All practice questions

Question 521Question →

A quadratic function $f$ is defined by $f(x) = a(x - 4)(x - k)$ , where $a$ and $k$ are constants. In the $xy$ -plane, the graph of $y = f(x)$ has its vertex at $(6, 12)$ . What is the value of $k$ ?

Show answer & explanation

Answer: 8

Answer

The value of

k

is

8

.

The given quadratic function is in the factored form

f(x) = a(x - 4)(x - k)

, meaning the x-intercepts of its graph are

x = 4

and

x = k

. Due to the symmetry of a parabola, the x-coordinate of the vertex is the midpoint of the x-intercepts. Since the vertex is at

(6, 12)

, its x-coordinate is

6

. Setting the midpoint of the intercepts equal to 6 gives the equation

\frac{4 + k}{2} = 6

. Multiplying both sides by 2 gives

4 + k = 12

, and subtracting 4 gives the correct answer

k = 8

.

Step-by-Step Solution

1

Identify the x-intercepts of the function

f(x) = a(x - 4)(x - k)

.

The x-intercepts are at

x = 4

and

x = k

.

For any quadratic function written in the factored form

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

, the values

r_1

and

r_2

correspond to the x-intercepts of the graph.

2

Relate the x-coordinate of the vertex to the x-intercepts using symmetry.

The axis of symmetry equation is

\frac{4 + k}{2} = 6

.

Because a parabola is symmetric, the x-coordinate of the vertex always lies exactly halfway between the two x-intercepts.

3

Solve the equation for

k

.

k = 8

.

Multiplying both sides of the equation

\frac{4 + k}{2} = 6

by 2 yields

4 + k = 12

. Subtracting 4 from both sides gives

k = 8

.

Key Concept

Symmetry of quadratic graphs and their vertices relative to their x-intercepts.

Alternative Method

Alternatively, substitute the vertex coordinates

(6, 12)

into the function:

12 = a(6 - 4)(6 - k) = 2a(6 - k)

. Since the vertex is the maximum point, the derivative

f'(x) = a(x - k) + a(x - 4)

must be equal to 0 at

x = 6

. This yields

a(6 - k) + a(6 - 4) = 0

. Since

a \neq 0

, dividing by

a

gives

(6 - k) + 2 = 0

, which simplifies to

8 - k = 0

, or

k = 8

.

Estimated Time:1m 30s

Question 522Question →

The function $f$ is defined by $f(x) = |x - 6| + 3$ . In the $xy$ -plane, the graph of the function $g$ is obtained by reflecting the graph of $f$ across the $x$ -axis and then translating it vertically upward by 11 units. If $g(a) = 0$ and $a > 0$ , what is the value of $a$ ?

Show answer & explanation

Answer: 14

Answer

The value of

a

is 14.

Reflecting

f(x) = |x - 6| + 3

across the

x

-axis negates the entire expression, giving

-f(x) = -|x - 6| - 3

. Translating this graph upward by 11 units adds 11 to the expression, which yields

g(x) = -|x - 6| - 3 + 11 = -|x - 6| + 8

. Setting

g(a) = 0

results in

-|a - 6| + 8 = 0

, which simplifies to

|a - 6| = 8

. The solutions to this equation are

a = 14

and

a = -2

. Since the problem specifies that

a > 0

, the correct answer is 14.

Step-by-Step Solution

1

Reflect the function

f(x)

across the

x

-axis.

-f(x) = -|x - 6| - 3

Reflecting a graph across the

x

-axis negates the entire function expression, transforming

y = f(x)

to

y = -f(x)

.

2

Translate the reflected function upward by 11 units to obtain

g(x)

.

g(x) = -|x - 6| + 8

Translating a function vertically upward by

k

units adds

k

to the expression, so

g(x) = -f(x) + 11 = -|x - 6| - 3 + 11 = -|x - 6| + 8

.

3

Set

g(a) = 0

and solve for

a

.

|a - 6| = 8 \implies a = 14

or

a = -2

Solving the equation

-|a - 6| + 8 = 0

requires isolating the absolute value term to get

|a - 6| = 8

. This splits into two cases:

a - 6 = 8

and

a - 6 = -8

.

4

Apply the constraint

a > 0

to identify the final answer.

a = 14

The problem specifies that

a

must be positive, which excludes the solution

a = -2

and leaves

a = 14

.

Key Concept

Function transformations including reflections across the axes and vertical translations.

Estimated Time:1m 30s

Question 523Question →

A straight ladder is leaning against a vertical wall. The base of the ladder is placed $9$ feet from the bottom of the wall. If the top of the ladder touches the wall at a height of $12$ feet above the ground, what is the length, in feet, of the ladder?

Show answer & explanation

Answer: 15

Answer

The length of the ladder is

15

feet.

By representing the scenario as a right triangle, the two perpendicular sides (legs) have lengths of

9

feet and

12

feet. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we find

9^2 + 12^2 = 81 + 144 = 225

. Taking the square root of

225

gives the hypotenuse length, which is

15

feet.

Step-by-Step Solution

1

Model the scenario using a right triangle.

A right triangle where the vertical leg is

12

feet (height on the wall), the horizontal leg is

9

feet (distance along the ground), and the hypotenuse is

c

(the ladder length).

The intersection of the vertical wall and the horizontal ground forms a right angle (

90^\circ

).

2

Set up the Pythagorean equation.

9^2 + 12^2 = c^2

The Pythagorean theorem states that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse.

3

Solve for the hypotenuse

c

.

c = \sqrt{81 + 144} = \sqrt{225} = 15

Simplifying the arithmetic yields

225

, and taking the square root gives the final length of

15

feet.

Key Concept

Pythagorean Theorem

Estimated Time:45s

Question 524Question →

The table below shows several values of $x$ and the corresponding values of the third-degree polynomial function $f$ .

$x$	$f(x)$
$-1$	$0$
$2$	$0$
$5$	$0$
$0$	$30$

If $f(x) = a(x-r_1)(x-r_2)(x-r_3)$ for all real numbers $x$ , where $a$ , $r_1$ , $r_2$ , and $r_3$ are constants, what is the value of $a$ ?

Show answer & explanation

Answer: 3

Answer

3

Since the function

f(x)

is a third-degree polynomial with roots at

x = -1

,

x = 2

, and

x = 5

, it can be factored as

f(x) = a(x+1)(x-2)(x-5)

. Evaluating this expression at

x = 0

gives

f(0) = a(1)(-2)(-5) = 10a

. From the table,

f(0) = 30

, so setting

10a = 30

yields

a = 3

.

Step-by-Step Solution

1

Identify the roots and factors of the polynomial

f(x)

using the table.

The roots are

x = -1

,

x = 2

, and

x = 5

, which correspond to the factors

(x+1)

,

(x-2)

, and

(x-5)

.

Points where

f(x) = 0

represent the

x

-intercepts (roots) of the function.

2

Write the general form of the cubic polynomial using its factors.

f(x) = a(x+1)(x-2)(x-5)

A third-degree polynomial with three distinct real roots can be factored completely as

a(x-r_1)(x-r_2)(x-r_3)

.

3

Substitute the point

(0, 30)

into the equation to solve for the constant coefficient

a

.

a = 3

The table gives

f(0) = 30

, which allows us to set up the equation

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

and solve for

a

.

Key Concept

Using the relationship between the factors, roots, and points on the graph of a polynomial function to determine its equation.

Question 525Question →

In triangle $ABC$ , point $D$ lies on side $BC$ such that the ratio of $BD$ to $DC$ is $1$ to $2$ . Point $E$ lies on segment $AD$ such that the ratio of $AE$ to $ED$ is $3$ to $1$ . A line passing through $E$ and parallel to $AC$ intersects side $AB$ at $F$ and side $BC$ at $G$ . If the area of triangle $ABC$ is $128$ , what is the area of triangle $BFG$ ?

Show answer & explanation

Answer: 32

Answer

32

The correct answer is 32. By establishing a coordinate system, we find that the similarity ratio of triangle

BFG

to triangle

ABC

is exactly

1/2

regardless of the triangle's shape. Since the ratio of the areas of similar triangles is the square of the similarity ratio, the area of triangle

BFG

is

(1/2)^2 = 1/4

of the area of triangle

ABC

, which is

128 \times 1/4 = 32

.

Step-by-Step Solution

1

Set up a coordinate system to represent the triangle's vertices.

Let

B = (0, 0)

and

C = (3, 0)

. Since

D

lies on side

BC

and

BD:DC = 1:2

, the coordinates of

D

are

(1, 0)

. Let

A = (a, b)

.

Setting up coordinates simplifies the proof by allowing algebraic verification of the ratio.

2

Calculate the coordinates of point

E

on segment

AD

.

Using the section formula with ratio

AE:ED = 3:1

,

E = \left(\frac{1(a) + 3(1)}{4}, \frac{1(b) + 3(0)}{4}\right) = \left(\frac{a+3}{4}, \frac{b}{4}\right)

.

Finding the coordinates of

E

is necessary to determine the equation of line

FG

.

3

Find the equation of line

FG

which is parallel to

AC

and passes through

E

.

The slope of

AC

is

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

. The equation of

FG

is

y - \frac{b}{4} = \frac{-b}{3-a}\left(x - \frac{a+3}{4}\right)

.

The line

FG

is parallel to

AC

, meaning they share the same slope.

4

Find the coordinates of

G

by setting

y = 0

in the equation of

FG

.

Setting

y = 0

yields

-\frac{b}{4} = \frac{-b}{3-a}\left(x_G - \frac{a+3}{4}\right) \implies x_G = 1.5

. Thus,

G = (1.5, 0)

.

Point

G

is the intersection of the line

FG

with side

BC

(the x-axis).

5

Determine the similarity ratio and calculate the area of triangle

BFG

.

Since

G

is the midpoint of

BC

, the similarity ratio of

\triangle BFG

to

\triangle BAC

is

k = \frac{1}{2}

. The ratio of their areas is

k^2 = \frac{1}{4}

. The area of

\triangle BFG

is

128 \times \frac{1}{4} = 32

.

Similar triangles have area ratios equal to the square of their similarity ratio.

Key Concept

The ratio of the areas of similar triangles is equal to the square of their similarity ratio.

Alternative Method

Using Menelaus's Theorem on triangle

ABD

and transversal

FEG

can also establish the midpoint relations directly without using coordinates.

Estimated Time:3m 0s

Question 526Question →

An acute angle $\theta$ satisfies the equation $\sin(\theta) = 0.6$ . What is the value of $\cos(\theta)$ ?

Show answer & explanation

Answer: 0.8

Answer

The value of

\cos(\theta)

is 0.8.

Using the fundamental Pythagorean trigonometric identity

\sin^2(\theta) + \cos^2(\theta) = 1

, we substitute the given value

\sin(\theta) = 0.6

to obtain

(0.6)^2 + \cos^2(\theta) = 1

, which simplifies to

0.36 + \cos^2(\theta) = 1

. Subtracting

0.36

from both sides yields

\cos^2(\theta) = 0.64

. Taking the positive square root because

\theta

is an acute angle gives

\cos(\theta) = 0.8

.

Step-by-Step Solution

1

State the Pythagorean trigonometric identity

\sin^2(\theta) + \cos^2(\theta) = 1

This identity relates the sine and cosine of any angle.

2

Substitute the value of

\sin(\theta)

into the identity

(0.6)^2 + \cos^2(\theta) = 1

We are given that

\sin(\theta) = 0.6

.

3

Solve for

\cos^2(\theta)

\cos^2(\theta) = 0.64

Subtracting

0.36

from

1

isolates the squared cosine term.

4

Take the square root of both sides

\cos(\theta) = 0.8

Since

\theta

is an acute angle, the value of

\cos(\theta)

must be positive.

Key Concept

Pythagorean Identity

Question 527Question →

In the $xy$ -plane, the graph of the quadratic function $f$ , defined by $f(x) = -(x - 4)^2 + c$ where $c$ is a constant, intersects the $x$ -axis at two points. If the distance between these two points is $10$ , what is the value of $c$ ?

Show answer & explanation

Answer: 25

Answer

25

The quadratic function is defined by

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

, which is in the vertex form

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

. The vertex of this parabola is at

(4, c)

, and the vertical line

x = 4

is its axis of symmetry. The distance between the two

x

-intercepts is

10

. By symmetry, the intercepts must be located

5

units to the left and right of the axis of symmetry, placing them at

x = 4 - 5 = -1

and

x = 4 + 5 = 9

. Since the graph intersects the

x

-axis at these points, substituting either

x

-coordinate into the function gives a

y

-value of

0

. Substituting

x = 9

yields

0 = -(9 - 4)^2 + c

, which simplifies to

0 = -25 + c

, so

c = 25

.

Step-by-Step Solution

1

Identify the axis of symmetry of the quadratic function.

The axis of symmetry is

x = 4

.

The function

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

is written in vertex form,

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

, where the vertex is

(h, k)

and the axis of symmetry is

x = h

.

2

Find the

x

-coordinates of the

x

-intercepts using the distance between them.

The

x

-intercepts are at

x = -1

and

x = 9

.

Since the parabola is symmetric about the line

x = 4

and the distance between the two intercepts is

10

, each intercept is

\frac{10}{2} = 5

units away from the axis of symmetry. Thus, the intercepts are at

4 - 5 = -1

and

4 + 5 = 9

.

3

Substitute one of the

x

-intercepts into the function to solve for the constant

c

.

c = 25

Since

(9, 0)

is on the graph,

f(9) = 0

. Substituting this gives

0 = -(9 - 4)^2 + c \implies 0 = -25 + c \implies c = 25

.

Key Concept

Using symmetry and the vertex form of a quadratic function to determine constants from key graphical features such as

x

-intercepts.

Question 528Question →

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ . If $\cos(A) = 3\sin(A)$ , what is the value of $\tan(B)$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is

3

.

Because angle

C

is

90^\circ

in right triangle

ABC

, the acute angles

A

and

B

are complementary (

A + B = 90^\circ

). By the co-function identities,

\sin(B) = \cos(A)

and

\cos(B) = \sin(A)

. The tangent of

B

is defined as

\tan(B) = \frac{\sin(B)}{\cos(B)}

. Substituting the co-function identities gives

\tan(B) = \frac{\cos(A)}{\sin(A)}

. Since we are given that

\cos(A) = 3\sin(A)

, we substitute this expression into the numerator to get

\tan(B) = \frac{3\sin(A)}{\sin(A)} = 3

.

Step-by-Step Solution

1

Determine the relationship between the acute angles in right triangle

ABC

.

\sin(B) = \cos(A)

and

\cos(B) = \sin(A)

Since angle

C

is

90^\circ

, the other two angles

A

and

B

must sum to

90^\circ

(they are complementary angles).

2

Express

\tan(B)

in terms of the trigonometric ratios of angle

A

.

\tan(B) = \frac{\cos(A)}{\sin(A)}

By definition, the tangent of angle

B

is the ratio of its sine to its cosine, which yields

\frac{\cos(A)}{\sin(A)}

after substituting the complementary angle relations.

3

Substitute the given relation

\cos(A) = 3\sin(A)

into the expression for

\tan(B)

.

\tan(B) = \frac{3\sin(A)}{\sin(A)}

Substituting the value of

\cos(A)

allows us to simplify the fraction by expressing both terms with

\sin(A)

.

4

Simplify the fraction to get the final numerical value.

\tan(B) = 3

The term

\sin(A)

cancels out from the numerator and denominator since

A

is an acute angle and

\sin(A) \neq 0

.

Key Concept

Co-function identities and trigonometric definitions in a right triangle.

Alternative Method

Alternatively, construct a right triangle where the side opposite to angle

A

has length

1

. Since

\cos(A) = 3\sin(A)

, the ratio of the adjacent side to the hypotenuse is

3

times the ratio of the opposite side to the hypotenuse, meaning the side adjacent to angle

A

must have length

3

. Because angle

B

is the complement of angle

A

, the side opposite to angle

B

is the side adjacent to angle

A

(which is

3

), and the side adjacent to angle

B

is the side opposite to angle

A

(which is

1

). Therefore,

\tan(B) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{1} = 3

.

Estimated Time:1m 30s

Question 529Question →

In rectangle $ABCD$ , the length of side $AB$ is $12$ and the length of side $BC$ is $9$ . Point $P$ lies on the diagonal $AC$ such that $AP = \frac{1}{3} AC$ . What is the area of triangle $BPD$ ?

Show answer & explanation

Answer: 18

Answer

18

The correct answer is 18. The area of the right triangle

ABD

is half of the area of rectangle

ABCD

, which is

\frac{12 \times 9}{2} = 54

. Since point

P

lies on diagonal

AC

such that

AP = \frac{1}{3} AC

, triangle

ABP

has a base of

AP

along line

AC

and shares vertex

B

with triangle

ABC

. Thus, its area is

\frac{1}{3}

of the area of triangle

ABC

, which is

\frac{54}{3} = 18

. Similarly, triangle

ADP

shares vertex

D

with triangle

ADC

and has base

AP

, so its area is

\frac{1}{3}

of the area of triangle

ADC

, which is

\frac{54}{3} = 18

. Because

AP = \frac{1}{3} AC < \frac{1}{2} AC

, point

P

lies inside triangle

ABD

. Therefore, the area of triangle

BPD

is the area of triangle

ABD

minus the areas of triangles

ABP

and

ADP

, which is

54 - 18 - 18 = 18

.

Step-by-Step Solution

1

Calculate the area of triangle

ABD

.

\text{Area}(\triangle ABD) = \frac{12 \times 9}{2} = 54

The diagonal

BD

divides the rectangle

ABCD

into two congruent right triangles, each with an area equal to half of the rectangle's total area.

2

Find the areas of triangles

ABP

and

ADP

using the ratio of

AP

to

AC

.

\text{Area}(\triangle ABP) = \frac{1}{3} \times \text{Area}(\triangle ABC) = 18

and

\text{Area}(\triangle ADP) = \frac{1}{3} \times \text{Area}(\triangle ADC) = 18

.

Triangles

ABP

and

ABC

share the same altitude from vertex

B

to the line containing diagonal

AC

. Therefore, the ratio of their areas is equal to the ratio of their bases, which is

\frac{AP}{AC} = \frac{1}{3}

. The same logic applies to triangles

ADP

and

ADC

with vertex

D

.

3

Subtract the areas of triangles

ABP

and

ADP

from the area of triangle

ABD

to find the area of triangle

BPD

.

\text{Area}(\triangle BPD) = 54 - 18 - 18 = 18

Since

AP = \frac{1}{3} AC

, which is less than half the length of the diagonal, point

P

lies strictly within the interior of triangle

ABD

. Thus, the area of triangle

ABD

is partitioned into the areas of triangles

ABP

,

ADP

, and

BPD

.

Key Concept

Partitioning the area of a polygon and using the ratio of bases for triangles sharing a vertex to compute sub-areas.

Alternative Method

Alternatively, place the rectangle in a coordinate system with

B

at the origin

(0,0)

,

C

at

(12,0)

,

A

at

(0,9)

, and

D

at

(12,9)

. The coordinates of point

P

on diagonal

AC

(from

(0,9)

to

(12,0)

) at one-third of the distance from

A

to

C

are

x = 0 + \frac{1}{3}(12 - 0) = 4

and

y = 9 + \frac{1}{3}(0 - 9) = 6

. The area of triangle

BPD

with vertices

B(0,0)

,

P(4,6)

, and

D(12,9)

can be found using the shoelace formula:

\text{Area} = \frac{1}{2} |0(6 - 9) + 4(9 - 0) + 12(0 - 6)| = \frac{1}{2} |36 - 72| = 18

.

Estimated Time:2m 0s

Question 530Question →

The table below shows several values of the function $f$ .

$x$	$f(x)$
$-5$	$12$
$-3$	$-2$
$-1$	$4$
$3$	$18$

The function $g$ is defined by $g(x) = \frac{1}{2}f(x + 4) + 3$ . What is the value of $g(-5)$ ?

Show answer & explanation

Answer: 5

Answer

The correct answer is 5.

To find the value of

g(-5)

, substitute

-5

for

x

in the definition of

g(x)

, which gives

g(-5) = \frac{1}{2}f(-5 + 4) + 3 = \frac{1}{2}f(-1) + 3

. According to the table,

f(-1) = 4

. Substituting this value into the expression yields

g(-5) = \frac{1}{2}(4) + 3 = 2 + 3 = 5

.

Step-by-Step Solution

1

Substitute the input value into the function definition.

g(-5) = \frac{1}{2}f(-5 + 4) + 3 = \frac{1}{2}f(-1) + 3

To evaluate

g(-5)

, replace all occurrences of

x

with

-5

in the definition of

g(x)

.

2

Find the value of

f(-1)

from the table.

f(-1) = 4

The table provides specific input-output pairs for the function

f

. When the input is

-1

, the output is

4

.

3

Calculate the final value.

g(-5) = 2 + 3 = 5

Substitute

4

for

f(-1)

in the expression and simplify the terms.

Key Concept

Evaluating transformed functions using tabular data and applying horizontal translations, vertical compressions, and vertical translations.

Question 531Question →

In right triangle $ABC$ , the hypotenuse $AC$ has length $25$ , and leg $AB$ has length $20$ . What is the length of leg $BC$ ?

Show answer & explanation

Answer: 15

Answer

15

For any right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse (

a^2 + b^2 = c^2

). In right triangle

ABC

, the hypotenuse is

AC = 25

and one of the legs is

AB = 20

. Substituting these values into the theorem gives

20^2 + BC^2 = 25^2

, which simplifies to

400 + BC^2 = 625

. Subtracting

400

from both sides results in

BC^2 = 225

. Taking the square root of both sides gives the length of leg

BC

as

15

.

Step-by-Step Solution

1

Set up the Pythagorean theorem equation for the right triangle.

AB^2 + BC^2 = AC^2

The Pythagorean theorem relates the lengths of the legs and the hypotenuse of any right triangle.

2

Substitute the known lengths of side

AB

and hypotenuse

AC

.

20^2 + BC^2 = 25^2

We plug in

20

for leg

AB

and

25

for hypotenuse

AC

.

3

Evaluate the squared terms.

400 + BC^2 = 625

Squaring

20

gives

400

, and squaring

25

gives

625

.

4

Isolate the variable term

BC^2

.

BC^2 = 225

Subtracting

400

from both sides of the equation yields

225

.

5

Find the length of side

BC

.

BC = 15

Taking the square root of

225

gives the positive length of leg

BC

.

Key Concept

Applying the Pythagorean theorem to calculate a missing leg length of a right triangle when the hypotenuse and one leg length are given.

Question 532Question →

In the $xy$ -plane, the vertex of the parabola defined by $y = a(x - 2)(x - 8)$ has a $y$ -coordinate of $-18$ , where $a$ is a positive constant. What is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

2

The correct answer is 2. The quadratic function is given in factored form as

y = a(x - 2)(x - 8)

. The x-intercepts of this parabola are at

x = 2

and

x = 8

. Because of the symmetry of a parabola, the x-coordinate of the vertex is the midpoint of the x-intercepts:

x = \frac{2 + 8}{2} = 5

. The y-coordinate of the vertex is given as

-18

, meaning the vertex is at the point

(5, -18)

. Substituting these coordinates into the equation gives

-18 = a(5 - 2)(5 - 8)

, which simplifies to

-18 = a(3)(-3) = -9a

. Solving for

a

yields

a = 2

.

Step-by-Step Solution

1

Identify the x-intercepts from the factored form equation

y = a(x - 2)(x - 8)

and find the x-coordinate of the vertex.

The x-intercepts are

x = 2

and

x = 8

. The x-coordinate of the vertex is the midpoint of the intercepts:

x = \frac{2 + 8}{2} = 5

.

The axis of symmetry of a parabola passes through its vertex and lies midway between its x-intercepts.

2

Substitute the coordinates of the vertex

(5, -18)

into the quadratic equation to solve for the constant

a

.

Substituting

x = 5

and

y = -18

yields

-18 = a(5 - 2)(5 - 8)

, which simplifies to

-18 = a(3)(-3)

, so

-18 = -9a

, giving

a = 2

.

Since the vertex is a point on the parabola, its coordinates must satisfy the equation of the parabola.

Key Concept

Using the symmetry of quadratic functions in factored form to find the vertex coordinates.

Question 533Question →

In the $xy$ -plane, the graph of the polynomial function $f$ has $x$ -intercepts at $(-3, 0)$ , $(1, 0)$ , and $(k, 0)$ , where $k$ is a constant greater than $1$ . The function is defined by $f(x) = -2(x + 3)(x - 1)(x - k)$ . If the graph of $f$ passes through the point $(2, 20)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of

k

is

4

.

Substituting the coordinates of the point

(2, 20)

into the function equation

f(x) = -2(x + 3)(x - 1)(x - k)

gives

20 = -2(2 + 3)(2 - 1)(2 - k)

. Simplifying this expression yields

20 = -10(2 - k)

. Dividing both sides by

-10

gives

-2 = 2 - k

, which simplifies to

k = 4

. Since

4 > 1

, this meets the condition that

k

is a constant greater than

1

.

Step-by-Step Solution

1

Substitute the coordinates of the point

(2, 20)

into the function definition.

20 = -2(2 + 3)(2 - 1)(2 - k)

Since the graph passes through the point

(2, 20)

, the coordinates satisfy the equation of the function.

2

Simplify the numerical factors on the right side of the equation.

20 = -10(2 - k)

Calculating

2+3 = 5

and

2-1 = 1

, then multiplying the constants:

-2 \times 5 \times 1 = -10

.

3

Solve the linear equation for

k

.

k = 4

Dividing both sides by

-10

yields

2 - k = -2

. Adding

k

to both sides and adding

2

to both sides gives

k = 4

.

Key Concept

Determining a constant root of a polynomial function by evaluating it at a given point on its graph.

Estimated Time:1m 30s

Question 534Question →

A wheel rotates through a central angle of $\frac{11\pi}{18}$ radians. What is the measure of this angle in degrees?

Show answer & explanation

Answer: 110

Answer

110

To convert radians to degrees, multiply the angle by

\frac{180}{\pi}

. Thus,

\frac{11\pi}{18} \times \frac{180}{\pi} = 110

. The measure of the angle is 110 degrees.

Step-by-Step Solution

1

Identify the conversion relationship between radians and degrees.

Multiply the angle in radians by

\frac{180}{\pi}

to convert to degrees.

Since

\pi

radians is equal to

180

degrees, the conversion factor is

\frac{180}{\pi}

.

2

Multiply the given radian measure of

\frac{11\pi}{18}

by the conversion factor.

110

Applying the conversion factor simplifies the expression by canceling

\pi

and dividing

180

by

18

to get

10

, which is then multiplied by

11

to get the final degree measure.

Key Concept

To convert an angle from radians to degrees, multiply the angle in radians by

\frac{180}{\pi}

.

Estimated Time:45s

Question 535Question →

An environmental science class recorded the daily PM2.5 air quality index (AQI) values for a city over a 7-day period. The recorded values for 6 of the days were $38$ , $42$ , $45$ , $49$ , $52$ , and $58$ . The AQI value for the 7th day, $x$ , is unknown. If the mean AQI value for the 7 days is $48$ , what is the median AQI value for the 7 days?

Show answer & explanation

Answer: 49

Answer

The median AQI value for the 7 days is 49.

The correct median is 49. Multiplying the mean of 48 by the 7 days yields a total sum of 336. Subtracting the sum of the 6 known days (284) gives the 7th day's value as 52. Ordering all 7 values from least to greatest (

38, 42, 45, 49, 52, 52, 58

) reveals that the 4th value (the median) is 49.

Step-by-Step Solution

1

Calculate the sum of all 7 AQI values using the given mean.

The total sum is

7 \times 48 = 336

.

The mean of a dataset is the sum of its values divided by the number of values, so the sum is the mean multiplied by the number of values.

2

Calculate the sum of the 6 known AQI values.

38 + 42 + 45 + 49 + 52 + 58 = 284

.

This determines the total contribution of the known days to the sum.

3

Find the 7th AQI value (

x

).

x = 336 - 284 = 52

.

Subtracting the sum of the 6 known values from the total sum gives the value of the 7th day.

4

Sort the complete list of 7 values in ascending order.

38, 42, 45, 49, 52, 52, 58

.

Finding the median requires arranging the data from least to greatest.

5

Identify the median.

The median is

49

.

For an odd number of values (7), the median is the middle value in the sorted list (the 4th value).

Key Concept

Calculating and comparing measures of center (mean and median) for a data distribution

Question 536Question →

A scientist is tracking the growth of a cell culture. The cell culture's area, in square millimeters, can be modeled by either a linear function or an exponential function. The table below shows the area of the cell culture at the end of Day 1 and Day 2.

Day	Area (square millimeters)
1	150
2	180

If the area of the cell culture grows exponentially, the area on Day 4 would be $E$ square millimeters. If the area grows linearly, the area on Day 4 would be $L$ square millimeters. What is the value of $E - L$ ?

Show answer & explanation

Answer: 19.2

Answer

19.2

For the linear model, the growth is

30

square millimeters per day, so the area on Day 4 is

150 + 3(30) = 240

square millimeters. For the exponential model, the growth factor is

1.2

per day, so the area on Day 4 is

150 \times (1.2)^3 = 259.2

square millimeters. The difference

E - L

is

259.2 - 240 = 19.2

.

Step-by-Step Solution

1

Determine the linear growth model and calculate the area on Day 4.

L = 240

Under a linear growth model, the area increases by a constant amount each day. The difference between Day 1 and Day 2 is

180 - 150 = 30

square millimeters. Extending this pattern, the area on Day 3 is

180 + 30 = 210

square millimeters, and the area on Day 4 is

210 + 30 = 240

square millimeters.

2

Determine the exponential growth model and calculate the area on Day 4.

E = 259.2

Under an exponential growth model, the area increases by a constant multiplier each day. The ratio of the area on Day 2 to Day 1 is

\frac{180}{150} = 1.2

. Extending this pattern, the area on Day 3 is

180 \times 1.2 = 216

square millimeters, and the area on Day 4 is

216 \times 1.2 = 259.2

square millimeters.

3

Calculate the difference between the two models.

19.2

Subtract the linear model value from the exponential model value:

E - L = 259.2 - 240 = 19.2

.

Key Concept

Distinguishing between linear growth (constant additive change) and exponential growth (constant multiplicative change).

Question 537Question →

In triangle $ABC$ , the angle at vertex $B$ is a right angle. The lengths of sides $AB$ and $BC$ are $12$ and $16$ , respectively. A point $D$ is chosen on the hypotenuse $AC$ such that $AD = 5$ . A line drawn through $D$ perpendicular to $AC$ intersects the line passing through $B$ and $C$ at point $G$ , such that $B$ lies between $G$ and $C$ . What is the length of segment $GD$ ?

Show answer & explanation

Answer: 11.25

Answer

The length of segment

GD

is 11.25.

By the Pythagorean theorem, the hypotenuse

AC

of right triangle

ABC

is

\sqrt{12^2 + 16^2} = 20

. Subtracting the length of

AD

from

AC

gives

DC = 20 - 5 = 15

. Because the line

GD

is perpendicular to

AC

, the angle

\angle GDC

is

90^\circ

. The triangles

\triangle GDC

and

\triangle ABC

share the angle at vertex

C

and both have a right angle, which means they are similar by Angle-Angle (AA) similarity:

\triangle GDC \sim \triangle ABC

. Using the ratio of corresponding sides, we have

\frac{GD}{AB} = \frac{DC}{BC}

, which translates to

\frac{GD}{12} = \frac{15}{16}

. Solving for

GD

yields

GD = 12 \times \frac{15}{16} = 11.25

.

Step-by-Step Solution

1

Calculate the length of the hypotenuse

AC

using the Pythagorean theorem in right triangle

ABC

.

AC = \sqrt{12^2 + 16^2} = 20

The length of

AC

is required to find the segment lengths on the hypotenuse.

2

Determine the length of segment

DC

.

DC = AC - AD = 20 - 5 = 15

The segment

DC

is a side of the similar triangle

\triangle GDC

that corresponds to side

BC

in

\triangle ABC

.

3

Establish the similarity between triangles

\triangle GDC

and

\triangle ABC

.

\triangle GDC \sim \triangle ABC

by AA similarity

Both triangles share the angle at vertex

C

, and both have a right angle (

\angle GDC = \angle ABC = 90^\circ

).

4

Set up the ratio of corresponding sides and solve for

GD

.

\frac{GD}{AB} = \frac{DC}{BC} \Rightarrow GD = 12 \times \frac{15}{16} = 11.25

The ratio of corresponding sides in similar triangles is equal.

Key Concept

Using right triangle similarity and the Pythagorean theorem to solve for unknown side lengths.

Question 538Question →

An agricultural drone sprays liquid fertilizer at a constant rate. The drone covers $3$ acres of cropland every $20$ minutes. If the drone operates continuously at this rate, how many hours will it take the drone to spray $54$ acres of cropland?

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6. Since the drone covers

3

acres every

20

minutes, it covers

9

acres per hour because there are three

20

-minute intervals in one hour (

3 \times 3 = 9

). Dividing the total area of

54

acres by the rate of

9

acres per hour gives

6

hours.

Step-by-Step Solution

1

Set up a proportion to solve for the total time in minutes,

t

.

\frac{3}{20} = \frac{54}{t}

Since the drone operates at a constant rate, the ratio of acres to minutes remains constant.

2

Solve for

t

in minutes.

t = \frac{54 \times 20}{3} = 360

minutes

Cross-multiply and solve for

t

to find the total time needed to cover the cropland.

3

Convert the total time from minutes to hours.

360 \div 60 = 6

hours

Since

1

hour equals

60

minutes, divide the total minutes by

60

to obtain the final time in hours.

Key Concept

Setting up rates and solving multi-step proportions with unit conversions.

Question 539Question →

For an acute angle $\theta$ , $\cos(\theta) = \frac{5}{13}$ . What is the value of $5\tan(\theta) + 13\sin(\theta)$ ?

Show answer & explanation

Answer: 24

Answer

24

Using the Pythagorean identity

\sin^2(\theta) + \cos^2(\theta) = 1

with

\cos(\theta) = \frac{5}{13}

gives

\sin(\theta) = \frac{12}{13}

because

\theta

is an acute angle. The quotient identity gives

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{12}{5}

. Substituting these ratios into the expression

5\tan(\theta) + 13\sin(\theta)

gives

5\left(\frac{12}{5}\right) + 13\left(\frac{12}{13}\right) = 12 + 12 = 24

.

Step-by-Step Solution

1

Find the value of

\sin(\theta)

using the Pythagorean identity.

\sin(\theta) = \frac{12}{13}

Since

\sin^2(\theta) + \cos^2(\theta) = 1

and

\theta

is an acute angle, the sine value is positive:

\sin(\theta) = \sqrt{1 - \left(\frac{5}{13}\right)^2} = \frac{12}{13}

.

2

Find the value of

\tan(\theta)

using the quotient identity.

\tan(\theta) = \frac{12}{5}

By definition,

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

. Substituting the known values yields

\tan(\theta) = \frac{12/13}{5/13} = \frac{12}{5}

.

3

Substitute the trigonometric ratios into the given expression and simplify.

24

Substituting the values of

\tan(\theta)

and

\sin(\theta)

into

5\tan(\theta) + 13\sin(\theta)

gives

5\left(\frac{12}{5}\right) + 13\left(\frac{12}{13}\right) = 12 + 12 = 24

.

Key Concept

Trigonometric ratios and identities, specifically the Pythagorean identity

\sin^2(\theta) + \cos^2(\theta) = 1

and the definition of tangent as

\frac{\sin(\theta)}{\cos(\theta)}

.

Alternative Method

Alternatively, draw a right triangle with an acute angle

\theta

. Since

\cos(\theta) = \frac{5}{13} = \frac{\text{adjacent}}{\text{hypotenuse}}

, label the adjacent side as 5 and the hypotenuse as 13. By the Pythagorean theorem, the opposite side is

\sqrt{13^2 - 5^2} = 12

. From this triangle,

\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{12}{13}

and

\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{5}

. Substituting these values into the expression gives

5\left(\frac{12}{5}\right) + 13\left(\frac{12}{13}\right) = 24

.

Estimated Time:1m 30s

Question 540Question →

For an acute angle $\theta$ , the equation $2\cos^2(\theta) - 5\sin(\theta) + 1 = 0$ is true. What is the value of $\sin(\theta)$ ?

Show answer & explanation

Answer: 0.5

Answer

The value of

\sin(\theta)

is

0.5

(or

\frac{1}{2}

)

By applying the Pythagorean identity

\cos^2(\theta) = 1 - \sin^2(\theta)

, the equation

2\cos^2(\theta) - 5\sin(\theta) + 1 = 0

can be rewritten entirely in terms of

\sin(\theta)

, yielding

-2\sin^2(\theta) - 5\sin(\theta) + 3 = 0

. Multiplying by

-1

gives the standard quadratic equation

2\sin^2(\theta) + 5\sin(\theta) - 3 = 0

, which factors as

(2\sin(\theta) - 1)(\sin(\theta) + 3) = 0

. Solving for

\sin(\theta)

yields

\sin(\theta) = 0.5

or

\sin(\theta) = -3

. Since the sine value of any angle must be in the range

[-1, 1]

and the sine of an acute angle must be positive,

\sin(\theta) = 0.5

is the only valid solution.

Step-by-Step Solution

1

Apply the Pythagorean identity to rewrite the cosine term.

2(1 - \sin^2(\theta)) - 5\sin(\theta) + 1 = 0

The equation contains both

\cos^2(\theta)

and

\sin(\theta)

. Substituting

\cos^2(\theta) = 1 - \sin^2(\theta)

allows the equation to be expressed in terms of a single trigonometric function,

\sin(\theta)

.

2

Distribute and simplify the equation into standard quadratic form.

2\sin^2(\theta) + 5\sin(\theta) - 3 = 0

Expanding the equation yields

2 - 2\sin^2(\theta) - 5\sin(\theta) + 1 = 0

, which simplifies to

-2\sin^2(\theta) - 5\sin(\theta) + 3 = 0

. Multiplying the entire equation by

-1

puts it into standard quadratic form

as^2 + bs + c = 0

.

3

Factor the quadratic expression.

(2\sin(\theta) - 1)(\sin(\theta) + 3) = 0

Finding two numbers that multiply to

-6

(from

2 \times -3

) and add to

5

leads to the factors

6

and

-1

. Splitting the middle term and factoring by grouping yields

(2\sin(\theta) - 1)(\sin(\theta) + 3) = 0

.

4

Determine the valid solution based on the angle's constraints.

\sin(\theta) = 0.5

Setting each factor to zero gives

\sin(\theta) = 0.5

or

\sin(\theta) = -3

. Since the sine of any real angle must be between

-1

and

1

,

\sin(\theta) = -3

is undefined. Furthermore, because

\theta

is an acute angle (

0^\circ < \theta < 90^\circ

), the sine value must be positive, confirming

\sin(\theta) = 0.5

.

Key Concept

Pythagorean identity and quadratic trigonometric equations

Estimated Time:2m 0s

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