

Key Concept

Linear Equations in Two Variables

Question 263Question →

In the $xy$ -plane, the graph of the linear equation $3x + by = 36$ , where $b$ is a constant, is a line with a slope of $-0.75$ . What is the value of $b$ ?

Show answer & explanation

Answer: 4

Answer

The value of

b

is

4

.

The linear equation

3x + by = 36

can be rewritten in slope-intercept form by subtracting

3x

from both sides to get

by = -3x + 36

, and then dividing all terms by

b

to get

y = -\frac{3}{b}x + \frac{36}{b}

. The slope of this line is the coefficient of

x

, which is

-\frac{3}{b}

. Setting this equal to the given slope of

-0.75

(or

-\frac{3}{4}

) gives

-\frac{3}{b} = -\frac{3}{4}

. Solving for

b

yields

b = 4

.

Step-by-Step Solution

1

Express the given linear equation in slope-intercept form.

y = -\frac{3}{b}x + \frac{36}{b}

To identify the slope of the line in terms of the constant

b

, we rewrite the equation

3x + by = 36

in the form

y = mx + d

.

2

Equate the expression for the slope to the given slope value.

-\frac{3}{b} = -0.75

The coefficient of

x

in the slope-intercept form represents the slope of the line, which is given as

-0.75

.

3

Solve the equation for

b

.

b = 4

Multiply both sides of the equation by

-1

to get

\frac{3}{b} = 0.75

. Since

0.75 = \frac{3}{4}

, we have

\frac{3}{b} = \frac{3}{4}

, which gives

b = 4

.

Key Concept

Converting a linear equation from standard form to slope-intercept form to determine its slope.

Estimated Time:1m 30s

Question 264Question →

In the quadratic equation $x^2 - 15x + k = 0$ , where $k$ is a constant, one of the real solutions is $4$ times the other solution. What is the value of $k$ ?

Show answer & explanation

Answer: 36

Answer

36

The correct answer is 36. By Vieta's formulas, the sum of the roots of the quadratic equation

x^2 - 15x + k = 0

is

15

, and the product of the roots is

k

. Let the roots be

r

and

4r

. Their sum is

r + 4r = 5r = 15

, which gives

r = 3

. The roots are

3

and

4(3) = 12

. The product of the roots is

3 \times 12 = 36

, which represents the constant term

k

.

Step-by-Step Solution

1

Express the relationship between the two roots.

Let the two roots be

r

and

4r

.

We are given that one of the real solutions is 4 times the other solution.

2

Use Vieta's formulas to set up an equation for the sum of the roots.

r + 4r = 15 \implies 5r = 15

For a quadratic equation in the form

x^2 - sx + p = 0

, the sum of the roots is equal to

s

.

3

Solve for the variable

r

and find both roots.

r = 3

, so the roots are

3

and

4(3) = 12

.

Dividing both sides of

5r = 15

by 5 gives

r = 3

. Substituting this back gives the two roots.

4

Calculate the product of the roots to find the value of

k

.

k = 3 \times 12 = 36

For a quadratic equation in the form

x^2 - sx + p = 0

, the product of the roots is equal to

p

, which in this case is

k

.

Key Concept

Using Vieta's formulas to find relationships between the coefficients and roots of a quadratic equation.

Question 265Question →

A commercial 3D printer uses a polymer filament to print prototype parts. The remaining mass of the filament spool, $M$ , in grams, is modeled by a linear function of the total printing time, $t$ , in minutes. When the printer has been running for $20$ minutes, the spool has $850$ grams of filament remaining. When the printer has been running for $50$ minutes, the spool has $685$ grams of filament remaining. According to the model, how many grams of filament does the printer use per minute?

Show answer & explanation

Answer: 5.5

Answer

The printer uses 5.5 grams of filament per minute.

The relationship between the printing time

t

and the remaining mass of the filament

M

is linear. The slope of this line represents the change in filament mass per unit of time. Based on the given coordinates

(20, 850)

and

(50, 685)

, the slope is calculated as

\frac{685 - 850}{50 - 20} = \frac{-165}{30} = -5.5

grams per minute. This means that the remaining mass decreases by

5.5

grams for each additional minute the printer runs, which corresponds to a usage rate of

5.5

grams of filament per minute.

Step-by-Step Solution

1

Identify the data points representing time and remaining filament mass from the context.

The two points are

(20, 850)

and

(50, 685)

.

These points allow us to calculate the rate at which the mass of the filament is changing over time.

2

Calculate the change in remaining mass and the change in printing time.

Change in mass is

685 - 850 = -165

grams. Change in time is

50 - 20 = 30

minutes.

Finding the differences in the dependent variable (mass) and independent variable (time) is the standard method to calculate a rate of change.

3

Calculate the rate of filament usage per minute.

\frac{-165\text{ grams}}{30\text{ minutes}} = -5.5

grams per minute. The rate of usage is the positive magnitude, which is

5.5

grams per minute.

The slope of the linear relationship is negative because the mass is decreasing, but the rate of consumption/usage is represented as a positive quantity.

Key Concept

Interpreting the slope of a linear function in context as a rate of change.

Question 266Question →

If $5^{a + 2} = 25^{b}$ and $a + b = 10$ , what is the value of $b$ ?

Show answer & explanation

Answer: 4

Answer

The correct answer is 4.

The correct answer is 4. By rewriting

25^b

as

(5^2)^b = 5^{2b}

, we can set the exponents equal to get

a + 2 = 2b

. Substituting

a = 10 - b

from the second equation into this relation yields

(10 - b) + 2 = 2b

. Simplifying this gives

12 = 3b

, which results in

b = 4

.

Step-by-Step Solution

1

Rewrite the exponential equation using a common base.

5^{a + 2} = 5^{2b}

Since

25 = 5^2

, we write

25^b

as

(5^2)^b = 5^{2b}

to match the base on the left side.

2

Equate the exponents from the base 5 terms.

a + 2 = 2b

When bases are equal, their exponents must be equal for the equation to hold.

3

Express

a

in terms of

b

using the linear equation

a + b = 10

.

a = 10 - b

This allows for substitution into the equation from Step 2 to solve for a single variable.

4

Substitute

a = 10 - b

into

a + 2 = 2b

and solve for

b

.

b = 4

Substituting gives

(10 - b) + 2 = 2b \implies 12 - b = 2b \implies 12 = 3b \implies b = 4

.

Key Concept

Solving systems of equations involving exponential equations by finding a common base.

Question 267Question →

In the $xy$ -plane, the graph of the quadratic function $f(x) = a(x-3)(x-11)$ , where $a$ is a constant, has a vertex at $(h, k)$ . The graph of the quadratic function $g(x) = -(x-h)^2 + q$ has its vertex at $(h, k+8)$ . If the graphs of $f$ and $g$ intersect at a point with an $x$ -coordinate of $5$ , what is the value of $a$ ?

Show answer & explanation

Answer: 1

Answer

1

The value of

a

is 1. By finding the vertex of

f(x)

at

(7, -16a)

and using the vertex of

g(x)

at

(7, -16a+8)

, we evaluate both functions at

x=5

. Setting

f(5) = g(5)

yields the linear equation

-12a = -16a + 4

, which simplifies to

a = 1

.

Step-by-Step Solution

1

Find the x-coordinate of the vertex of

f(x)

h = 7

The x-coordinate of the vertex of a quadratic function of the form

f(x) = a(x-r_1)(x-r_2)

is the midpoint of its x-intercepts,

r_1 = 3

and

r_2 = 11

.

2

Find the y-coordinate of the vertex of

f(x)

in terms of

a

k = -16a

Evaluate the function at the vertex x-coordinate:

f(7) = a(7-3)(7-11) = -16a

.

3

Write the equation of

g(x)

using the given vertex information

g(x) = -(x-7)^2 + k+8

The vertex of

g(x)

is given as

(h, k+8)

. Since

h = 7

, the vertex form of

g(x)

is

g(x) = -(x-7)^2 + k+8

.

4

Set up the equation for the intersection point at

x = 5

f(5) = g(5)

Since the graphs of

f

and

g

intersect at

x=5

, their y-values must be equal at this point.

5

Substitute expressions and solve for

a

a = 1

Evaluate both sides:

f(5) = a(5-3)(5-11) = -12a

, and

g(5) = -(5-7)^2 + k+8 = -4 + k+8 = k+4

. Substituting

k = -16a

gives

-12a = -16a + 4 \implies 4a = 4 \implies a = 1

.

Key Concept

Quadratic Functions and Graphs

Question 268Question →

Consider the system of linear equations below:

\begin{aligned} 3x + 4y &= 8 \\ ax + 8y &= 15 \end{aligned}

If the system of equations has no solution, and $a$ is a constant, what is the value of $a$ ?

Show answer & explanation

Answer: 6

Answer

The correct answer is 6.

For a system of linear equations to have no solution, the lines representing the equations must be parallel, which requires them to have the same slope but different y-intercepts. Writing

3x + 4y = 8

in slope-intercept form gives

y = -\frac{3}{4}x + 2

, so its slope is

-\frac{3}{4}

. Writing

ax + 8y = 15

in slope-intercept form gives

y = -\frac{a}{8}x + \frac{15}{8}

, so its slope is

-\frac{a}{8}

. Equating the slopes gives

-\frac{3}{4} = -\frac{a}{8}

. Solving for

a

gives

a = 6

. Since the y-intercepts (

2

and

\frac{15}{8}

) are different, the lines are parallel and have no intersection points.

Step-by-Step Solution

1

Convert the first equation

3x + 4y = 8

to slope-intercept form.

y = -\frac{3}{4}x + 2

To find the slope of the first line, which is

-\frac{3}{4}

.

2

Convert the second equation

ax + 8y = 15

to slope-intercept form.

y = -\frac{a}{8}x + \frac{15}{8}

To find the slope of the second line in terms of

a

, which is

-\frac{a}{8}

.

3

Equate the slopes of the two lines.

-\frac{3}{4} = -\frac{a}{8}

Parallel lines have the same slope, and a system with parallel lines has no solution.

4

Solve the equation for

a

.

a = 6

Isolating the variable

a

yields the value of the constant.

Key Concept

A system of linear equations has no solution if the lines represented by the equations are parallel, meaning they have the same slope but different y-intercepts.

Question 269Question →

The table below shows some values for the linear function $f$ .

$x$	$f(x)$
$2$	$5$
$5$	$14$
$8$	$23$

If the function $g$ is defined by $g(x) = f(x) + k$ , where $k$ is a constant, and the $x$ -intercept of the graph of $y = g(x)$ in the $xy$ -plane is $-2$ , what is the value of $k$ ?

Show answer & explanation

Answer: 7

Answer

The value of

k

is

7

.

To find the value of

k

, we first determine the linear function

f(x)

using the values in the table. The slope of

f

is calculated as

\frac{14 - 5}{5 - 2} = 3

. Substituting the point

(2, 5)

into the slope-intercept form

f(x) = 3x + b

gives

5 = 3(2) + b

, which simplifies to

b = -1

. Therefore,

f(x) = 3x - 1

. The function

g

is defined as

g(x) = f(x) + k = 3x - 1 + k

. Since the

x

-intercept of the graph of

y = g(x)

is

-2

, we must have

g(-2) = 0

. Substituting

x = -2

into the equation for

g(x)

gives

3(-2) - 1 + k = 0

, which simplifies to

-7 + k = 0

. Solving for

k

yields

k = 7

.

Step-by-Step Solution

1

Determine the slope of the linear function

f

.

The slope of

f

is

3

.

Using the points

(2, 5)

and

(5, 14)

from the table, the slope is calculated as

m = \frac{14 - 5}{5 - 2} = \frac{9}{3} = 3

.

2

Find the equation for

f(x)

.

f(x) = 3x - 1

Using the slope-intercept form

f(x) = 3x + b

and the point

(2, 5)

, we solve for

b

:

5 = 3(2) + b

, which gives

b = -1

.

3

Set up the equation for

g(x)

and use the

x

-intercept to find the value of

k

.

k = 7

Since

g(x) = f(x) + k

, we have

g(x) = 3x - 1 + k

. The

x

-intercept of

g(x)

is

-2

, so

g(-2) = 0

. This gives

3(-2) - 1 + k = 0

, which simplifies to

-7 + k = 0

, meaning

k = 7

.

Key Concept

Finding the equation of a linear function from tabular data and applying translations and intercepts.

Question 270Question →

Consider the system of equations below:

\begin{aligned} y &= 2x^2 + kx + 9 \\ y &= -2x + 1 \end{aligned}

In the system of equations above, $k$ is a positive constant. If the system has exactly one real solution, what is the value of $k$ ?

Show answer & explanation

Answer: 6

Answer

The value of the positive constant

k

is 6.

By substituting the linear equation into the quadratic equation, we obtain the equation

2x^2 + (k+2)x + 8 = 0

. For the system to have exactly one real solution, the discriminant of this quadratic equation,

\Delta = b^2 - 4ac

, must be equal to zero. Substituting

a = 2

,

b = k+2

, and

c = 8

into the discriminant formula gives

(k+2)^2 - 4(2)(8) = 0

, which simplifies to

(k+2)^2 = 64

. Solving for

k

gives

k+2 = 8

or

k+2 = -8

, resulting in

k = 6

or

k = -10

. Since

k

is specified to be a positive constant, the value of

k

must be 6.

Step-by-Step Solution

1

Substitute the expression for

y

from the linear equation into the quadratic equation.

2x^2 + kx + 9 = -2x + 1

Setting the two expressions for

y

equal to each other allows us to solve for the

x

-coordinates of the intersection points.

2

Rearrange the equation into the standard quadratic form

ax^2 + bx + c = 0

.

2x^2 + (k+2)x + 8 = 0

Grouping the terms by power of

x

lets us identify the coefficients

a = 2

,

b = k+2

, and

c = 8

.

3

Set the discriminant of the quadratic equation to zero.

(k+2)^2 - 4(2)(8) = 0

, which simplifies to

(k+2)^2 - 64 = 0

A quadratic equation has exactly one real solution if and only if its discriminant,

b^2 - 4ac

, is equal to zero.

4

Solve for the positive constant

k

.

k = 6

Solving

(k+2)^2 = 64

yields

k+2 = 8

or

k+2 = -8

, which means

k = 6

or

k = -10

. Since

k

must be a positive constant, we select 6.

Key Concept

Determining the number of solutions of a quadratic-linear system using the discriminant.

Alternative Method

Alternatively, the quadratic equation

2x^2 + (k+2)x + 8 = 0

has exactly one real solution when it is a perfect square trinomial. Factoring out the leading coefficient gives

2(x^2 + \frac{k+2}{2}x + 4) = 0

. For the term inside the parentheses to be a perfect square trinomial of the form

x^2 + 2dx + d^2

, we must have

d^2 = 4

, which means

d = \pm 2

. Thus, the middle coefficient

\frac{k+2}{2}

must equal

2d = \pm 4

. Since

k

is positive, we set

\frac{k+2}{2} = 4

, which solves to

k = 6

.

Estimated Time:1m 30s

Question 271Question →

A researcher monitors the temperature of a chemical compound during an experiment. The temperature, $T$ , in degrees Celsius, of the compound $m$ minutes after heating begins can be modeled by a linear relationship. The temperature increases by $4.5$ degrees Celsius every $2$ minutes. If the temperature of the compound was $-12$ degrees Celsius when heating began, after how many minutes will the temperature of the compound reach $15$ degrees Celsius?

Show answer & explanation

Answer: 12

Answer

The temperature of the compound will reach

15

degrees Celsius after

12

minutes.

The correct answer is

12

because the temperature increases at a constant rate of

2.25

degrees Celsius per minute. Starting from an initial temperature of

-12

degrees Celsius, the equation is

T = 2.25m - 12

. Setting

T = 15

gives

15 = 2.25m - 12

, which simplifies to

27 = 2.25m

, resulting in

m = 12

.

Step-by-Step Solution

1

Calculate the constant rate of temperature increase per minute.

2.25

degrees Celsius per minute

This rate represents the slope of the linear relationship between time and temperature.

2

Formulate the linear equation representing the temperature

T

as a function of the elapsed minutes

m

.

T = 2.25m - 12

The slope is

2.25

and the vertical intercept is the initial temperature of

-12

degrees Celsius.

3

Set

T = 15

and solve the linear equation for

m

.

m = 12

This determines the exact number of minutes needed for the temperature to reach

15

degrees Celsius.

Key Concept

Interpreting and solving linear equations modeled from context

Question 272Question →

To restore a depleted wetland, conservationists pump water into a basin. The volume of water in the basin, $W$ , in thousands of gallons, $t$ hours after the pumping begins is modeled by the equation $W = 12.5t + 85$ . According to the model, how many hours of pumping are required for the volume of water in the basin to increase by $150$ thousand gallons?

Show answer & explanation

Answer: 12

Answer

12

The linear equation is given in slope-intercept form,

W = mt + b

, where

m = 12.5

is the slope and

b = 85

is the y-intercept. In this context, the slope

m = 12.5

represents the rate at which water is pumped into the basin, which is

12.5

thousand gallons per hour. To find the number of hours required for the volume of water to increase by

150

thousand gallons, divide the total increase by the rate:

\frac{150}{12.5} = 12

hours.

Step-by-Step Solution

1

Identify the rate of water volume increase per hour from the equation.

The rate is

12.5

thousand gallons per hour.

In the linear equation

W = 12.5t + 85

, the coefficient of

t

(the slope) represents the rate of change of the water volume with respect to time.

2

Set up a relation to find the time

t

for a volume increase of

150

thousand gallons.

12.5t = 150

The change in volume is equal to the rate of change multiplied by the time elapsed.

3

Solve the equation for

t

.

t = 12

Dividing both sides of the equation

12.5t = 150

by

12.5

isolates

t

.

Key Concept

Interpreting the slope of a linear equation in context as a rate of change.

Estimated Time:1m 30s

Question 273Question →

In the $xy$ -plane, a system of equations consists of the circle with equation $(x - 7)^2 + (y - 4)^2 = 1$ and the line $y = kx$ , where $k$ is a positive constant. If the system has exactly one real solution and $k > 0.5$ , what is the value of $k$ ?

Show answer & explanation

Answer: 0.75

Answer

The correct answer is 3/4 (or 0.75).

The correct answer is

3/4

(or

0.75

). Substituting the line

y = kx

into the circle equation and setting the discriminant of the resulting quadratic equation to zero yields a quadratic in

k

:

48k^2 - 56k + 15 = 0

. Solving this equation gives two positive tangent slopes:

k = 5/12

and

k = 3/4

. Since the problem specifies that

k > 0.5

, we choose

k = 3/4

, which equals

0.75

. Alternatively, using geometry, the distance from the center

(7,4)

to the line

kx - y = 0

must equal the radius

1

. This gives

\frac{|7k-4|}{\sqrt{k^2+1}} = 1

. Squaring both sides yields

(7k-4)^2 = k^2+1

, which simplifies to

48k^2 - 56k + 15 = 0

, giving the same values of

k

.

Step-by-Step Solution

1

Substitute the linear equation

y = kx

into the circle's quadratic equation

(x - 7)^2 + (y - 4)^2 = 1

.

(x - 7)^2 + (kx - 4)^2 = 1

To find the points of intersection between the line and the circle, substitute the expression for

y

from the linear equation into the circle's equation.

2

Expand the terms and write the equation in the standard quadratic form

Ax^2 + Bx + C = 0

.

(1 + k^2)x^2 - (14 + 8k)x + 64 = 0

Expanding

(x - 7)^2

yields

x^2 - 14x + 49

and expanding

(kx - 4)^2

yields

k^2x^2 - 8kx + 16

. Grouping the terms by powers of

x

and subtracting

1

from both sides gives the standard form.

3

Set the discriminant of the quadratic equation to

0

to satisfy the condition of exactly one real solution.

\Delta = [-(14 + 8k)]^2 - 4(1 + k^2)(64) = 0

A quadratic equation has exactly one real solution if and only if its discriminant,

B^2 - 4AC

, is equal to zero. Geometrically, this means the line is tangent to the circle.

4

Simplify the discriminant equation and solve for

k

.

48k^2 - 56k + 15 = 0

, which factors as

(12k - 5)(4k - 3) = 0

. Thus,

k = \frac{5}{12}

or

k = \frac{3}{4}

.

Simplifying

(14 + 8k)^2 - 256(1 + k^2) = 0

gives

196 + 224k + 64k^2 - 256 - 256k^2 = 0

, which simplifies to

-192k^2 + 224k - 60 = 0

. Dividing the entire equation by

-4

yields the quadratic equation in

k

.

5

Select the value of

k

that satisfies the given condition

k > 0.5

.

k = \frac{3}{4}

(or

0.75

)

Since

\frac{5}{12} \approx 0.417

is less than

0.5

and

\frac{3}{4} = 0.75

is greater than

0.5

, the correct value is

3/4

.

Key Concept

Solving a nonlinear system of equations involving a circle and a line by setting the discriminant of the substituted quadratic equation to zero to find the slope of the tangent lines.

Estimated Time:2m 30s

Question 274Question →

If $\left(\frac{1}{8}\right)^{2x - 3} = 16^{x - 2}$ , what is the value of $x$ ?

Show answer & explanation

Answer: 1.7

Answer

1.7 (or 17/10)

The correct answer is 1.7 (or 17/10). By writing

\frac{1}{8}

as

2^{-3}

and

16

as

2^4

, the equation becomes

(2^{-3})^{2x - 3} = (2^4)^{x - 2}

. Applying exponent rules yields

2^{-6x + 9} = 2^{4x - 8}

. Equating the exponents gives

-6x + 9 = 4x - 8

, which simplifies to

10x = 17

, resulting in

x = 1.7

.

Step-by-Step Solution

1

Express both bases as powers of

2

.

(2^{-3})^{2x - 3} = (2^4)^{x - 2}

To solve the exponential equation, both sides must have a common base. Both

\frac{1}{8}

and

16

can be rewritten as powers of

2

.

2

Apply the power-of-a-power exponent rule

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

to simplify the exponents.

2^{-3(2x - 3)} = 2^{4(x - 2)}

, which simplifies to

2^{-6x + 9} = 2^{4x - 8}

This allows us to write each side as a single exponential term with a base of

2

.

3

Set the exponents equal to each other.

-6x + 9 = 4x - 8

Since the bases are equal, their exponents must be equal for the equation to hold.

4

Solve the linear equation for

x

.

10x = 17

, which gives

x = 1.7

Isolate the variable

x

by algebraic manipulation.

Key Concept

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

Alternative Method

Alternatively, you can apply the definition of logarithms to both sides of the equation. Taking the logarithm base 2 of both sides gives

\log_2\left(\left(\frac{1}{8}\right)^{2x-3}\right) = \log_2\left(16^{x-2}\right)

. Using the power property of logarithms, this simplifies to

(2x - 3)\log_2\left(2^{-3}\right) = (x - 2)\log_2\left(2^4\right)

, which simplifies directly to

-3(2x - 3) = 4(x - 2)

. Solving this linear equation yields the same result,

x = 1.7

.

Estimated Time:1m 30s

Question 275Question →

The profit $P(x)$ , in dollars, a company makes from selling $x$ units of a product is given by the function $P(x) = -3x^2 + bx - 1,500$ , where $b$ is a positive constant. If the maximum profit the company can make is $1,200$ dollars, what is the value of $b$ ?

Show answer & explanation

Answer: 180

Answer

180

The maximum profit of the quadratic profit function occurs at its vertex. The x-coordinate of the vertex of

P(x) = -3x^2 + bx - 1,500

is given by

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

. Substituting this into the profit equation and setting it equal to

1,200

yields

-3\left(\frac{b}{6}\right)^2 + b\left(\frac{b}{6}\right) - 1,500 = 1,200

. Simplifying this expression results in

\frac{b^2}{12} = 2,700

, which gives

b^2 = 32,400

. Since

b

is positive,

b = 180

.

Step-by-Step Solution

1

Identify the x-coordinate of the vertex of the quadratic function in terms of

b

.

The x-coordinate of the vertex is

x = \frac{b}{6}

.

A quadratic function in the form

f(x) = ax^2 + bx + c

has its maximum value at the vertex when the coefficient of the squared term,

a

, is negative. The x-coordinate of the vertex is given by

x = -\frac{b}{2a}

. For this function,

a = -3

, so the x-coordinate of the vertex is

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

.

2

Substitute

x = \frac{b}{6}

back into the profit equation

P(x)

and set it equal to the maximum profit of

1,200

dollars.

-3\left(\frac{b}{6}\right)^2 + b\left(\frac{b}{6}\right) - 1,500 = 1,200

The maximum profit of

1,200

dollars is the value of the function at the vertex.

3

Simplify the equation to solve for the positive constant

b

.

b = 180

Simplifying the term

-3\left(\frac{b}{6}\right)^2

gives

-3\left(\frac{b^2}{36}\right) = -\frac{b^2}{12}

. Simplifying the term

b\left(\frac{b}{6}\right)

gives

\frac{b^2}{6}

. Substituting these back in gives

-\frac{b^2}{12} + \frac{b^2}{6} - 1,500 = 1,200

. Combining the

b^2

terms gives

\frac{b^2}{12} - 1,500 = 1,200

. Adding

1,500

to both sides gives

\frac{b^2}{12} = 2,700

. Multiplying both sides by

12

yields

b^2 = 32,400

. Taking the square root of both sides gives

b = \pm 180

. Since

b

must be a positive constant,

b = 180

.

Key Concept

Finding the maximum value of a quadratic function by using the vertex formula

x = -\frac{b}{2a}

and evaluating the function at that point.

Question 276Question →

For all $x > 3$ , the expression $\frac{2x^2 - 11x + 15}{x - 3} + \frac{3x^2 - 12}{x - 2}$ is equivalent to $ax + b$ , where $a$ and $b$ are constants. What is the value of $a + b$ ?

Show answer & explanation

Answer: 6

Answer

The value of

a + b

is

6

.

The correct answer is

6

. By factoring the numerators of both rational expressions, we can simplify them. The first term is

\frac{(2x-5)(x-3)}{x-3} = 2x-5

. The second term is

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

. Adding these simplified terms gives

(2x-5) + (3x+6) = 5x+1

. Comparing this to

ax+b

yields

a=5

and

b=1

, so

a+b = 5+1=6

.

Step-by-Step Solution

1

Factor the numerator of the first rational expression and simplify.

\frac{2x^2 - 11x + 15}{x - 3} = 2x - 5

Since

2x^2 - 11x + 15 = (2x - 5)(x - 3)

, the factor

(x - 3)

divides out for all

x > 3

.

2

Factor the numerator of the second rational expression and simplify.

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

Since

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

, the factor

(x - 2)

divides out for all

x > 3

.

3

Combine the simplified terms by adding them.

5x + 1

Adding

(2x - 5)

and

(3x + 6)

yields

(2x + 3x) + (-5 + 6) = 5x + 1

.

4

Identify the values of

a

and

b

and calculate their sum.

a = 5

,

b = 1

, and

a + b = 6

Comparing

5x + 1

with

ax + b

gives

a = 5

and

b = 1

, so

a + b = 5 + 1 = 6

.

Key Concept

Simplifying rational expressions by factoring the numerator and dividing out common factors.

Question 277Question →

For what value of $x$ is the equation $8^{x + 1} = 16^x$ true?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

The correct answer is 3. To find the value of

x

, express both 8 and 16 as powers of 2:

8 = 2^3

and

16 = 2^4

. Substituting these values into the equation gives

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

. Applying the power of a power rule,

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

, yields

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

, which simplifies to

2^{3x + 3} = 2^{4x}

. Since the bases are now the same, their exponents must be equal:

3x + 3 = 4x

. Subtracting

3x

from both sides gives

x = 3

.

Step-by-Step Solution

1

Express the bases 8 and 16 as powers of 2.

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

Writing both sides of the equation with a common base allows for the equating of exponents.

2

Apply the power of a power rule

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

to simplify the exponents.

2^{3x + 3} = 2^{4x}

This simplifies each side of the equation to a single base with a single exponent.

3

Equate the exponents and solve the resulting linear equation.

3x + 3 = 4x

, which simplifies to

x = 3

Since the bases are equal, their exponents must be equal for the equation to hold true.

Key Concept

Solving exponential equations by expressing both sides with a common base and equating the exponents.

Alternative Method

An alternative approach is to write the equation in terms of base 4. While 8 is not an integer power of 4, we can write

8 = 4^{1.5}

and

16 = 4^2

. The equation becomes

(4^{1.5})^{x+1} = (4^2)^x

, which simplifies to

1.5(x+1) = 2x

. Solving this gives

1.5x + 1.5 = 2x \implies 0.5x = 1.5 \implies x = 3

.

Estimated Time:45s

Question 278Question →

The graph of the quadratic function $f$ in the $xy$ -plane has a vertex at $(4, 7)$ . If the function $g$ is defined by $g(x) = f(x) + 5$ , what is the $y$ -coordinate of the vertex of the graph of $y = g(x)$ ?

Show answer & explanation

Answer: 12

Answer

The correct answer is 12.

The vertex of the graph of the quadratic function

f

is

(4, 7)

. The function

g(x) = f(x) + 5

adds

5

to every output value of

f(x)

, which shifts the entire graph vertically upward by

5

units. Consequently, the vertex of the graph of

g

is translated from

(4, 7)

to

(4, 7 + 5)

, which simplifies to

(4, 12)

. Therefore, the

y

-coordinate of the vertex of the graph of

y = g(x)

is

12

.

Step-by-Step Solution

1

Identify the y-coordinate of the vertex of the graph of the function f.

The y-coordinate of the vertex of the graph of f is 7.

The vertex of the graph of f is given as (4, 7).

2

Determine the vertical translation from f(x) to g(x).

The graph of g(x) is shifted upward by 5 units relative to the graph of f(x).

The term + 5 in the definition g(x) = f(x) + 5 increases all y-values of the function by 5.

3

Calculate the y-coordinate of the vertex of the graph of g.

The new y-coordinate is 7 + 5 = 12.

Applying the vertical shift of 5 units to the original y-coordinate of 7 gives the new vertex's y-coordinate.

Key Concept

Vertical translations of quadratic functions and their graphs

Question 279Question →

A quadratic function $f$ is defined by $f(x) = a(x - h)^2 + k$ , where $a$ , $h$ , and $k$ are constants. The graph of $y = f(x)$ in the $xy$ -plane passes through the points $(0, 5)$ and $(4, 5)$ . If the minimum value of $f(x)$ for $0 \le x \le 3$ is $1$ , what is the value of $f(6)$ ?

Show answer & explanation

Answer: 17

Answer

The value of

f(6)

is

17

.

The correct answer is

17

. Since the quadratic function passes through

(0, 5)

and

(4, 5)

, its axis of symmetry is the line

x = 2

, which means the vertex

x

-coordinate is

h = 2

. For the interval

0 \le x \le 3

, this vertex is within the bounds. An upward-opening parabola has its minimum value at its vertex, so the minimum value of

1

must be the

y

-coordinate of the vertex, giving

k = 1

. The function can then be written as

f(x) = a(x - 2)^2 + 1

. Substituting

(0, 5)

into this equation gives

5 = a(0 - 2)^2 + 1

, which simplifies to

4a = 4

, meaning

a = 1

. The fully determined function is

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

. Evaluating this at

x = 6

yields

f(6) = (6 - 2)^2 + 1 = 17

.

Step-by-Step Solution

1

Determine the axis of symmetry of the parabola.

h = 2

Since the points

(0, 5)

and

(4, 5)

have the same

y

-value, they are symmetric across the vertical line of symmetry of the parabola. The

x

-coordinate of the vertex

h

is the average of the

x

-coordinates of these two points:

h = \frac{0 + 4}{2} = 2

.

2

Determine the

y

-coordinate of the vertex

k

using the minimum value over the interval

[0, 3]

.

k = 1

The vertex

x

-coordinate

x = 2

lies within the interval

0 \le x \le 3

. For a quadratic function that opens upward, the minimum value occurs at the vertex. Thus, the minimum value on this interval is the vertex

y

-coordinate, so

k = 1

.

3

Substitute the point

(0, 5)

into the vertex form of the equation to solve for

a

.

a = 1

Using the vertex form

f(x) = a(x - 2)^2 + 1

, substitute

x = 0

and

f(x) = 5

:

5 = a(0 - 2)^2 + 1 \implies 5 = 4a + 1 \implies 4a = 4 \implies a = 1

.

4

Evaluate

f(6)

using the complete quadratic function formula.

f(6) = 17

Substitute

x = 6

into the equation

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

:

f(6) = (6 - 2)^2 + 1 = 4^2 + 1 = 16 + 1 = 17

.

Key Concept

Using symmetry properties and interval extrema to determine the equation of a quadratic function in vertex form.

Question 280Question →

\begin{aligned} y + 11 &= x^2 \\ y - 3x &= 7 \end{aligned}

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

Show answer & explanation

Answer: 31

Answer

The correct answer is 31.

Solving the system of equations by expressing

y

in terms of

x

from the linear equation gives

y = 3x + 7

. Substituting this expression for

y

in the first equation yields

(3x + 7) + 11 = x^2

, which simplifies to

x^2 - 3x - 18 = 0

. Factoring the quadratic expression gives

(x - 6)(x + 3) = 0

, so

x = 6

or

x = -3

. Since the problem specifies that

x > 0

, the

x

-value of the solution must be

6

. Substituting

x = 6

back into the linear equation gives

y = 3(6) + 7 = 25

. The sum of

x

and

y

is

6 + 25 = 31

.

Step-by-Step Solution

1

Express

y

in terms of

x

from the linear equation.

y = 3x + 7

This isolates the variable

y

to facilitate substitution.

2

Substitute the expression for

y

into the quadratic equation.

(3x + 7) + 11 = x^2

This eliminates the variable

y

, yielding a single equation in terms of

x

.

3

Rearrange the equation into standard quadratic form.

x^2 - 3x - 18 = 0

Standard form (

ax^2 + bx + c = 0

) is required to solve by factoring.

4

Factor the quadratic equation.

(x - 6)(x + 3) = 0

Finding factors helps determine the possible values of

x

.

5

Find the values of

x

and apply the positive constraint.

x = 6

The equation yields

x = 6

and

x = -3

. The condition

x > 0

restricts the solution to

x = 6

.

6

Calculate the corresponding value of

y

.

y = 25

Substituting

x = 6

into the linear equation gives

y = 3(6) + 7 = 25

.

7

Calculate the sum of

x

and

y

.

31

The problem asks for the value of

x + y

.

Key Concept

Solving systems of nonlinear equations algebraically using substitution and quadratic factoring.

Alternative Method

Alternatively, solve the linear equation for

x

to get

x = \frac{y - 7}{3}

and substitute this into the quadratic equation to solve for

y

first. This approach is more complex because it introduces fractional terms.

Estimated Time:1m 30s

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