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 282Question →

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 283Question →

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 284Question →

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 285Question →

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 286Question →

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 287Question →

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 288Question →

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 289Question →

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 290Question →

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 291Question →

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 292Question →

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 293Question →

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

Show answer & explanation

Answer: 4

Answer

The value of

d

is 4.

First, the equation of the function

f(x)

is determined in vertex form. Since the vertex of

f

is

(3, 12)

, we write

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

. Substituting the point

(1, 0)

into this equation gives

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

, which yields

a = -3

. Thus,

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

. The transformation

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

shifts the graph of

f

left by

d

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

g

is

(3 - d, 8)

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

x

-coordinate must be negative, meaning

3 - d < 0

, or

d > 3

. The

y

-intercept of

g

is

(0, 5)

, so

g(0) = 5

. Since

g(0) = f(d) - 4

, we have

f(d) = 9

. Substituting

d

into

f(x)

gives

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

, which simplifies to

(d - 3)^2 = 1

. Solving for

d

gives

d = 4

or

d = 2

. Since

d > 3

, the value of

d

must be 4.

Step-by-Step Solution

1

Write the function

f(x)

in vertex form and substitute the point

(1, 0)

to solve for

a

.

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

The vertex

(h, k)

is given as

(3, 12)

, and the point

(1, 0)

lies on the graph.

2

Determine the vertex of

g(x)

based on the horizontal and vertical translations of

f(x)

.

The vertex of

g

is

(3 - d, 8)

.

The transformation

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

shifts the vertex of

f(x)

left by

d

units and down by 4 units.

3

Establish the constraint on

d

using the quadrant of the vertex of

g

.

d > 3

For the vertex

(3 - d, 8)

to lie in the second quadrant, the

x

-coordinate must be negative.

4

Set up an equation for

d

using the

y

-intercept of

g(x)

.

f(d) = 9

The

y

-intercept is

(0, 5)

, so

g(0) = 5

. Substituting this into

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

gives

5 = f(d) - 4

.

5

Solve

f(d) = 9

for

d

.

d = 2

or

d = 4

Substituting

d

into

f(x)

gives

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

, which simplifies to

(d - 3)^2 = 1

.

6

Select the correct value of

d

that satisfies the quadrant constraint.

d = 4

Since

d > 3

, the value

d = 2

is discarded, leaving

d = 4

as the only valid solution.

Key Concept

Vertex form and transformations of quadratic functions

Question 294Question →

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

Show answer & explanation

Answer: 7

Answer

7

The expression simplifies to

4x^2 y^1

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

16x^6 y^4

, and simplifying the radical in the denominator to get

4x^4 y^3

. Dividing the terms yields

4x^2 y^1

. Identifying the coefficients and exponents gives

a = 4

,

b = 2

, and

c = 1

. The sum of these values is

4 + 2 + 1 = 7

.

Step-by-Step Solution

1

Simplify the numerator of the expression.

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

Apply the power of a product rule

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

and the power of a power rule

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

.

2

Simplify the denominator of the expression.

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

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

\sqrt{16} = 4

,

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

, and

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

.

3

Divide the numerator by the denominator.

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

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

\frac{16}{4} = 4

,

x^{6-4} = x^2

, and

y^{4-3} = y^1

.

4

Identify the values of

a

,

b

, and

c

, and sum them.

a = 4

,

b = 2

,

c = 1

, and

a + b + c = 7

Match the simplified form

4x^2 y^1

to

ax^b y^c

to find

a

,

b

, and

c

, then calculate their sum.

Key Concept

Simplifying equivalent algebraic expressions using exponent rules and radical properties.

Estimated Time:1m 30s

Question 295Question →

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

Show answer & explanation

Answer: 4

Answer

4

The correct answer is 4. Adding the term

\frac{2}{x}

to both sides of the equation yields

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

. Cross-multiplying the terms gives

x^2 = 16

. Solving for

x

by taking the square root of both sides gives

x = 4

or

x = -4

. Since the question specifies that

x > 0

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

Step-by-Step Solution

1

Add

\frac{2}{x}

to both sides of the equation.

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

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

2

Cross-multiply the terms.

x^2 = 16

To eliminate the denominators and form a quadratic equation.

3

Solve for

x

by taking the square root of both sides.

x = 4

or

x = -4

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

4

Apply the constraint

x > 0

.

x = 4

The question specifies that

x

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

Key Concept

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

Question 296Question →

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

Show answer & explanation

Answer: 16

Answer

The maximum

y

-value achieved by the parabola is 16.

The slopes of the lines connecting the

x

-intercepts

P(x_1, 0)

and

Q(x_2, 0)

to the

y

-intercept

C(0, 12)

determine their coordinates. The slope of the line through

P

and

C

is

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

, which gives

x_1 = -6

. The slope of the line through

Q

and

C

is

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

, which gives

x_2 = 2

. The factored form of the parabola is

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

. Using the

y

-intercept

C(0, 12)

, we find

12 = a(6)(-2)

, meaning

a = -1

. The

x

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

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

. Substituting

x = -2

into the equation gives

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

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

y

-value.

Step-by-Step Solution

1

Find the coordinates of

P(x_1, 0)

and

Q(x_2, 0)

using the slopes of the lines from these points to the

y

-intercept

C(0, 12)

.

P = (-6, 0)

and

Q = (2, 0)

The slope formula between

(x_1, 0)

and

(0, 12)

is

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

, which gives

x_1 = -6

. Similarly, for the second line,

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

, which gives

x_2 = 2

.

2

Write the quadratic equation in factored form using the

x

-intercepts.

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

Since the parabola intersects the

x

-axis at

(-6, 0)

and

(2, 0)

, these points are the roots of the quadratic function.

3

Solve for the leading coefficient

a

by substituting the

y

-intercept

(0, 12)

into the equation.

a = -1

Substituting

x = 0

and

y = 12

gives

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

, which simplifies to

12 = -12a

, so

a = -1

.

4

Determine the

x

-coordinate of the vertex.

x = -2

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

x

-intercepts:

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

.

5

Calculate the maximum

y

-value by evaluating the function at the

x

-coordinate of the vertex.

y = 16

Substituting

x = -2

into

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

gives

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

.

Key Concept

Using

x

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

Question 297Question →

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

Show answer & explanation

Answer: 5

Answer

5

The system of equations representing the scenario is

4b + 6p = 46

for compost and

8b + 5p = 57

for sand, where

b

represents the number of bags of basic mixture and

p

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

2

gives

8b + 12p = 92

. Subtracting the second equation from this yields

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

, which simplifies to

7p = 35

. Dividing by

7

gives

p = 5

. Thus, the landscaper purchased

5

bags of premium mixture.

Step-by-Step Solution

1

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

Let

b

represent the number of basic mixture bags and

p

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

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

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

2

Multiply the first equation by

2

to facilitate the elimination method.

8b + 12p = 92

Aligning the coefficients of

b

allows us to eliminate

b

by subtracting the two equations.

3

Subtract the second equation from the new equation.

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

This isolates the variable

p

by eliminating the variable

b

.

4

Solve for

p

.

p = 5

Dividing both sides of the equation by

7

gives the final number of premium mixture bags.

Key Concept

Solving systems of linear equations in context

Question 298Question →

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

Show answer & explanation

Answer: 4

Answer

4

The relationship between the total mass of the flour,

M

, and the time,

t

, is linear and can be represented by the equation

M = mt + b

, where

m

is the rate at which flour is added and

b

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

(8, 14)

and

(20, 29)

, the slope

m

is calculated as

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

kilograms per minute. Substituting

m = 1.25

and the point

(8, 14)

into the linear equation gives

14 = 1.25(8) + b

, which simplifies to

14 = 10 + b

. Solving for

b

yields

b = 4

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

4

kilograms.

Step-by-Step Solution

1

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

The rate is

1.25

kilograms per minute.

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

2

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

The initial mass is

4

kilograms.

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

t = 0

minutes.

Key Concept

Interpreting the y-intercept of a linear relationship in context

Question 299Question →

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

Show answer & explanation

Answer: 16

Answer

The value of

k

is 16.

The value of

k

is 16 because when

k = 16

, the points on the graph are

(16, 35)

and

(32, 79)

. The slope of the line is

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

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

y = 2.75x + b

. Using the point

(16, 35)

, we get

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

, which matches the given

y

-intercept of

-9

.

Step-by-Step Solution

1

Calculate the slope of the line in terms of

k

using the two given points

(k, 2k + 3)

and

(2k, 5k - 1)

.

The slope

m

is

\frac{3k - 4}{k}

.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

.

2

Use the

y

-intercept of

-9

, which corresponds to the point

(0, -9)

, along with the point

(k, 2k + 3)

to write another expression for the slope.

The slope

m

is

\frac{2k + 12}{k}

.

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

y

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

3

Equate the two slope expressions and solve for

k

.

k = 16

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

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

. Multiplying by

k

on both sides yields

3k - 4 = 2k + 12

, which simplifies to

k = 16

.

Key Concept

Linear Functions and Graphs

Question 300Question →

In the $xy$ -plane, a line passes through the point $(5, -1)$ and has a slope of $\frac{2}{5}$ . If the line also passes through the point $(15, p)$ , what is the value of $p$ ?

Show answer & explanation

Answer: 3

Answer

The value of

p

is

3

.

To find the value of

p

, the equation of the line can be established using the point-slope form:

y - y_1 = m(x - x_1)

. Substituting the given slope

m = \frac{2}{5}

and the point

(5, -1)

gives

y - (-1) = \frac{2}{5}(x - 5)

. Simplifying this yields

y + 1 = \frac{2}{5}x - 2

, which reduces to

y = \frac{2}{5}x - 3

. Substituting the point

(15, p)

into this equation gives

p = \frac{2}{5}(15) - 3 = 6 - 3 = 3

.

Step-by-Step Solution

1

Determine the equation of the line using point-slope form.

y = \frac{2}{5}x - 3

The equation of a line with slope

m

passing through a point

(x_1, y_1)

is

y - y_1 = m(x - x_1)

. Substituting

m = \frac{2}{5}

and the point

(5, -1)

gives

y - (-1) = \frac{2}{5}(x - 5)

. Simplifying this equation results in

y + 1 = \frac{2}{5}x - 2

, which becomes

y = \frac{2}{5}x - 3

.

2

Substitute the point

(15, p)

into the linear equation.

p = 3

Since the line passes through the point

(15, p)

, the coordinates must satisfy the equation of the line. Substituting

x = 15

and

y = p

into

y = \frac{2}{5}x - 3

gives

p = \frac{2}{5}(15) - 3

, which simplifies to

p = 6 - 3 = 3

.

Key Concept

Using the slope and a point on a line to find another coordinate along the same line.

Answer

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