All practice questions

Question 2341Question →

In the standard $(x, y)$ coordinate plane, the circle defined by $x^2 + y^2 = 26$ and the line defined by $y = x - 4$ intersect at two points. What is the distance between these two points of intersection?

2\sqrt{10}

4\sqrt{2}

6

6\sqrt{2}

12

Show answer & explanation

Answer: $6\sqrt{2}$

Answer

6\sqrt{2}

Solving the system of equations by substituting

y = x - 4

into

x^2 + y^2 = 26

yields

2x^2 - 8x - 10 = 0

, which simplifies to

x^2 - 4x - 5 = 0

. Factoring gives

(x-5)(x+1) = 0

, leading to the points

(5, 1)

and

(-1, -5)

. The distance between these points is

\sqrt{6^2 + 6^2} = 6\sqrt{2}

.

Step-by-Step Solution

1

Substitute the linear equation into the circle equation.

x^2 + (x - 4)^2 = 26

This reduces the system to a single equation in terms of

x

.

2

Expand the squared binomial and simplify the quadratic equation.

2x^2 - 8x - 10 = 0

, which simplifies to

x^2 - 4x - 5 = 0

.

Expanding

(x - 4)^2

yields

x^2 - 8x + 16

. Setting the equation to zero allows us to solve for

x

.

3

Factor the quadratic equation to find the

x

-coordinates.

x = 5

or

x = -1

.

The factored form is

(x - 5)(x + 1) = 0

.

4

Substitute the

x

-values back into the linear equation to find the corresponding

y

-coordinates.

The intersection points are

(5, 1)

and

(-1, -5)

.

For

x = 5

,

y = 5 - 4 = 1

. For

x = -1

,

y = -1 - 4 = -5

.

5

Apply the distance formula to find the straight-line distance between the two points.

6\sqrt{2}

The distance is

\sqrt{(5 - (-1))^2 + (1 - (-5))^2} = \sqrt{6^2 + 6^2} = \sqrt{72} = 6\sqrt{2}

.

Key Concept

Solving systems of linear and non-linear (circular) equations by substitution and finding the distance between intersection points.

Question 2342Question →

In $\triangle ABC$ , the measure of $\angle B$ is $80^\circ$ and the measure of $\angle C$ is $40^\circ$ . A point $D$ lies on side $BC$ such that $AD$ bisects $\angle BAC$ , and a point $E$ lies on side $AC$ such that $AD = AE$ . What is the measure, in degrees, of $\angle CDE$ ?

Show answer & explanation

Answer: 35

Answer

35

The correct answer is

35

. By first finding that

\angle BAC = 60^\circ

, we use the angle bisector

AD

to find

\angle CAD = 30^\circ

. In

\triangle ADC

, we find the interior angle

\angle ADC = 110^\circ

. In the isosceles triangle

\triangle ADE

with

AD=AE

, the base angles are

\angle ADE = \angle AED = 75^\circ

. Finally, subtracting

\angle ADE

from

\angle ADC

gives

\angle CDE = 35^\circ

.

Step-by-Step Solution

1

Find the measure of the third angle of the main triangle,

\angle BAC

.

\angle BAC = 60^\circ

The sum of the interior angles of any triangle is

180^\circ

. Therefore,

\angle BAC = 180^\circ - \angle B - \angle C = 180^\circ - 80^\circ - 40^\circ = 60^\circ

.

2

Determine the measure of the bisected angle

\angle CAD

.

\angle CAD = 30^\circ

Since

AD

bisects

\angle BAC

, it divides the angle into two equal parts:

\angle BAD = \angle CAD = \frac{60^\circ}{2} = 30^\circ

.

3

Calculate the interior angle

\angle ADC

in

\triangle ADC

.

\angle ADC = 110^\circ

In

\triangle ADC

, the sum of angles is

180^\circ

. Therefore,

\angle ADC = 180^\circ - \angle CAD - \angle C = 180^\circ - 30^\circ - 40^\circ = 110^\circ

.

4

Find the base angles of the isosceles triangle

ADE

.

\angle ADE = 75^\circ

Since

AD = AE

,

\triangle ADE

is an isosceles triangle with vertex angle

\angle DAE = 30^\circ

. The two base angles,

\angle ADE

and

\angle AED

, are equal. Thus,

\angle ADE = \frac{180^\circ - 30^\circ}{2} = 75^\circ

.

5

Determine the final angle

\angle CDE

by subtraction.

\angle CDE = 35^\circ

Since point

E

lies on side

AC

, ray

DE

lies between rays

DA

and

DC

. Therefore,

\angle ADC = \angle ADE + \angle CDE

. Rearranging gives

\angle CDE = \angle ADC - \angle ADE = 110^\circ - 75^\circ = 35^\circ

.

Key Concept

Applying triangle angle sum theorem, angle bisector properties, and isosceles triangle base angle properties to perform multi-step angle tracing.

Alternative Method

Use the exterior angle theorem on

\triangle ADC

at vertex

D

:

\angle ADB = \angle CAD + \angle C = 30^\circ + 40^\circ = 70^\circ

. Then, since

E

is on

AC

,

A

,

E

, and

C

are collinear. In

\triangle ADE

, the exterior angle at

E

is

\angle DEC = \angle DAE + \angle ADE = 30^\circ + 75^\circ = 105^\circ

. In

\triangle DEC

, the sum of angles is

180^\circ

, so

\angle CDE = 180^\circ - 105^\circ - 40^\circ = 35^\circ

.

Estimated Time:2m 30s

Question 2343Question →

In the standard $(x,y)$ coordinate plane, a triangle has vertices at $A(1, 2)$ , $B(9, 2)$ , and $C(5, 8)$ . A horizontal line defined by the equation $y = k$ divides the area of the triangle into two regions of equal area. What is the value of $k$ ?

5

8 - 3\sqrt{2}

2 + 3\sqrt{2}

8 - 2\sqrt{3}

4

Show answer & explanation

Answer: $8 - 3\sqrt{2}$

Answer

8 - 3\sqrt{2}

The correct answer is

8 - 3\sqrt{2}

. Because the horizontal line

y=k

is parallel to the base of the triangle, it creates a smaller top triangle similar to the original one. The area of the original triangle is

24

and the area of the smaller triangle is

12

, yielding an area ratio of

\frac{1}{2}

. The height of the original triangle is

6

, and the height of the smaller triangle is

8-k

. Since the ratio of the areas of similar triangles is the square of the ratio of their heights, we write

\frac{1}{2} = \left(\frac{8-k}{6}\right)^2

. Solving this equation yields

k = 8 - 3\sqrt{2}

.

Step-by-Step Solution

1

Calculate the area of the entire triangle

ABC

.

The base

\overline{AB}

is horizontal along

y = 2

with a length of

9 - 1 = 8

. The height is the vertical distance from

y = 2

to the vertex

C(5, 8)

, which is

8 - 2 = 6

. The area of

\triangle ABC

is

\frac{1}{2} \times 8 \times 6 = 24

.

Finding the total area is necessary to determine the target area of the divided regions.

2

Find the area of the smaller triangle formed above the line

y = k

.

The line

y = k

divides the triangle into a smaller top triangle and a bottom trapezoid. Since both regions have equal areas, the area of the smaller top triangle is

\frac{24}{2} = 12

.

The line divides the original area of

24

into two equal parts of

12

each.

3

Set up the area ratio equation using the properties of similar triangles.

The smaller top triangle is similar to

\triangle ABC

because its base is parallel to

\overline{AB}

. The ratio of their areas is the square of the ratio of their heights:

\frac{12}{24} = \left(\frac{8 - k}{6}\right)^2 \implies \frac{1}{2} = \left(\frac{8 - k}{6}\right)^2

.

For similar figures, the area ratio is equal to the square of the scale factor.

4

Solve the ratio equation for

k

.

Taking the square root of both sides gives

\frac{1}{\sqrt{2}} = \frac{8 - k}{6}

, which simplifies to

\frac{\sqrt{2}}{2} = \frac{8 - k}{6}

. Multiplying both sides by

6

gives

3\sqrt{2} = 8 - k

, which yields

k = 8 - 3\sqrt{2}

.

This isolates

k

to find the exact vertical coordinate of the dividing line.

Key Concept

Using properties of similar figures to determine areas and coordinates on the coordinate plane.

Question 2344Question →

A drone starts at position $P$ on a grid. The drone's path is programmed with a sequence of two movements: first, it is translated $6$ units to the right and $5$ units up, and then its position is reflected across the $y$ -axis. If the drone's final position is $(-4, 8)$ , what were the coordinates of its starting position $P$ ?

(10, 3)

(-2, 13)

(-2, 3)

(10, 13)

(-10, -13)

Show answer & explanation

Answer: $(-2, 3)$

Answer

The correct answer is

(-2, 3)

because working backward from the final position

(-4, 8)

by undoing the reflection across the

y

-axis gives

(4, 8)

, and then undoing the translation by subtracting

6

from the

x

-coordinate and

5

from the

y

-coordinate yields

(-2, 3)

.

The correct answer is

(-2, 3)

because we can find the starting position by working backward from the final position. The last transformation applied was a reflection across the

y

-axis. Undoing this reflection maps the final coordinates

(-4, 8)

to

(4, 8)

because reflecting across the

y

-axis changes the sign of the

x

-coordinate. The first transformation was a translation of

6

units to the right and

5

units up. To undo this translation, we subtract

6

from the

x

-coordinate and subtract

5

from the

y

-coordinate of

(4, 8)

, giving

(4 - 6, 8 - 5) = (-2, 3)

.

Step-by-Step Solution

1

Identify the final transformation and undo it. The final transformation is a reflection across the

y

-axis, which maps

(x, y) \rightarrow (-x, y)

.

The coordinates before reflection are

(4, 8)

.

Undoing a reflection across the

y

-axis on

(-4, 8)

means changing the sign of the

x

-coordinate:

-(-4) = 4

, while keeping the

y

-coordinate the same.

2

Identify the first transformation and undo it. The first transformation is a translation of

6

units right and

5

units up, which maps

(x, y) \rightarrow (x+6, y+5)

.

The starting coordinates

P

are

(-2, 3)

.

To undo the translation, subtract

6

from the

x

-coordinate and

5

from the

y

-coordinate of the intermediate point

(4, 8)

:

4 - 6 = -2

and

8 - 5 = 3

.

Key Concept

Transformations in the Coordinate Plane

Alternative Method

Instead of working backward, you can test the options by applying the transformations forward. For

(-2, 3)

, translating 6 units right and 5 units up gives

(-2+6, 3+5) = (4, 8)

. Reflecting across the

y

-axis then gives

(-4, 8)

, which matches the final position.

Estimated Time:1m 15s

Question 2345Question →

In a certain triangle, the ratio of the measure of the first angle to the measure of the second angle is $1:2$ . The measure of the third angle is $20^\circ$ less than the measure of the second angle. What is the measure of the largest angle in the triangle?

40^\circ

64^\circ

80^\circ

100^\circ

120^\circ

Show answer & explanation

Answer: $80^\circ$

Answer

80^\circ

The correct answer is

80^\circ

. By representing the first angle as

x

, the second angle is

2x

, and the third angle is

2x - 20

. The sum of the interior angles in a triangle is always

180^\circ

, which gives the equation

x + 2x + (2x - 20) = 180

. Simplifying this results in

5x - 20 = 180

, which yields

5x = 200

and

x = 40

. Substituting this value back into the expressions for the three angles gives measures of

40^\circ

,

80^\circ

, and

60^\circ

. Comparing these values shows that the largest angle is

80^\circ

.

Step-by-Step Solution

1

Define the measures of the first and second angles using a single variable based on their ratio.

Let the first angle be

x

and the second angle be

2x

.

Since the ratio of the first angle to the second angle is

1:2

, we can represent them as

x

and

2x

respectively.

2

Express the measure of the third angle in terms of the same variable.

The third angle is

2x - 20

.

The problem states the third angle is

20^\circ

less than the second angle, which has a measure of

2x

.

3

Set up an equation using the triangle angle sum theorem and solve for

x

.

x + 2x + (2x - 20) = 180 \implies 5x - 20 = 180 \implies 5x = 200 \implies x = 40

.

The sum of the measures of the interior angles of any triangle is always

180^\circ

.

4

Calculate the measures of all three angles to determine which is the largest.

First angle =

40^\circ

, Second angle =

2(40^\circ) = 80^\circ

, Third angle =

2(40^\circ) - 20^\circ = 60^\circ

. The largest angle is

80^\circ

.

We must substitute

x = 40

back into our expressions for each angle to find their actual degree measures and identify the largest one.

Key Concept

The interior angles of a triangle always sum to

180^\circ

. Ratios and word problems can be modeled algebraically to determine unknown angle measures.

Estimated Time:1m 15s

Question 2346Question →

In the standard $(x, y)$ coordinate plane, a line $L$ passes through the points $(a, a^2)$ and $(b, b^2)$ , where $a$ and $b$ are distinct real numbers. The slope of line $L$ is $8$ . If the midpoint of the line segment connecting these two points lies on the line $y = 5x - 1$ , what is the $y$ -coordinate of this midpoint?

Show answer & explanation

Answer: 19

Answer

The

y

-coordinate of the midpoint is

19

.

Applying the slope formula to the points

(a, a^2)

and

(b, b^2)

yields \frac{b^2-a^2}{b-a} = a+b = 8. The

x

-coordinate of the midpoint is \frac{a+b}{2} = 4. Substituting this value into the equation

y = 5x - 1

gives

y = 5(4) - 1 = 19

.

Step-by-Step Solution

1

Express the slope of line

L

in terms of

a

and

b

and set it equal to the given slope.

a + b = 8

The slope formula is

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

. For the points

(a, a^2)

and

(b, b^2)

, the slope is \frac{b^2 - a^2}{b - a}. Factoring the numerator gives \frac{(b-a)(b+a)}{b-a} = a + b. Since the slope is given as

8

, we establish

a + b = 8

.

2

Find the

x

-coordinate of the midpoint of the segment connecting the two points.

x_M = 4

The midpoint formula for the

x

-coordinate is

x_M = \frac{x_1 + x_2}{2}

. For our points,

x_M = \frac{a+b}{2}

. Substituting the value

a+b = 8

gives

x_M = \frac{8}{2} = 4

.

3

Determine the

y

-coordinate of the midpoint using the line equation.

y_M = 19

The midpoint lies on the line

y = 5x - 1

. Substituting

x_M = 4

into this equation gives

y_M = 5(4) - 1 = 19

.

Key Concept

Slope of a Line

Alternative Method

Let

a

and

b

be the real roots of the quadratic equation

t^2 - 8t + 13 = 0

. By Vieta's formulas,

a+b = 8

and

ab = 13

. The

x

-coordinate of the midpoint is \frac{a+b}{2} = 4, and the

y

-coordinate is \frac{a^2+b^2}{2} = \frac{(a+b)^2 - 2ab}{2} = \frac{64 - 26}{2} = 19. Since the point

(4, 19)

satisfies

y = 5x - 1

, this confirms the existence of valid real coordinates

(a, a^2)

and

(b, b^2)

that produce the midpoint on the line.

Estimated Time:2m 0s

Question 2347Question →

A municipal water reservoir is being drained at a constant rate. After $4$ hours of draining, the reservoir contains $18,000$ gallons of water. After $7$ hours of draining, it contains $13,500$ gallons of water. How many hours after the draining process begins will the reservoir be completely empty?

Show answer & explanation

Answer: 16

Answer

The reservoir will be completely empty

16

hours after draining begins.

The correct answer is

16

hours. The constant draining rate is calculated as the change in volume divided by the change in time:

\frac{13,500 - 18,000}{7 - 4} = -1,500

gallons per hour. Since there are

18,000

gallons left at the

4

-hour mark, it will take an additional

\frac{18,000}{1,500} = 12

hours to completely empty the reservoir. The total time from the start is

4 + 12 = 16

hours.

Step-by-Step Solution

1

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

The rate is

-1,500

gallons per hour.

The rate of change is the change in volume divided by the change in time:

\frac{13,500 - 18,000}{7 - 4} = \frac{-4,500}{3} = -1,500

gallons per hour.

2

Determine the remaining time needed to empty the reservoir after the

7

-hour mark.

It will take an additional

9

hours.

At

7

hours, the reservoir contains

13,500

gallons. Draining at a rate of

1,500

gallons per hour, the remaining time is

\frac{13,500}{1,500} = 9

hours.

3

Calculate the total elapsed time since the draining process began.

16

hours

Adding the initial

7

hours of draining to the additional

9

hours needed gives

7 + 9 = 16

hours.

Key Concept

Linear word problems and finding intercepts

Alternative Method

We can model the volume of water

W

as a linear function of time

t

using the slope-intercept form

W(t) = mt + b

. Substituting the rate

m = -1,500

and the point

(4, 18,000)

gives

18,000 = -1,500(4) + b

, which yields the

y

-intercept (initial volume)

b = 24,000

gallons. The linear model is

W(t) = -1,500t + 24,000

. Setting

W(t) = 0

to find when the reservoir is empty gives

0 = -1,500t + 24,000

, which simplifies to

t = \frac{24,000}{1,500} = 16

hours.

Estimated Time:1m 30s

Question 2348Question →

In $\triangle ABC$ , point $D$ lies on side $BC$ . The segment $AD$ divides the interior angle $\angle BAC$ into two angles, $\angle BAD$ and $\angle DAC$ , whose measures are in the ratio $3:2$ , respectively. The measures of the interior angles $\angle B$ and $\angle C$ are in the ratio $5:4$ , respectively. If the measure of $\angle ADC$ is $104^\circ$ , what is the measure of $\angle BAC$ ?

54^\circ

72^\circ

76^\circ

90^\circ

108^\circ

Show answer & explanation

Answer: $90^\circ$

Answer

90^\circ

The correct answer is

90^\circ

. By expressing the angles in terms of variables using their ratios, we set up two independent linear equations:

3x + 5y = 104

(from the exterior angle theorem on

\triangle ABD

) and

2x + 4y = 76

(from the sum of angles in

\triangle ADC

). Solving this system yields

x = 18

. Since

\angle BAC

is composed of

\angle BAD

and

\angle DAC

, its measure is

3x + 2x = 5x = 5(18^\circ) = 90^\circ

.

Step-by-Step Solution

1

Define variables for the partitioned angles and the base angles using the given ratios.

Let the measures of

\angle BAD

and

\angle DAC

be

3x

and

2x

respectively, so that

\angle BAC = 5x

. Let the measures of

\angle B

and

\angle C

be

5y

and

4y

respectively.

Ratios express quantities as multiples of a common variable, which simplifies setting up equations.

2

Apply the exterior angle theorem to

\triangle ABD

at vertex

D

.

The exterior angle

\angle ADC = \angle BAD + \angle B \Rightarrow 104^\circ = 3x + 5y

.

The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

3

Apply the angle sum theorem to

\triangle ADC

.

\angle DAC + \angle C + \angle ADC = 180^\circ \Rightarrow 2x + 4y + 104^\circ = 180^\circ \Rightarrow 2x + 4y = 76^\circ

.

The sum of the measures of the interior angles of any triangle is always

180^\circ

.

4

Solve the system of linear equations: (1)

3x + 5y = 104

and (2)

2x + 4y = 76

.

Multiply equation (1) by 2 and equation (2) by 3 to align the coefficients of

x

:

6x + 10y = 208

6x + 12y = 228

Subtract the first aligned equation from the second:

2y = 20 \Rightarrow y = 10

.
Substitute

y = 10

back into equation (2):

2x + 4(10) = 76 \Rightarrow 2x = 36 \Rightarrow x = 18

.

Solving the system of linear equations determines the values of the variables

x

and

y

.

5

Calculate the measure of

\angle BAC

.

\angle BAC = 5x = 5(18^\circ) = 90^\circ

.

We defined the total measure of

\angle BAC

as the sum of its two partitioned parts,

3x + 2x = 5x

.

Key Concept

Using triangle angle properties and exterior angle theorems to set up and solve systems of linear equations.

Question 2349Question →

In the standard $(x, y)$ coordinate plane, point $P$ is rotated $90^\circ$ clockwise about the origin, and then translated $3$ units to the left and $4$ units up to form the image point $P'(1, 2)$ . What are the coordinates of the pre-image point $P$ ?

(-1, 3)

(-6, -2)

(2, 4)

(-2, -4)

(1, -3)

Show answer & explanation

Answer: $(2, 4)$

Answer

The coordinates of the pre-image point

P

are

(2, 4)

.

To find the coordinates of the pre-image point, we must undo the transformations in reverse order. First, we undo the translation of

3

units left and

4

units up by translating the image point

(1, 2)

by

3

units right and

4

units down, yielding the intermediate point

(4, -2)

. Next, we undo the

90^\circ

clockwise rotation by rotating this intermediate point

90^\circ

counterclockwise about the origin. Using the rotation rule

(x, y) \rightarrow (-y, x)

on

(4, -2)

gives the pre-image point

(2, 4)

.

Step-by-Step Solution

1

Identify the sequence of forward transformations and set up the path to work backward from the final image point

P'(1, 2)

to the pre-image point

P

.

The forward sequence is:

P \xrightarrow{\text{Rotate } 90^\circ \text{ clockwise}} P_{\text{mid}} \xrightarrow{\text{Translate left 3, up 4}} P'(1, 2)

. To find

P

, we must apply the inverse transformations in reverse order:

P'(1, 2) \xrightarrow{\text{Translate right 3, down 4}} P_{\text{mid}} \xrightarrow{\text{Rotate } 90^\circ \text{ counterclockwise}} P

.

Working backward with inverse operations is the standard mathematical procedure to retrieve a pre-image from its transformed image.

2

Apply the inverse translation to the image point

P'(1, 2)

to find the coordinates of the intermediate point

P_{\text{mid}}

.

To undo a translation of

3

units left and

4

units up, translate

3

units right and

4

units down:

x = 1 + 3 = 4

, and

y = 2 - 4 = -2

. Thus,

P_{\text{mid}} = (4, -2)

.

Applying the opposite direction and magnitude to the coordinates reverses the translation effect.

3

Apply the inverse rotation to the intermediate point

P_{\text{mid}}(4, -2)

to find the coordinates of the pre-image point

P

.

The inverse of a

90^\circ

clockwise rotation is a

90^\circ

counterclockwise rotation about the origin. The rule for a

90^\circ

counterclockwise rotation is

(x, y) \rightarrow (-y, x)

. Applying this to

(4, -2)

gives

P(-(-2), 4) = P(2, 4)

.

Rotating

90^\circ

counterclockwise reverses the initial

90^\circ

clockwise rotation, returning the point to its original pre-image coordinates.

Key Concept

Determining the coordinates of a pre-image by reversing a composite transformation in the coordinate plane.

Question 2350Question →

A company's weekly revenue, $R(x)$ , and weekly cost, $C(x)$ , in dollars, are modeled by the functions $R(x) = (2x^2 + 3)^2$ and $C(x) = x^2(x^3 - 5)$ , where $x$ represents the number of units produced and sold. Which of the following expressions represents the company's weekly profit, $P(x) = R(x) - C(x)$ ?

-x^5 + 4x^4 + 7x^2 + 9

-x^6 + 4x^4 + 17x^2 + 9

-x^5 + 4x^4 + 17x^2 + 9

-x^5 + 4x^4 + 5x^2 + 9

-x^6 + 4x^4 + 7x^2 + 9

Show answer & explanation

Answer: $-x^5 + 4x^4 + 17x^2 + 9$

Answer

The expression representing the weekly profit is

-x^5 + 4x^4 + 17x^2 + 9

.

To find the profit function, we subtract the cost function from the revenue function. First, expanding

(2x^2 + 3)^2

yields

4x^4 + 12x^2 + 9

. Second, distributing

x^2

over

(x^3 - 5)

yields

x^5 - 5x^2

. Subtracting these two functions gives

(4x^4 + 12x^2 + 9) - (x^5 - 5x^2)

. Distributing the subtraction sign changes the signs of the cost function to

-x^5 + 5x^2

. Combining like terms (

12x^2 + 5x^2 = 17x^2

) and organizing the expression in descending order yields

-x^5 + 4x^4 + 17x^2 + 9

.

Step-by-Step Solution

1

Expand the revenue function

R(x) = (2x^2 + 3)^2

using the binomial square identity

(a+b)^2 = a^2 + 2ab + b^2

.

R(x) = (2x^2)^2 + 2(2x^2)(3) + 3^2 = 4x^4 + 12x^2 + 9

This simplifies the revenue expression into a standard polynomial form.

2

Expand the cost function

C(x) = x^2(x^3 - 5)

by distributing

x^2

to both terms inside the parentheses and applying the product rule for exponents

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

.

C(x) = x^2(x^3) - x^2(5) = x^5 - 5x^2

This simplifies the cost expression into a standard polynomial form.

3

Subtract

C(x)

from

R(x)

by setting up

P(x) = R(x) - C(x)

and distributing the negative sign to all terms of

C(x)

.

P(x) = (4x^4 + 12x^2 + 9) - (x^5 - 5x^2) = 4x^4 + 12x^2 + 9 - x^5 + 5x^2

Profit is revenue minus cost, and distributing the negative sign correctly is essential.

4

Combine like terms and write the final expression in descending order of exponents.

P(x) = -x^5 + 4x^4 + 17x^2 + 9

To present the polynomial in standard form.

Key Concept

Polynomial operations, including expanding binomial squares, distributing monomial terms, and subtracting polynomials with careful attention to sign distribution.

Question 2351Question →

A right triangle is plotted in the standard $(x, y)$ coordinate plane. The vertices of the triangle are located at $(-3, -2)$ , $(5, -2)$ , and $(-3, 4)$ . What is the length of the hypotenuse of the triangle?

\frac{4}{3}

2\sqrt{7}

\frac{3}{5}

10

14

Show answer & explanation

Answer: $10$

Answer

10

The distance between the vertices

(-3, -2)

and

(5, -2)

along the horizontal line

y = -2

is

5 - (-3) = 8

units. The distance between the vertices

(-3, -2)

and

(-3, 4)

along the vertical line

x = -3

is

4 - (-2) = 6

units. Because horizontal and vertical segments meet at a right angle, they form the legs of a right triangle. Applying the Pythagorean theorem, the length of the hypotenuse is the square root of the sum of the squares of the legs:

\sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10

.

Step-by-Step Solution

1

Determine the lengths of the two perpendicular legs of the right triangle on the coordinate plane.

The horizontal leg has a length of

5 - (-3) = 8

units, and the vertical leg has a length of

4 - (-2) = 6

units.

Because the segment between

(-3, -2)

and

(5, -2)

is horizontal (constant

y = -2

) and the segment between

(-3, -2)

and

(-3, 4)

is vertical (constant

x = -3

).

2

Apply the Pythagorean theorem to calculate the length of the hypotenuse.

The hypotenuse length is

\sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10

units.

The Pythagorean theorem states that for any right triangle with legs

a

and

b

and hypotenuse

c

,

a^2 + b^2 = c^2

.

Key Concept

Finding the lengths of segments and applying the Pythagorean theorem on the coordinate plane.

Question 2352Question →

The ratio of the number of students in a school's science club to the number of students in its art club is $3:5$ . There are $8$ more students in the art club than in the science club. If $x$ students from the art club leave to join the science club, the ratio of the number of science club members to the number of art club members becomes $3:1$ . What is the value of $x$ ?

4

6

12

24

48

Show answer & explanation

Answer: 12

Answer

12

The correct answer is 12. Let the initial number of students in the science and art clubs be

3k

and

5k

, respectively. Since there are 8 more students in the art club, we set up the equation

5k = 3k + 8

, which simplifies to

2k = 8

, giving

k = 4

. This means there are initially 12 science club members and 20 art club members. When

x

students transfer from the art club to the science club, the new sizes are

12 + x

and

20 - x

. Setting up the new ratio gives

\frac{12 + x}{20 - x} = 3

. Solving this equation yields

12 + x = 60 - 3x

, which simplifies to

4x = 48

, resulting in

x = 12

.

Step-by-Step Solution

1

Represent the initial number of students in each club using the given ratio.

Let the number of science club members be

3k

and the number of art club members be

5k

, where

k

is a positive constant.

The ratio of science club members to art club members is

3:5

.

2

Set up and solve a linear equation to find the value of

k

and the starting number of members in each club.

5k = 3k + 8 \Rightarrow 2k = 8 \Rightarrow k = 4

. Thus, the science club has

3(4) = 12

members and the art club has

5(4) = 20

members.

There are

8

more students in the art club than in the science club, so the difference between the two groups is

8

.

3

Write the new member counts after

x

students transfer and set up the ratio equation.

The new science club size is

12 + x

and the new art club size is

20 - x

. The new ratio equation is

\frac{12 + x}{20 - x} = \frac{3}{1}

.

The

x

students leave the art club (subtracting

x

) and join the science club (adding

x

), resulting in a new ratio of

3:1

.

4

Solve the rational equation for

x

.

12 + x = 3(20 - x) \Rightarrow 12 + x = 60 - 3x \Rightarrow 4x = 48 \Rightarrow x = 12

.

Cross-multiplying and isolating

x

yields the number of students who transferred.

Key Concept

Translating verbal statements involving ratios and changes in quantities into solvable linear algebraic equations.

Estimated Time:1m 30s

Question 2353Question →

A geometric sequence has a first term of $2^3$ and a common ratio of $2^2$ . What is the value of the 5th term of this sequence?

24

512

2,048

8,192

16,777,216

Show answer & explanation

Answer: $2,048$

Answer

The 5th term of the geometric sequence is

2,048

.

To find the 5th term of a geometric sequence, we use the formula

a_n = a_1 \cdot r^{n-1}

. Substituting the given first term

a_1 = 2^3 = 8

and the common ratio

r = 2^2 = 4

for

n = 5

gives

a_5 = 2^3 \cdot (2^2)^4

. Applying the power of a power rule,

(2^2)^4 = 2^{2 \times 4} = 2^8

. Then, multiplying the bases by adding the exponents gives

2^3 \cdot 2^8 = 2^{3+8} = 2^{11}

, which evaluates to

2,048

.

Step-by-Step Solution

1

Identify the given values and the formula for the

n

-th term of a geometric sequence.

The first term is

a_1 = 2^3 = 8

, the common ratio is

r = 2^2 = 4

, and we need to find the term for

n = 5

using the formula

a_n = a_1 \cdot r^{n-1}

.

Knowing the correct formula is necessary to compute the specific term of a geometric sequence.

2

Substitute the values into the formula to express the 5th term in terms of base 2.

a_5 = 2^3 \cdot (2^2)^{5-1} = 2^3 \cdot (2^2)^4

This substitutes the specific term number and sequence parameters into the general term formula.

3

Simplify the exponential expression and calculate the final numerical value.

a_5 = 2^3 \cdot 2^8 = 2^{3+8} = 2^{11} = 2,048

Applying exponent rules (multiplying powers of a power and adding exponents when multiplying like bases) allows us to evaluate the expression to a single number.

Key Concept

Finding a specific term in a geometric sequence using exponential properties

Estimated Time:1m 0s

Question 2354Question →

A circle in the standard $(x,y)$ coordinate plane is defined by the equation $(x + 3)^2 + (y - 4)^2 = 25$ . What are the coordinates of the center and the length of the radius of this circle?

Center:

(3, -4)

; Radius:

5

Center:

(-3, 4)

; Radius:

25

Center:

(-3, 4)

; Radius:

5

Center:

(3, -4)

; Radius:

25

Center:

(-3, -4)

; Radius:

5

Show answer & explanation

Answer: Center: $(-3, 4)$ ; Radius: $5$

Answer

Center:

(-3, 4)

; Radius:

5

The standard form of a circle's equation is

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

, where the center is

(h, k)

and the radius is

r

. Matching

(x + 3)^2 + (y - 4)^2 = 25

to this standard form, we find

h = -3

,

k = 4

, and

r^2 = 25

(which gives

r = 5

). Therefore, the center is

(-3, 4)

and the radius is

5

.

Step-by-Step Solution

1

Identify the standard form of a circle's equation.

The standard equation of a circle with center

(h, k)

and radius

r

is

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

.

This formula provides the template to match the given equation and extract the parameters.

2

Rewrite the given equation

(x + 3)^2 + (y - 4)^2 = 25

to match the signs and exponents of the standard form.

The equation can be written as

(x - (-3))^2 + (y - 4)^2 = 5^2

.

Rewriting the terms helps to identify the exact values of

h

,

k

, and

r

without sign confusion.

3

Extract the center

(h, k)

and radius

r

from the rewritten equation.

Comparing the terms shows

h = -3

,

k = 4

, and

r = 5

, giving a center of

(-3, 4)

and a radius of

5

.

These extracted values are the final answer.

Key Concept

Equations and Graphs of Circles

Question 2355Question →

A parabola is defined by the equation $y = x^2 - 4x + 3$ , and a line is defined by the equation $y = -x + 7$ . The parabola and the line intersect at two points in the standard $(x, y)$ coordinate plane. What is the distance between these two points of intersection?

5

\sqrt{34}

5\sqrt{2}

3\sqrt{2}

50

Show answer & explanation

Answer: $5\sqrt{2}$

Answer

5\sqrt{2}

The correct answer is the value representing the straight-line distance between the two intersection points. Setting the equations equal to each other gives

x^2 - 3x - 4 = 0

, which factors to

(x - 4)(x + 1) = 0

. This yields

x = 4

and

x = -1

. Evaluating these in

y = -x + 7

yields the points

(4, 3)

and

(-1, 8)

. The distance between them is

\sqrt{(4 - (-1))^2 + (3 - 8)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}

.

Step-by-Step Solution

1

Set the equations of the parabola and line equal to each other to find the

x

-coordinates of the intersection points.

x^2 - 4x + 3 = -x + 7

Intersection points must satisfy both equations simultaneously.

2

Rearrange the equation into standard quadratic form and factor it to solve for

x

.

x^2 - 3x - 4 = 0 \implies (x - 4)(x + 1) = 0 \implies x = 4

or

x = -1

Factoring the quadratic equation gives the roots which correspond to the

x

-coordinates of the intersection points.

3

Substitute the

x

-values back into the linear equation to determine the corresponding

y

-coordinates.

For

x = 4

,

y = -(4) + 7 = 3

, yielding point

(4, 3)

. For

x = -1

,

y = -(-1) + 7 = 8

, yielding point

(-1, 8)

.

Substituting into the simpler linear equation provides the

y

-coordinates of the intersection points.

4

Apply the distance formula to calculate the distance between

(4, 3)

and

(-1, 8)

.

d = \sqrt{(4 - (-1))^2 + (3 - 8)^2} = \sqrt{5^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}

The distance formula calculates the straight-line distance between two coordinates in the coordinate plane.

Key Concept

Systems of Linear and Non-Linear Equations

Question 2356Question →

A recipe calls for $\frac{1}{2}$ cup of sugar on the first day of a fermentation process. Each day after that, the ratio of the sugar added on that day to the sugar added on the previous day is $1:3$ . What is the total amount of sugar, in cups, added during the first 3 days?

\frac{13}{18}

\frac{3}{26}

\frac{2}{9}

\frac{5}{2}

\frac{21}{32}

Show answer & explanation

Answer: $\frac{13}{18}$

Answer

The total amount of sugar added is

\frac{13}{18}

cups.

The correct answer is

\frac{13}{18}

. The sugar added on Day 1 is

\frac{1}{2}

cup. Since the ratio between consecutive days is

1:3

, the common ratio is

r = \frac{1}{3}

. This gives a Day 2 amount of

\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}

cup, and a Day 3 amount of

\frac{1}{6} \times \frac{1}{3} = \frac{1}{18}

cup. Summing these three amounts using a common denominator of 18 yields

\frac{9}{18} + \frac{3}{18} + \frac{1}{18} = \frac{13}{18}

cups.

Step-by-Step Solution

1

Identify the type of sequence and the first term.

The first term is

a_1 = \frac{1}{2}

. The sequence is geometric because the ratio between the amounts added on consecutive days is constant.

The problem states that the ratio of sugar added on consecutive days is

1:3

, which establishes a common ratio for a geometric sequence.

2

Determine the common ratio and find the terms for the second and third days.

The common ratio is

r = \frac{1}{3}

. The second term is

a_2 = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}

, and the third term is

a_3 = \frac{1}{6} \times \frac{1}{3} = \frac{1}{18}

.

Each subsequent term of a geometric sequence is found by multiplying the previous term by the common ratio.

3

Calculate the sum of the first three terms.

The total sum is

S_3 = \frac{1}{2} + \frac{1}{6} + \frac{1}{18} = \frac{9}{18} + \frac{3}{18} + \frac{1}{18} = \frac{13}{18}

.

Finding the total amount requires summing the individual amounts added over the three days using a common denominator of 18.

Key Concept

Sum of a finite geometric series

Alternative Method

Instead of calculating and adding the individual terms, you can use the sum of a finite geometric series formula:

S_n = \frac{a_1(1-r^n)}{1-r}

. Substituting

a_1 = \frac{1}{2}

,

r = \frac{1}{3}

, and

n=3

yields:

S_3 = \frac{\frac{1}{2}\left(1 - \left(\frac{1}{3}\right)^3\right)}{1 - \frac{1}{3}} = \frac{\frac{1}{2}\left(1 - \frac{1}{27}\right)}{\frac{2}{3}} = \frac{\frac{1}{2}\left(\frac{26}{27}\right)}{\frac{2}{3}} = \frac{13}{27} \times \frac{3}{2} = \frac{13}{18}

.

Estimated Time:1m 30s

Question 2357Question →

A circle in the standard $(x, y)$ coordinate plane is defined by the equation $(x - 9)^2 + (y + 4)^2 = 81$ . What is the radius of this circle?

Show answer & explanation

Answer: 9

Answer

The radius of the circle is 9.

The standard equation of a circle in the coordinate plane is

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

, where

r

is the radius. For the equation

(x - 9)^2 + (y + 4)^2 = 81

, the right-hand side represents

r^2

, so

r^2 = 81

. Taking the positive square root of both sides gives

r = \sqrt{81} = 9

.

Step-by-Step Solution

1

Identify the standard form of the circle equation.

The standard equation of a circle is

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

, where

(h, k)

is the center and

r

is the radius.

This allows us to relate the terms of the given equation to the components of the circle.

2

Match the given equation to the standard form.

By comparing

(x - 9)^2 + (y + 4)^2 = 81

to

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

, we find that

r^2 = 81

.

The constant on the right side of the standard equation represents the square of the radius.

3

Solve for the radius

r

.

r = \sqrt{81} = 9

.

Taking the square root of

r^2

gives the actual radius of the circle.

Key Concept

Identifying the radius from the standard equation of a circle

Question 2358Question →

An artist is designing a rectangular stained-glass window. The total area of the window, in square inches, is represented by the polynomial $8x^2 + 2x - 3$ . The area of the central blue glass section, in square inches, is represented by the product $(2x - 3)(x - 2)$ . The remaining portion of the window is made of clear glass. Which of the following expressions represents the area, in square inches, of the clear glass section?

6x^2 + 9x - 9

6x^2 - 5x + 3

6x^2 + 9x + 3

8x^2 + 7x - 9

6x^2 + 2x - 9

Show answer & explanation

Answer: $6x^2 + 9x - 9$

Answer

The expression representing the area of the clear glass section is

6x^2 + 9x - 9

.

The correct expression is

6x^2 + 9x - 9

. The area of the clear glass is found by expanding

(2x - 3)(x - 2)

to get

2x^2 - 7x + 6

and then subtracting this from the total area:

(8x^2 + 2x - 3) - (2x^2 - 7x + 6) = 8x^2 + 2x - 3 - 2x^2 + 7x - 6 = 6x^2 + 9x - 9

.

Step-by-Step Solution

1

Expand the product representing the area of the blue glass section using binomial multiplication.

(2x - 3)(x - 2) = 2x^2 - 4x - 3x + 6 = 2x^2 - 7x + 6

Converting the factored area of the blue glass section to standard form is necessary before subtraction.

2

Set up the subtraction of the blue glass area from the total area.

(8x^2 + 2x - 3) - (2x^2 - 7x + 6)

The clear glass area is found by subtracting the blue glass area from the total window area.

3

Distribute the negative sign to each term of the second polynomial and combine like terms.

8x^2 + 2x - 3 - 2x^2 + 7x - 6 = (8x^2 - 2x^2) + (2x + 7x) + (-3 - 6) = 6x^2 + 9x - 9

Distributing the subtraction sign changes the signs of all terms inside the second set of parentheses, allowing for correct simplification.

Key Concept

Operations on Polynomials

Estimated Time:1m 30s

Question 2359Question →

A rectangular garden has a straight walking path that connects two opposite corners. The length of the path is $170$ meters, and the width of the garden is $80$ meters. What is the length, in meters, of the garden?

90

120

150

188

250

Show answer & explanation

Answer: 150

Answer

The length of the garden is

150

meters.

The diagonal path, width, and length of the rectangular garden form a right triangle where the path is the hypotenuse. According to the Pythagorean theorem, the square of the length plus the square of the width equals the square of the path:

\text{length}^2 + 80^2 = 170^2

. This simplifies to

\text{length}^2 + 6,400 = 28,900

. Subtracting

6,400

from both sides gives

\text{length}^2 = 22,500

. Taking the square root of

22,500

yields

150

meters.

Step-by-Step Solution

1

Identify the right triangle formed by the length, width, and diagonal path of the garden.

The width (

80

meters) and the unknown length are the legs, while the diagonal path (

170

meters) is the hypotenuse.

The diagonal of a rectangle forms two congruent right triangles with the rectangle's sides.

2

Set up the Pythagorean equation to solve for the unknown leg.

\text{length}^2 + 80^2 = 170^2

The Pythagorean theorem states that

a^2 + b^2 = c^2

, where

c

is the hypotenuse.

3

Calculate the squares of the known lengths.

80^2 = 6,400

and

170^2 = 28,900

Evaluate the exponents to simplify the equation.

4

Isolate the squared unknown variable.

\text{length}^2 = 28,900 - 6,400 = 22,500

Subtract

6,400

from both sides of the equation.

5

Take the square root of both sides to find the length.

\text{length} = \sqrt{22,500} = 150

The square root operation reverses the squaring of the variable.

Key Concept

Applying the Pythagorean theorem to find the length of an unknown leg in a right triangle.

Estimated Time:1m 0s

Question 2360Question →

In the standard $(x, y)$ coordinate plane, the midpoint of a line segment with endpoints $(1, -2)$ and $(7, 10)$ is the center of a circle. If the point $(8, 7)$ lies on the circle, what is the radius of the circle?

7

25

3\sqrt{5}

5

\sqrt{26}

Show answer & explanation

Answer: 5

Answer

The radius of the circle is 5.

The midpoint of the line segment with endpoints

(1, -2)

and

(7, 10)

is calculated as

(\frac{1+7}{2}, \frac{-2+10}{2}) = (4, 4)

, which is the center of the circle. The distance from the center

(4, 4)

to the point

(8, 7)

on the circle is the radius, which is

\sqrt{(8-4)^2 + (7-4)^2} = \sqrt{16+9} = 5

.

Step-by-Step Solution

1

Find the midpoint of the line segment with endpoints

(1, -2)

and

(7, 10)

to determine the center of the circle.

The center of the circle is

(4, 4)

.

The midpoint of a segment with endpoints

(x_1, y_1)

and

(x_2, y_2)

is given by

(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})

. Calculating this gives

(\frac{1+7}{2}, \frac{-2+10}{2}) = (4, 4)

.

2

Calculate the distance between the center of the circle

(4, 4)

and the point

(8, 7)

on the circle to find the radius.

The radius of the circle is 5.

The distance formula is

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

. Substituting the coordinates gives

\sqrt{(8-4)^2 + (7-4)^2} = \sqrt{4^2 + 3^2} = \sqrt{16+9} = \sqrt{25} = 5

.

Key Concept

Finding the midpoint of a line segment to determine a circle's center and using the distance formula to calculate its radius.

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Common Mistakes

Answer

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Key Concept

Common Mistakes

Answer

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Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

Answer

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