All practice questions

Question 1481Question →

A company allocates a total budget of $\$4,000$ between its marketing and research departments. The amount allocated to marketing is $\$400$ more than three times the amount allocated to research. How many dollars are allocated to research?

Show answer & explanation

Answer: 900

Answer

The amount allocated to research is

900

dollars.

If

x

represents the research budget, the marketing budget is

3x + 400

. Summing both department allocations gives

x + (3x + 400) = 4,000

. Simplifying this linear equation gives

4x + 400 = 4,000

, leading to

4x = 3,600

and

x = 900

.

Step-by-Step Solution

1

Define the unknown variable and express both allocations algebraically.

Let

x

be the research budget. The marketing budget is

3x + 400

.

The marketing allocation is defined relative to the research allocation.

2

Formulate a linear equation representing the combined budget.

x + (3x + 400) = 4000

The total budget allocated across both departments is

\$4,000

.

3

Solve the linear equation for

x

.

4x + 400 = 4000 \implies 4x = 3600 \implies x = 900

Combine like terms, isolate the variable term by subtracting

400

, and divide by

4

.

Key Concept

Setting up and solving a linear equation in one variable from a real-life word problem context.

Question 1482Question →

In a circle centered at point $O$ , the ratio of the area of sector $AOB$ to the area of the entire circle is $3:8$ . If the total perimeter of sector $AOB$ is $24 + 9\pi$ , what is the radius of the circle?

Show answer & explanation

Answer: 12

Answer

The radius of the circle is 12.

Since the ratio of the sector area to the total circle area is

3:8

, the sector accounts for

\frac{3}{8}

of the entire circumference

2\pi r

, making the arc length

L = \frac{3}{8}(2\pi r) = \frac{3}{4}\pi r

. The perimeter of the sector is

2r + L = 2r + \frac{3}{4}\pi r

. Equating this to

24 + 9\pi

gives

2r = 24

, so

r = 12

.

Step-by-Step Solution

1

Relate sector area ratio to arc length

Arc length

L = \frac{3}{4}\pi r

The fraction of the circle occupied by the sector is

\frac{3}{8}

, so the arc length is

\frac{3}{8}

of the circle's circumference

2\pi r

.

2

Formulate the perimeter expression for sector

AOB

Perimeter =

2r + \frac{3}{4}\pi r

The boundary of a sector includes two straight radii of length

r

plus the curved arc length

L

.

3

Equate to given perimeter and solve for

r

r = 12

Comparing

2r + \frac{3}{4}\pi r

to

24 + 9\pi

, setting

2r = 24

yields

r = 12

, which also satisfies

\frac{3}{4}\pi (12) = 9\pi

.

Key Concept

Perimeter of a circle sector and fractional relationship between sector area, central angle, and arc length.

Question 1483Question →

If

x

is a positive real number satisfying the equation

\sqrt[3]{\frac{x^3 \sqrt{x\sqrt{x}}}{x^{-1/4}}} = 16

what is the value of

x

?

Show answer & explanation

Answer: 8

Answer

8

By converting all radicals into fractional exponents and systematically applying exponent rules, the expression under the cube root simplifies to

x^4

. Taking the cube root gives

x^{4/3} = 16

. Solving for

x

by raising both sides to

3/4

yields

x = 16^{3/4} = (2^4)^{3/4} = 2^3 = 8

.

Step-by-Step Solution

1

Express the inner nested radical using fractional exponents

\sqrt{x\sqrt{x}} = \sqrt{x \cdot x^{1/2}} = \sqrt{x^{3/2}} = x^{3/4}

Applying the product and power rules of exponents:

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

and

(x^a)^b = x^{ab}

.

2

Simplify the numerator inside the outer radical

x^3 \cdot x^{3/4} = x^{3 + 3/4} = x^{15/4}

Multiplying exponential terms with the same base requires adding their exponents.

3

Divide by the negative exponent in the denominator

\frac{x^{15/4}}{x^{-1/4}} = x^{15/4 - (-1/4)} = x^{16/4} = x^4

Dividing exponential terms with the same base requires subtracting the denominator exponent from the numerator exponent.

4

Apply the outer cube root to the simplified expression

\sqrt[3]{x^4} = (x^4)^{1/3} = x^{4/3}

The

n

-th root of an expression is equivalent to raising that expression to the power of

1/n

.

5

Solve the resulting exponential equation for

x

x^{4/3} = 16 \implies x = 16^{3/4} = (2^4)^{3/4} = 2^3 = 8

Raise both sides of

x^{4/3} = 16

to the power of

3/4

to isolate

x

.

Key Concept

Simplifying nested algebraic radicals and solving equations with fractional exponents using exponent rules.

Question 1484Question →

A bookstore sells hardcover books for $\$15$ each and paperback books for $\$10$ each. A customer purchased a total of $12$ books and spent $\$150$ in total. Let $h$ represent the number of hardcover books purchased and $p$ represent the number of paperback books purchased. Which of the following equations correctly model this situation? Select all such equations.

Select all that apply

h + p = 12

15h + 10p = 150

3h + 2p = 30

10h + 15p = 150

15h + 10p = 12

Show answer & explanation

Answer: $h + p = 12$ ; $15h + 10p = 150$ ; $3h + 2p = 30$

Answer

The correct equations are

h + p = 12

,

15h + 10p = 150

, and

3h + 2p = 30

.

The system of equations that models this situation requires one equation for the total count of books (

h + p = 12

) and one for the total monetary expenditure (

15h + 10p = 150

). Dividing the cost equation

15h + 10p = 150

by 5 gives another mathematically equivalent relation

3h + 2p = 30

. Thus, all three of these statements accurately represent the given conditions.

Step-by-Step Solution

1

Formulate the total quantity equation

h + p = 12

The total number of books bought is 12, which is the sum of hardcover books

h

and paperback books

p

.

2

Formulate the total cost equation

15h + 10p = 150

Hardcover books cost

\$15

each and paperbacks cost

\$10

each, yielding a total cost of

\$150

.

3

Simplify the total cost equation by dividing by the greatest common divisor

3h + 2p = 30

Dividing all coefficients in

15h + 10p = 150

by

5

produces an equivalent simplified linear equation.

Key Concept

Linear Modeling and Equivalent Equations

Estimated Time:1m 0s

Question 1485Question →

In a geometric plane, line $L_1$ is parallel to line $L_2$ . Points $A$ and $C$ lie on line $L_1$ , and points $B$ and $D$ lie on line $L_2$ . Line segments $AB$ and $CD$ intersect at point $X$ located between lines $L_1$ and $L_2$ . If measure of $\angle XAC = 42^\circ$ and measure of $\angle XDB = 35^\circ$ , what is the measure, in degrees, of $\angle AXC$ ?

Show answer & explanation

Answer: 103

Answer

The measure of

\angle AXC

is

103^\circ

.

Line

L_1

is parallel to line

L_2

, meaning segment

AC

is parallel to segment

BD

. Transversal line

CD

intersects both parallel lines, creating alternate interior angles

\angle XCA

and

\angle XDB

. Hence,

\angle XCA = \angle XDB = 35^\circ

. Inside triangle

ACX

, the three interior angles must sum to

180^\circ

. Substituting the values yields

\angle AXC = 180^\circ - (42^\circ + 35^\circ) = 103^\circ

.

Step-by-Step Solution

1

Identify parallel lines and the transversal line

Line segment

CD

acts as a transversal line intersecting parallel lines

L_1

and

L_2

.

Points

A

and

C

lie on line

L_1

while points

B

and

D

lie on line

L_2

with

L_1 \parallel L_2

.

2

Apply the alternate interior angles theorem

\angle XCA = \angle XDB = 35^\circ

When a transversal intersects two parallel lines, alternate interior angles are equal.

3

Calculate the target angle using the sum of interior angles in a triangle

\angle AXC = 180^\circ - (42^\circ + 35^\circ) = 103^\circ

The sum of interior angles in triangle

ACX

is

180^\circ

.

Key Concept

Properties of parallel lines intersected by a transversal and the triangle angle sum theorem.

Question 1486Question →

In the $xy$ -plane, point $P$ has coordinates $(3, -4)$ . Point $P$ is reflected across the $y$ -axis to form point $Q$ . Point $Q$ is then translated upward by $2$ units to form point $R$ . Which of the following statements must be true? Select all that apply.

Select all that apply

The coordinates of point

Q

are

(-3, -4)

.

Point

R

lies in Quadrant III.

The distance between point

P

and point

Q

is

8

units.

The

y

-coordinate of point

R

is

2

.

Show answer & explanation

Answer: The coordinates of point $Q$ are $(-3, -4)$ .; Point $R$ lies in Quadrant III.

Answer

The correct statements are that the coordinates of point

Q

are

(-3, -4)

and that point

R

lies in Quadrant III.

Reflecting

(3, -4)

across the

y

-axis yields

(-3, -4)

for point

Q

. Translating

(-3, -4)

upward by

2

units gives

(-3, -2)

for point

R

. Because both coordinates of

R

are negative, it resides in Quadrant III.

Step-by-Step Solution

1

Perform the reflection across the

y

-axis

Point

Q = (-3, -4)

Reflecting across the

y

-axis negates the

x

-coordinate while keeping the

y

-coordinate the same.

2

Perform the vertical translation upward by

2

units

Point

R = (-3, -4 + 2) = (-3, -2)

Translating upward adds

2

to the

y

-coordinate of point

Q

.

3

Determine the quadrant of point

R

Quadrant III

Points with

x < 0

and

y < 0

are located in Quadrant III.

4

Calculate the distance between point

P(3, -4)

and point

Q(-3, -4)

Distance =

|3 - (-3)| = 6

units

Points

P

and

Q

lie on the horizontal line

y = -4

, so the distance is the difference in their

x

-coordinates.

Key Concept

Coordinate transformations including axis reflections and translations in the Cartesian plane

Question 1487Question →

In circle $O$ , points $P$ and $Q$ lie on the circumference such that the ratio of the minor arc length $PQ$ to the radius $r$ of the circle is $\frac{5\pi}{6}$ . If the area of sector $POQ$ is $30\pi$ , what is the total perimeter of sector $POQ$ ?

12\sqrt{2} + 5\sqrt{2}\pi

12 + 5\pi

5\sqrt{2}\pi

12 + 10\pi

12\sqrt{2} + 12\sqrt{2}\pi

Show answer & explanation

Answer: $12\sqrt{2} + 5\sqrt{2}\pi$

Answer

12\sqrt{2} + 5\sqrt{2}\pi

The central angle

\theta

in radians is equal to the ratio of minor arc length to radius, so

\theta = \frac{5\pi}{6}

. Using the area of a sector formula

A = \frac{1}{2}r^2\theta

, we substitute

A = 30\pi

to get

\frac{1}{2}r^2\left(\frac{5\pi}{6}\right) = 30\pi

, which simplifies to

r^2 = 72

, or

r = 6\sqrt{2}

. The arc length is then

s = r\theta = (6\sqrt{2})\left(\frac{5\pi}{6}\right) = 5\sqrt{2}\pi

. The total perimeter of the sector includes both bounding radii and the arc length:

2r + s = 12\sqrt{2} + 5\sqrt{2}\pi

.

Step-by-Step Solution

1

Relate the central angle in radians to the given ratio of arc length to radius.

The central angle

\theta

in radians is given by

\theta = \frac{\text{arc length}}{r} = \frac{5\pi}{6}

.

By definition of radian measure, arc length

s = r\theta

, so

\frac{s}{r} = \theta

.

2

Use the sector area formula to solve for the radius

r

.

Setting

\frac{1}{2}r^2\left(\frac{5\pi}{6}\right) = 30\pi

yields

\frac{5\pi}{12}r^2 = 30\pi \implies r^2 = 72 \implies r = 6\sqrt{2}

.

The area of a sector with central angle

\theta

(in radians) is

A = \frac{1}{2}r^2\theta

.

3

Calculate the minor arc length

s

.

Arc length

s = r\theta = (6\sqrt{2})\left(\frac{5\pi}{6}\right) = 5\sqrt{2}\pi

.

Multiplying the radius by the central angle in radians gives the length of the subtended arc.

4

Calculate the total perimeter of sector

POQ

.

Perimeter

= 2r + s = 2(6\sqrt{2}) + 5\sqrt{2}\pi = 12\sqrt{2} + 5\sqrt{2}\pi

.

The perimeter of a sector consists of the two bounding radii plus the arc length.

Key Concept

Relationship between radian measure, sector area, arc length, and sector perimeter

Estimated Time:2m 0s

Question 1488Question →

A triangle has integer side lengths $a$ , $b$ , and $c$ such that its perimeter is $18$ . Which of the following could be the area of the triangle? Select all such values.

Select all that apply

12

6\sqrt{6}

9\sqrt{3}

4\sqrt{15}

8\sqrt{5}

Show answer & explanation

Answer: $12$ ; $6\sqrt{6}$ ; $9\sqrt{3}$

Answer

The possible areas of the triangle are

12

,

6\sqrt{6}

, and

9\sqrt{3}

.

To form a valid triangle with integer sides and a perimeter of 18, each side length must be a positive integer strictly less than 9 by the triangle inequality theorem. Evaluating Heron's formula

A = \sqrt{s(s-a)(s-b)(s-c)}

with

s=9

for all valid side combinations

(a,b,c)

produces the set of areas

\{3\sqrt{7}, 6\sqrt{3}, 3\sqrt{15}, 12, 6\sqrt{5}, 6\sqrt{6}, 9\sqrt{3}\}

. Among the options given, the values 12 (from side lengths 8, 5, 5),

6\sqrt{6}

(from side lengths 7, 6, 5), and

9\sqrt{3}

(from side lengths 6, 6, 6) are correct.

Step-by-Step Solution

1

Determine the structural bounds on side lengths using the Triangle Inequality Theorem.

For any side length

x

,

x < \frac{\text{Perimeter}}{2} = 9

. Thus,

a, b, c \in \{1, 2, 3, 4, 5, 6, 7, 8\}

.

The sum of any two side lengths must be strictly greater than the third side length.

2

List all unique integer side length combinations

(a, b, c)

where

a \ge b \ge c

and

a + b + c = 18

.

The valid triples are

(8, 8, 2)

,

(8, 7, 3)

,

(8, 6, 4)

,

(8, 5, 5)

,

(7, 7, 4)

,

(7, 6, 5)

, and

(6, 6, 6)

.

Exhaustive enumeration of integer partitions of 18 satisfying

a \le 8

.

3

Calculate the semi-perimeter

s

of the triangle.

s = \frac{18}{2} = 9

.

Required parameter for Heron's formula

A = \sqrt{s(s-a)(s-b)(s-c)}

.

4

Compute the area for candidate triples using Heron's formula.

For

(8, 5, 5)

:

A = \sqrt{9(1)(4)(4)} = 12

.
For

(7, 6, 5)

:

A = \sqrt{9(2)(3)(4)} = 6\sqrt{6}

.
For

(6, 6, 6)

:

A = \sqrt{9(3)(3)(3)} = 9\sqrt{3}

.

Direct evaluation of geometric area for the valid triangles.

Key Concept

Triangle Inequality Theorem and Area calculation via Heron's Formula

Question 1489Question →

A dataset of 25 student test scores has a mean of $70$ , a standard deviation of $8$ , and an interquartile range ( $\text{IQR}$ ) of $12$ . A instructor creates a modified dataset by multiplying each original test score by $1.5$ and then adding $10$ to the result. What is the standard deviation of the modified dataset?

8

12

18

22

28

Show answer & explanation

Answer: $12$

Answer

12

For any dataset, if each value

x

is transformed into

a \cdot x + b

, where

a

and

b

are constants, the new standard deviation is given by

|a| \cdot \sigma_{\text{old}}

. In this problem,

a = 1.5

and

b = 10

. Therefore, the new standard deviation is

1.5 \times 8 = 12

. The addition of

10

affects measures of center (such as the mean and median) but has no effect on measures of dispersion (such as standard deviation, IQR, and range).

Step-by-Step Solution

1

Analyze the effect of multiplying each data point by a constant on standard deviation.

Multiplying every value in a dataset by a constant

c = 1.5

multiplies the standard deviation

\sigma

by

|c|

. The new intermediate standard deviation becomes

1.5 \times 8 = 12

.

Standard deviation measures dispersion around the mean, so scaling all data points by a factor scales the distance between each point and the mean by that same factor.

2

Analyze the effect of adding a constant to each data point on standard deviation.

Adding a constant

k = 10

to every value shifts the entire distribution without changing the distances between data points or the relative spread around the mean. Thus, standard deviation remains

12

.

Adding a constant shifts both the individual values and the mean by the same amount, leaving

(x_i - \bar{x})

unchanged.

Key Concept

Linear Transformations on Measures of Dispersion

Question 1490Question →

If $x$ is an integer that satisfies both $|2x - 9| \le 15$ and $\frac{5 - 2x}{-3} \le 3$ , what is the product of the smallest and largest possible values of $x$ ?

-21

84

35

-36

21

Show answer & explanation

Answer: $-21$

Answer

The product of the smallest and largest possible integer values of

x

is

-21

.

Solving

|2x - 9| \le 15

gives

-3 \le x \le 12

. Solving

\frac{5 - 2x}{-3} \le 3

requires reversing the inequality sign twice (first when multiplying by

-3

, then when dividing by

-2

), which yields

x \le 7

. Taking the intersection of both intervals gives

-3 \le x \le 7

. The smallest integer in this range is

-3

and the largest is

7

, giving a product of

(-3) \times 7 = -21

.

Step-by-Step Solution

1

Solve the absolute value inequality

|2x - 9| \le 15

.

-15 \le 2x - 9 \le 15 \implies -6 \le 2x \le 24 \implies -3 \le x \le 12

.

An absolute value inequality

|u| \le a

(for

a \ge 0

) is equivalent to the compound inequality

-a \le u \le a

.

2

Solve the linear inequality

\frac{5 - 2x}{-3} \le 3

.

Multiply by

-3

and flip the inequality direction:

5 - 2x \ge -9

. Subtract

5

:

-2x \ge -14

. Divide by

-2

and flip the inequality direction again:

x \le 7

.

Multiplying or dividing an inequality by a negative number reverses the direction of the inequality sign.

3

Determine the intersection of the two solution sets.

Combining

-3 \le x \le 12

and

x \le 7

gives

-3 \le x \le 7

.

The value of

x

must satisfy both conditions simultaneously.

4

Identify the smallest and largest integer values of

x

and calculate their product.

Smallest integer

= -3

, largest integer

= 7

. Product

= (-3) \times 7 = -21

.

Both endpoints

-3

and

7

are included in the closed interval

[-3, 7]

.

Key Concept

Solving compound linear inequalities involving absolute values and applying the rule for reversing inequality signs when multiplying or dividing by negative numbers.

Estimated Time:2m 0s

Question 1491Question →

In the $xy$ -plane, line $k$ passes through the point $(1, -2)$ and is perpendicular to line $m$ , which is defined by the equation $3x - 4y = 12$ . Line $k$ intersects line $n$ , defined by the equation $y = 2x + 1$ , at point $P$ . What is the distance between point $P$ and the point $(3.5, 3)$ ?

Show answer & explanation

Answer: 5

Answer

5

Rewriting the equation of line

m

,

3x - 4y = 12

, in slope-intercept form gives

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

, so line

m

has a slope of

\frac{3}{4}

. Since line

k

is perpendicular to line

m

, its slope is the negative reciprocal,

-\frac{4}{3}

. Using the point

(1, -2)

, line

k

has the equation

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

. Setting this equal to the equation of line

n

(

y = 2x + 1

) yields

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

, which solves to

x = -0.5

and

y = 0

, giving the intersection point

P(-0.5, 0)

. Finally, calculating the distance between

P(-0.5, 0)

and

(3.5, 3)

using the distance formula gives

\sqrt{(3.5 - (-0.5))^2 + (3 - 0)^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5

.

Step-by-Step Solution

1

Determine the slope of line

m

The slope of line

m

is

\frac{3}{4}

Rewriting

3x - 4y = 12

in slope-intercept form gives

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

, so the slope is

\frac{3}{4}

.

2

Determine the slope of line

k

The slope of line

k

is

-\frac{4}{3}

Perpendicular lines have slopes that are negative reciprocals of each other.

3

Write the equation of line

k

The equation of line

k

is

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

Using point-slope form with

(1, -2)

gives

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

, which simplifies to

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

.

4

Find the coordinates of intersection point

P

Point

P

has coordinates

(-0.5, 0)

Setting

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

yields

-\frac{10}{3}x = \frac{5}{3} \implies x = -0.5

. Substituting

x = -0.5

into

y = 2x + 1

yields

y = 0

.

5

Calculate the distance between

P(-0.5, 0)

and

(3.5, 3)

The distance is

5

Applying the distance formula yields

d = \sqrt{(3.5 - (-0.5))^2 + (3 - 0)^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5

.

Key Concept

Perpendicular line slopes, finding intersection of two lines, and applying the distance formula.

Question 1492Question →

If $x$ is a real number that satisfies the inequality $3|x - 4| + 2|x + 5| \le 26$ , what is the maximum possible value of $|x - 7|$ ?

Show answer & explanation

Answer: 11

Answer

11

Solving the piecewise linear inequality

3|x - 4| + 2|x + 5| \le 26

yields the interval

[-4, 5.6]

. The distance function

|x - 7|

reaches its maximum at the endpoint farthest from

7

, which is

x = -4

. Evaluating

|-4 - 7|

yields 11.

Step-by-Step Solution

1

Identify the critical points of the absolute value terms.

The critical points are

x = -5

(where

x + 5 = 0

) and

x = 4

(where

x - 4 = 0

).

Critical points mark where the linear expressions inside the absolute values change sign.

2

Analyze the inequality piecewise across the three regions defined by the critical points.

For

x < -5

:

3(4 - x) + 2(-5 - x) \le 26 \implies 2 - 5x \le 26 \implies x \ge -4.8

. This produces no solution since

x

cannot be simultaneously

< -5

and

\ge -4.8

.
For

-5 \le x < 4

:

3(4 - x) + 2(x + 5) \le 26 \implies 22 - x \le 26 \implies x \ge -4

, yielding

-4 \le x < 4

.
For

x \ge 4

:

3(x - 4) + 2(x + 5) \le 26 \implies 5x - 2 \le 26 \implies x \le 5.6

, yielding

4 \le x \le 5.6

.

Expanding absolute value terms according to their regional sign definitions removes the absolute values.

3

Combine the valid regional solutions to establish the complete solution interval for

x

.

The set of all satisfying real numbers is

x \in [-4, 5.6]

.

Taking the union of the non-empty piecewise solution intervals yields the total solution set.

4

Find the maximum value of

|x - 7|

over

x \in [-4, 5.6]

.

At

x = -4

,

|-4 - 7| = |-11| = 11

. At

x = 5.6

,

|5.6 - 7| = |-1.4| = 1.4

. The maximum possible value is 11.

The expression

|x - 7|

measures distance from

7

. The maximum distance on a closed interval occurs at the endpoint farthest from

7

, which is

x = -4

.

Key Concept

Piecewise analysis of linear absolute value inequalities and optimization of absolute value distance functions.

Estimated Time:2m 30s

Question 1493Question →

In right triangle $PQR$ , the measure of angle $P$ is $30^\circ$ , the measure of angle $Q$ is $60^\circ$ , and the length of the hypotenuse $PR$ is $12$ . Which of the following statements must be true? Select all that apply.

Select all that apply

The length of the shortest side is

6

.

The length of the side opposite the

60^\circ

angle is

6\sqrt{3}

.

The length of the side opposite the

60^\circ

angle is

6\sqrt{2}

.

The area of triangle

PQR

is

36\sqrt{3}

.

The perimeter of triangle

PQR

is

18\sqrt{3}

.

Show answer & explanation

Answer: The length of the shortest side is $6$ .; The length of the side opposite the $60^\circ$ angle is $6\sqrt{3}$ .

Answer

The statements confirming that the shortest side has length

6

and that the side opposite the

60^\circ

angle has length

6\sqrt{3}

are both correct.

In any

30^\circ-60^\circ-90^\circ

right triangle, the sides opposite the

30^\circ

,

60^\circ

, and

90^\circ

angles are in the ratio

1 : \sqrt{3} : 2

. With a hypotenuse of

12

, the side opposite

30^\circ

is

\frac{12}{2} = 6

, and the side opposite

60^\circ

is

6\sqrt{3}

. Thus, both statements specifying these values are correct.

Step-by-Step Solution

1

Identify the side length ratio for a

30^\circ-60^\circ-90^\circ

right triangle.

The side lengths follow the standard ratio

1 : \sqrt{3} : 2

, corresponding to the sides opposite the

30^\circ

,

60^\circ

, and

90^\circ

angles respectively.

Special right triangle properties establish fixed proportional relationships between sides based on interior angles.

2

Calculate the length of the shortest side (opposite the

30^\circ

angle).

Shortest side =

\frac{\text{Hypotenuse}}{2} = \frac{12}{2} = 6

.

Since the hypotenuse corresponds to

2x = 12

,

x = 6

.

3

Calculate the length of the longer leg (opposite the

60^\circ

angle).

Longer leg =

x\sqrt{3} = 6\sqrt{3}

.

The side opposite

60^\circ

is

\sqrt{3}

times the side opposite

30^\circ

.

Key Concept

Side length ratios of a

30^\circ-60^\circ-90^\circ

special right triangle (

1 : \sqrt{3} : 2

).

Question 1494Question →

For all real numbers $p$ and $q$ , the custom operation $\odot$ is defined by $p \odot q = 2p^2 - 3q$ . What is the value of $3 \odot (-4)$ ?

Show answer & explanation

Answer: 30

Answer

The value of

3 \odot (-4)

is

30

.

By applying the definition of the custom operation

p \odot q = 2p^2 - 3q

with

p = 3

and

q = -4

, we obtain

2(3)^2 - 3(-4) = 2(9) + 12 = 18 + 12 = 30

.

Step-by-Step Solution

1

Substitute

p = 3

and

q = -4

into the expression

2p^2 - 3q

.

2(3)^2 - 3(-4)

The custom operation defines how to process the two inputs

p

and

q

.

2

Simplify the powers and products according to order of operations.

2(9) - (-12) = 18 + 12

Exponents must be calculated before multiplication, and multiplying two negative numbers yields a positive value.

3

Add the terms together to get the final numerical result.

30

18 + 12 = 30

.

Key Concept

Custom Symbol Operations

Question 1495Question →

Consider the linear equation in

x

shown below, where

k

is a real constant:

\frac{x - k}{2} - \frac{2x + 1}{3} = \frac{k(x + 1)}{6} - 1

Which of the following statements must be true? Select all such statements.

Select all that apply

If

k = 1

, then

x = 0

is the unique solution to the equation.

If

k = -1

, the equation has infinitely many solutions.

If

k = 0

, the solution to the equation is

x = 4

.

If

k = 3

, the solution to the equation is

x = 2

.

If

k = 2

, the solution to the equation is

x = -\frac{4}{3}

.

Show answer & explanation

Answer: If $k = 1$ , then $x = 0$ is the unique solution to the equation.; If $k = 0$ , the solution to the equation is $x = 4$ .; If $k = 2$ , the solution to the equation is $x = -\frac{4}{3}$ .

Answer

The correct statements are: if

k = 1

, then

x = 0

is the unique solution; if

k = 0

, the solution is

x = 4

; and if

k = 2

, the solution is

x = -\frac{4}{3}

.

Multiplying the equation by

6

yields

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

. Expanding and collecting terms in

x

leads to

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

. Testing the proposed values of

k

shows that when

k = 1

,

2x = 0

gives

x = 0

; when

k = 0

,

x = 4

; and when

k = 2

,

3x = -4

gives

x = -\frac{4}{3}

. Therefore, the statements corresponding to

k = 1

,

k = 0

, and

k = 2

are all correct.

Step-by-Step Solution

1

Clear fractions by multiplying the entire equation by the least common denominator,

6

.

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

Eliminating fractions simplifies algebraic expansion and grouping of like terms.

2

Expand all expressions and combine like terms on both sides.

3x - 3k - 4x - 2 = kx + k - 6 \implies -x - 3k - 2 = kx + k - 6

Distribute terms carefully to prevent sign errors.

3

Isolate terms containing

x

on one side and parameter/constant terms on the other side.

-x - kx = k + 3k - 6 + 2 \implies -x(k + 1) = 4k - 4 \implies (k + 1)x = 4(1 - k)

Factoring out

x

provides the general solution form

x = \frac{4(1 - k)}{k + 1}

for

k \neq -1

.

4

Evaluate each specified value of

k

against

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

.

For

k = 1

:

2x = 0 \implies x = 0

.
For

k = -1

:

0x = 8 \implies

no solution.
For

k = 0

:

1x = 4 \implies x = 4

.
For

k = 3

:

4x = -8 \implies x = -2

.
For

k = 2

:

3x = -4 \implies x = -\frac{4}{3}

.

Direct substitution verifies which given statements are true.

Key Concept

Solving linear equations with parameters and analyzing existence and uniqueness of solutions.

Estimated Time:2m 30s

Question 1496Question →

Consider the following system of linear equations in

x

,

y

, and

z

, where

c

is a real constant:

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

x + \frac{1}{3}y - z = 2

6x - y + z = c

If the system has at least one solution $(x, y, z)$ , what is the value of $7x - \frac{2}{3}y$ ?

18

16

24

7

2

Show answer & explanation

Answer: 18

Answer

18

To find the value of

7x - \frac{2}{3}y

without individual values for

x, y,

and

z

, we express

7x - \frac{2}{3}y

as a linear combination

m(\text{Eq. 1}) + n(\text{Eq. 2})

. Matching the

z

-coefficients requires

2m - n = 0 \implies n = 2m

. Matching the

y

-coefficients yields

-m + \frac{1}{3}(2m) = -\frac{2}{3} \implies m = 2

, which gives

n = 4

. Verifying the

x

-coefficient gives

2\left(\frac{3}{2}\right) + 4(1) = 7

. Applying these multipliers to the right-hand sides gives

2(5) + 4(2) = 10 + 8 = 18

.

Step-by-Step Solution

1

Identify the target expression

7x - \frac{2}{3}y

as a linear combination of the first two equations

Express

m\left(\frac{3}{2}x - y + 2z\right) + n\left(x + \frac{1}{3}y - z\right) = 7x - \frac{2}{3}y + 0z

Because the system is dependent when consistent, individual variable values cannot be uniquely determined, but specific linear combinations independent of

z

can be evaluated.

2

Set up a system of equations for the scalar multipliers

m

and

n

Equating coefficients of

z

:

2m - n = 0 \implies n = 2m

. Equating coefficients of

y

:

-m + \frac{1}{3}n = -\frac{2}{3}

.

Eliminating

z

requires the net coefficient of

z

to equal 0.

3

Solve for

m

and

n

Substitute

n = 2m

into the

y

-coefficient equation:

-m + \frac{2}{3}m = -\frac{1}{3}m = -\frac{2}{3} \implies m = 2

, which gives

n = 4

.

Determining the exact linear multipliers needed to match the target expression.

4

Verify

x

-coefficient consistency and compute the target value

x

-coefficient:

2\left(\frac{3}{2}\right) + 4(1) = 3 + 4 = 7

. Value:

2(5) + 4(2) = 10 + 8 = 18

.

Applying the scalars

m = 2

and

n = 4

to the right-hand side constants gives the exact numerical value of

7x - \frac{2}{3}y

.

Key Concept

Linear combinations of dependent systems of equations

Alternative Method

Multiply the first equation by 2 to clear fractions:

3x - 2y + 4z = 10

. Multiply the second equation by 3 to clear fractions:

3x + y - 3z = 6

. Eliminate

z

by forming

3(3x - 2y + 4z) + 4(3x + y - 3z) = 3(10) + 4(6) \implies 21x - 2y = 54

. Dividing both sides of

21x - 2y = 54

by 3 directly gives

7x - \frac{2}{3}y = 18

.

Estimated Time:2m 0s

Question 1497Question →

In circle $O$ , sector $AOB$ has an area of $45\pi$ square units and arc $AB$ has a length of $6\pi$ units. What is the perimeter of sector $AOB$ ?

15 + 6\pi

30 + 6\pi

45 + 6\pi

60 + 6\pi

30\pi

Show answer & explanation

Answer: $30 + 6\pi$

Answer

30 + 6\pi

The area of a sector is given by

\frac{1}{2} r L

, where

r

is the radius and

L

is the arc length. Substituting

L = 6\pi

and Area

= 45\pi

yields

45\pi = \frac{1}{2} r (6\pi)

, which simplifies to

45\pi = 3\pi r

, so

r = 15

. The perimeter of the sector is

L + 2r = 6\pi + 2(15) = 30 + 6\pi

. Thus, the option equal to

30 + 6\pi

is correct.

Step-by-Step Solution

1

Express sector area and arc length in terms of radius

r

and central angle

\theta

in degrees.

Sector Area =

\frac{\theta}{360^\circ} \times \pi r^2 = 45\pi

and Arc Length =

\frac{\theta}{360^\circ} \times 2\pi r = 6\pi

.

Relating both formulas allows solving for the radius directly using the ratio of area to arc length.

2

Divide the sector area formula by the arc length formula to find radius

r

.

\frac{\text{Sector Area}}{\text{Arc Length}} = \frac{\frac{\theta}{360^\circ} \pi r^2}{\frac{\theta}{360^\circ} 2\pi r} = \frac{r}{2} = \frac{45\pi}{6\pi} = 7.5 \implies r = 15.

The central angle fraction

\frac{\theta}{360^\circ}

and

\pi

cancel out, leaving

\frac{r}{2} = 7.5

.

3

Calculate the total perimeter of sector

AOB

.

\text{Perimeter} = \text{Arc Length} + 2r = 6\pi + 2(15) = 30 + 6\pi.

The perimeter of a sector consists of the outer arc length plus the two straight boundary radii (

OA

and

OB

).

Key Concept

The relationship between sector area, arc length, radius, and sector perimeter

Question 1498Question →

A researcher recorded seven daily temperature readings (in degrees Celsius): $14, 5, 22, 9, 25, 2,$ and $17$ . What is the median of these seven temperature readings?

9

14

13.5

17

25

Show answer & explanation

Answer: $14$

Answer

The median of the temperature readings is

14

.

To find the median of a set of numbers, first arrange them in ascending order:

2, 5, 9, 14, 17, 22, 25

. Since there are

7

numbers, the median is the middle value in the

4\text{th}

position, which is

14

.

Step-by-Step Solution

1

Arrange the given dataset in ascending numerical order.

The ordered list of

7

temperatures is:

2, 5, 9, 14, 17, 22, 25

.

To calculate the median of a dataset, the values must first be ordered from smallest to largest.

2

Identify the middle position for an odd number of data values (

n = 7

).

The position of the median is

\frac{7 + 1}{2} = 4\text{th}

position.

For a dataset with

n

elements where

n

is odd, the median is located at position

\frac{n+1}{2}

.

3

Extract the value located at the

4\text{th}

position in the ordered list.

The

4\text{th}

term is

14

.

The fourth number in

2, 5, 9, 14, 17, 22, 25

is

14

.

Key Concept

Median of a Dataset

Estimated Time:45s

Question 1499Question →

In $\triangle ABC$ , the lengths of sides $AB$ , $BC$ , and $AC$ are $13$ , $14$ , and $15$ , respectively. A line segment $DE$ is drawn parallel to side $BC$ , with point $D$ lying on side $AB$ and point $E$ lying on side $AC$ . If the perimeter of $\triangle ADE$ is equal to the perimeter of quadrilateral $DBCE$ , what is the area of $\triangle ADE$ ?

Show answer & explanation

Answer: 47.25

Answer

47.25

The area of the original triangle

\triangle ABC

is computed as

84

using Heron's formula. By defining the linear scale factor

k

between

\triangle ADE

and

\triangle ABC

, the perimeters of

\triangle ADE

and quadrilateral

DBCE

are expressed as

42k

and

42 - 14k

, respectively. Setting these equal yields

k = 0.75

. The area of

\triangle ADE

is then

k^2 \times 84 = 0.5625 \times 84 = 47.25

.

Step-by-Step Solution

1

Calculate the perimeter and area of the main triangle

\triangle ABC

.

The perimeter of

\triangle ABC

is

13 + 14 + 15 = 42

. Using Heron's formula with semi-perimeter

s = 21

,

\text{Area}(\triangle ABC) = \sqrt{21(21-13)(21-14)(21-15)} = \sqrt{21 \times 8 \times 7 \times 6} = 84

.

Finding the area and perimeter of the full triangle sets the required baseline for proportional scaling.

2

Set up expressions for the perimeters of

\triangle ADE

and quadrilateral

DBCE

using a scale factor

k

.

Because

DE \parallel BC

,

\triangle ADE \sim \triangle ABC

with scale factor

k = \frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}

. Thus,

\text{Perimeter}(\triangle ADE) = 13k + 15k + 14k = 42k

. The segments

DB = 13(1-k)

and

EC = 15(1-k)

, so

\text{Perimeter}(DBCE) = 13(1-k) + 14 + 15(1-k) + 14k = 42 - 14k

.

Parallel lines create similar triangles, which allows all perimeter segment lengths to be represented in terms of one variable

k

.

3

Solve for the scale factor

k

by equating the two perimeters.

42k = 42 - 14k \implies 56k = 42 \implies k = \frac{42}{56} = \frac{3}{4} = 0.75

.

Equating the perimeters satisfies the condition specified in the question stem.

4

Calculate the area of

\triangle ADE

using the square of the linear scale factor.

\text{Area}(\triangle ADE) = k^2 \times \text{Area}(\triangle ABC) = \left(\frac{3}{4}\right)^2 \times 84 = \frac{9}{16} \times 84 = \frac{189}{4} = 47.25

.

The area ratio of similar geometric figures is proportional to the square of their linear scale factor.

Key Concept

Properties of Similar Triangles, Area via Heron's Formula, and Perimeter Scaling

Question 1500Question →

A manufacturing facility uses Machine A and Machine B to process standard orders of raw materials. Operating independently at its constant rate, Machine A requires $x$ hours to process one standard order, where $x > 0$ . Machine B operates at a constant rate and requires $x + 2$ hours to process one standard order. When both machines operate simultaneously at their respective constant rates for $6$ hours, the total number of standard orders processed is $2$ fewer than the number of standard orders Machine A would process operating alone for $18$ hours. What is the value of $x$ ?

2

3

4

6

8

Show answer & explanation

Answer: 4

Answer

4

The rate of Machine A is

\frac{1}{x}

orders per hour, and the rate of Machine B is

\frac{1}{x+2}

orders per hour. In 18 hours, Machine A processes

\frac{18}{x}

orders. Working together for 6 hours, both machines process

6\left(\frac{1}{x} + \frac{1}{x+2}\right)

orders. Setting up the difference:

\frac{18}{x} - 6\left(\frac{1}{x} + \frac{1}{x+2}\right) = 2

, which simplifies to

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

. Multiplying both sides by

x(x+2)

yields

12(x+2) - 6x = 2x^2 + 4x

, leading to

2x^2 - 2x - 24 = 0

or

x^2 - x - 12 = 0

. Factoring gives

(x-4)(x+3) = 0

. Because time must be positive,

x = 4

.

Step-by-Step Solution

1

Express the individual work rates of Machine A and Machine B.

Machine A completes

\frac{1}{x}

orders per hour; Machine B completes

\frac{1}{x+2}

orders per hour.

Work rate is the reciprocal of the total time required to complete one unit of work.

2

Formulate the equation based on the total orders processed in the given time frames.

18 \cdot \frac{1}{x} - 6\left(\frac{1}{x} + \frac{1}{x+2}\right) = 2

Machine A alone in 18 hours processes

\frac{18}{x}

orders. Together in 6 hours, they process

6\left(\frac{1}{x} + \frac{1}{x+2}\right)

orders, which is 2 orders less.

3

Simplify the algebraic equation.

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

Subtracting

6 \cdot \frac{1}{x}

from

18 \cdot \frac{1}{x}

yields

\frac{12}{x}

.

4

Clear the denominators by multiplying through by

x(x+2)

and solve the resulting quadratic equation.

12(x+2) - 6x = 2x(x+2) \implies 6x + 24 = 2x^2 + 4x \implies 2x^2 - 2x - 24 = 0 \implies x^2 - x - 12 = 0 \implies (x-4)(x+3) = 0

Clearing denominators transforms the rational equation into a standard quadratic equation.

5

Select the physically meaningful solution for time

x

.

x = 4

hours (rejecting

x = -3

since

x > 0

).

Time must be positive.

Key Concept

Algebraic Work-Rate Modeling and Quadratic Solution

Estimated Time:2m 0s

Answer

Step-by-Step Solution

Key Concept

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Answer

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Common Mistakes

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Common Mistakes

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