All practice questions

Question 2141Question →

A line graphed in the standard $(x,y)$ coordinate plane has an $x$ -intercept of $6$ and a $y$ -intercept of $-3$ . What is the slope of this line?

Show answer & explanation

Answer: 0.5

Answer

The slope of the line is

0.5

(or

\frac{1}{2}

).

The correct slope is

0.5

. By identifying the

x

-intercept as

(6, 0)

and the

y

-intercept as

(0, -3)

, we can apply the slope formula

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

to find

m = \frac{-3 - 0}{0 - 6} = \frac{-3}{-6} = 0.5

.

Step-by-Step Solution

1

Identify the coordinates of the intercepts on the coordinate plane.

The points are

(6, 0)

and

(0, -3)

.

An

x

-intercept of

6

means the line crosses the

x

-axis at

(6, 0)

. A

y

-intercept of

-3

means the line crosses the

y

-axis at

(0, -3)

.

2

Apply the slope formula with the identified coordinates.

m = \frac{-3 - 0}{0 - 6} = \frac{-3}{-6} = 0.5

The slope formula is defined as

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

for any two points on a line.

Key Concept

Calculating the slope of a line from its intercepts

Question 2142Question →

In the standard $(x, y)$ coordinate plane, the points $A(1, 2)$ and $B(9, 8)$ are the endpoints of a diameter of a circle $C$ . A line $L$ passes through the center of $C$ and is perpendicular to segment $AB$ . A point $P(x, y)$ lies on line $L$ such that the distance from $P$ to the center of $C$ is equal to the radius of $C$ . If the $x$ -coordinate of $P$ is greater than the $x$ -coordinate of the center of $C$ , what is the $y$ -coordinate of $P$ ?

Show answer & explanation

Answer: 1

Answer

The y-coordinate of the point P is 1.

The correct answer is 1. The center of circle C is the midpoint of the diameter AB, which is calculated as M(5, 5). The radius is half the length of AB, which is 5. The line L passing through M perpendicular to AB has a slope of -4/3. Points on this line at a distance of 5 from M are found by changing the coordinates by (+3, -4) or (-3, +4), yielding (8, 1) and (2, 9). Since the x-coordinate must be greater than the center's x-coordinate of 5, the correct point is (8, 1), which has a y-coordinate of 1.

Step-by-Step Solution

1

Calculate the center of the circle C by finding the midpoint of the diameter AB.

The center is M(5, 5).

The center of a circle is the midpoint of any of its diameters.

2

Calculate the radius of circle C by finding half the distance between A(1, 2) and B(9, 8).

The radius is 5.

The distance formula gives the diameter length as 10, and the radius is half the diameter.

3

Find the slope of line L perpendicular to AB.

The slope of L is -4/3.

The slope of AB is 3/4, and perpendicular lines have slopes that are negative reciprocals of each other.

4

Determine the coordinates of point P using the distance from the center and the slope of line L.

The possible points are (8, 1) and (2, 9).

Moving a distance of 5 along a line with slope -4/3 from (5, 5) results in a change of +/-3 in the x-coordinate and -/+4 in the y-coordinate.

5

Apply the constraint that the x-coordinate of P must be greater than the x-coordinate of the center (5).

P is (8, 1), so the y-coordinate is 1.

Comparing the two candidate points, only (8, 1) has an x-coordinate greater than 5.

Key Concept

Applying midpoint, distance, and perpendicular slope relationships in coordinate geometry to locate points.

Estimated Time:2m 30s

Question 2143Question →

For a certain real number $x$ , the equation $0.4(3x - 5) - 0.15(2x + 8) = 1.3$ is true. What is the value of $4x - 3$ ?

Show answer & explanation

Answer: 17

Answer

The value of the expression is 17.

Distributing the decimals results in

1.2x - 2 - 0.3x - 1.2 = 1.3

. Combining like terms yields

0.9x - 3.2 = 1.3

. Adding

3.2

to both sides results in

0.9x = 4.5

, which simplifies to

x = 5

after dividing by

0.9

. Substituting

x = 5

into the expression

4x - 3

gives

4(5) - 3 = 17

.

Step-by-Step Solution

1

Distribute the decimal factors through the parentheses on the left side of the equation.

1.2x - 2 - 0.3x - 1.2 = 1.3

To eliminate the parentheses and set up terms for simplification.

2

Combine the variable terms and constant terms on the left side of the equation.

0.9x - 3.2 = 1.3

To group like terms and simplify the equation.

3

Isolate the variable term by adding

3.2

to both sides of the equation.

0.9x = 4.5

To gather all constant terms on the right side of the equation.

4

Divide both sides of the equation by

0.9

to solve for

x

.

x = 5

To find the numerical value of the variable.

5

Substitute the value of

x

into the requested expression

4x - 3

.

4(5) - 3 = 17

To evaluate the specific expression requested by the question.

Key Concept

Solving multi-step linear equations with decimals and evaluating algebraic expressions

Question 2144Question →

In the standard $(x, y)$ coordinate plane, the lines with equations $y = \frac{1}{2}x + 1$ , $y = -x + 7$ , and $y = k$ enclose a triangular region with an area of $6$ square units. What is a possible value of $k$ ?

2

3

4

5

7

Show answer & explanation

Answer: 5

Answer

The correct value of k is 5.

The correct answer is 5. Finding the intersection of the two boundary lines yields the vertex

P(4, 3)

. Calculating the intersection points of the horizontal line

y = k

with the boundary lines gives the

x

-coordinates

2k - 2

and

7 - k

. The distance between these coordinates represents the base of the triangle,

3|k - 3|

, while the height is the vertical distance

|k - 3|

. Substituting these into the triangle area formula

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

results in

6 = \frac{3}{2}(k - 3)^2

. Solving this quadratic equation gives

(k - 3)^2 = 4

, which leads to

k = 5

or

k = 1

. Therefore, 5 is the correct possible value.

Step-by-Step Solution

1

Find the intersection point of the two non-horizontal lines by setting their equations equal to each other.

\frac{1}{2}x + 1 = -x + 7 \Rightarrow \frac{3}{2}x = 6 \Rightarrow x = 4

. Substituting

x = 4

back into either equation gives

y = 3

. The intersection point is

P(4, 3)

.

The intersection point serves as the third vertex of the triangle, and its

y

-coordinate helps determine the height of the triangle relative to the horizontal boundary line

y = k

.

2

Find the

x

-coordinates of the intersection points between the horizontal line

y = k

and the other two lines.

For

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

, setting

y = k

gives

k = \frac{1}{2}x + 1 \Rightarrow x = 2k - 2

. For

y = -x + 7

, setting

y = k

gives

k = -x + 7 \Rightarrow x = 7 - k

.

These two points define the base of the triangle along the line

y = k

.

3

Express the base length and height of the triangle in terms of

k

, then set up the area equation.

The base is the distance between the two

x

-coordinates:

|(2k - 2) - (7 - k)| = |3k - 9| = 3|k - 3|

. The height is the vertical distance from the line

y = k

to the point

P(4, 3)

:

|k - 3|

. The area is

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{3}{2}(k - 3)^2

.

This sets up a solvable algebraic equation using the given area of

6

square units.

4

Solve the area equation for

k

.

\frac{3}{2}(k - 3)^2 = 6 \Rightarrow (k - 3)^2 = 4 \Rightarrow k - 3 = \pm 2

. Thus,

k = 5

or

k = 1

. Since

5

is among the options, it is the correct choice.

This yields the possible values of

k

that satisfy the geometric conditions.

Key Concept

Linear Equations and Graphing

Alternative Method

Instead of solving the algebraic quadratic equation, you can test the given choices for

k

. For example, if you test the value

5

, the line is

y = 5

. The intersection of

y = 5

and

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

is

(8, 5)

, and the intersection of

y = 5

and

y = -x + 7

is

(2, 5)

. The base of the triangle is the horizontal distance from

x = 2

to

x = 8

, which is

6

. The height of the triangle is the vertical distance from

y = 3

(the intersection vertex) to

y = 5

, which is

2

. The area is

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

. This matches the problem statement, confirming that

5

is the correct answer.

Estimated Time:2m 0s

Question 2145Question →

In the standard $(x,y)$ coordinate plane, a region in the first quadrant is bounded by the $x$ -axis, the $y$ -axis, and the line with equation $ax + by = c$ , where $a$ , $b$ , and $c$ are positive constants. The line passes through the point $(8, 18)$ . If the area of this region is minimized when $a = 3$ , what is the value of $c$ ?

Show answer & explanation

Answer: 48

Answer

48

Substituting the given point and

a = 3

into the equation yields

c = 24 + 18b

. The area of the triangle formed by the intercepts is

A = \frac{c^2}{6b}

. Substituting

c

gives

A = 6(9b + 24 + \frac{16}{b})

. Using AM-GM, the minimum occurs when

9b = \frac{16}{b}

, resulting in

b = \frac{4}{3}

. Using this value, we find

c = 48

.

Step-by-Step Solution

1

Substitute the point

(8, 18)

and

a = 3

into the equation

ax + by = c

.

24 + 18b = c

This establishes a relationship between the constants

b

and

c

based on the given point that lies on the line.

2

Calculate the

x

-intercept and

y

-intercept of the line.

x

-intercept is at

x = \frac{c}{3}

, and

y

-intercept is at

y = \frac{c}{b}

.

The boundary of the region in the first quadrant is defined by these coordinate intercepts.

3

Formulate the area

A

of the right triangle bounded by the axes and the line.

A = \frac{c^2}{6b}

The area of a right triangle with vertices at the origin and the intercepts is

\frac{1}{2} \cdot \text{base} \cdot \text{height}

.

4

Substitute

c = 24 + 18b

into the area formula and simplify.

A = 6\left(9b + 24 + \frac{16}{b}\right)

Expressing the area as a single-variable function of

b

allows us to find its minimum value.

5

Apply the AM-GM inequality to minimize the variable term

9b + \frac{16}{b}

.

b = \frac{4}{3}

minimizes the expression.

The sum of two positive terms is minimized when the terms are equal, so

9b = \frac{16}{b} \implies b^2 = \frac{16}{9} \implies b = \frac{4}{3}

.

6

Calculate the value of

c

using the minimizing value of

b

.

c = 48

Substituting

b = \frac{4}{3}

into the relation

c = 24 + 18b

yields the constant value

c

for the minimum area.

Key Concept

Minimizing the area bounded by a line and the coordinate axes using linear equation forms and algebraic minimization.

Alternative Method

Instead of using the AM-GM inequality, you can find the minimum by taking the derivative of the area function

A(b) = 54b + 144 + \frac{96}{b}

with respect to

b

. Setting the derivative

A'(b) = 54 - \frac{96}{b^2} = 0

yields

b^2 = \frac{96}{54} = \frac{16}{9}

, which gives

b = \frac{4}{3}

for

b > 0

.

Estimated Time:3m 0s

Question 2146Question →

In the standard $(x, y)$ coordinate plane, an ellipse is centered at the origin $(0, 0)$ and has vertices at $(-5, 0)$ and $(5, 0)$ . If the distance between the two foci of the ellipse is $8$ , what is the length of the minor axis of the ellipse?

Show answer & explanation

Answer: 6

Answer

The correct answer is 6.

The correct answer is 6 because the ellipse has a horizontal major axis with

a = 5

and focal distance

c = 4

. Using the relationship

c^2 = a^2 - b^2

, we solve for the semi-minor axis

b

to get

b = 3

. The total length of the minor axis is

2b = 6

.

Step-by-Step Solution

1

Determine the semi-major axis length

a

.

a = 5

The vertices are at

(\pm 5, 0)

, which are

5

units from the center

(0, 0)

along the major axis.

2

Determine the distance from the center to each focus

c

.

c = 4

The distance between the two foci is

2c = 8

, so the distance from the center to a focus is

c = 4

.

3

Find the semi-minor axis length

b

.

b = 3

Using the relation

c^2 = a^2 - b^2

for ellipses, we get

4^2 = 5^2 - b^2 \implies b^2 = 9 \implies b = 3

.

4

Calculate the full length of the minor axis.

6

The length of the minor axis is

2b = 2(3) = 6

.

Key Concept

The relationship between the semi-major axis, semi-minor axis, and focal distance of an ellipse.

Estimated Time:1m 30s

Question 2147Question →

In the standard $(x,y)$ coordinate plane, the point $P(3, -4)$ is translated $5$ units to the left and $2$ units up to map onto point $P'$ . What are the coordinates of $P'$ ?

(-2, -6)

(8, -2)

(-2, -2)

(8, -6)

(1, 1)

Show answer & explanation

Answer: $(-2, -2)$

Answer

The coordinates of point

P'

are

(-2, -2)

.

To find the coordinates of the image point

P'

after translation, apply the shifts to the coordinates of the pre-image point

P(3, -4)

. Translating

5

units left subtracts

5

from the x-coordinate:

3 - 5 = -2

. Translating

2

units up adds

2

to the y-coordinate:

-4 + 2 = -2

. Thus, the coordinates of

P'

are

(-2, -2)

.

Step-by-Step Solution

1

Determine the effect of translating

5

units to the left on the x-coordinate.

Subtract

5

from the x-coordinate:

3 - 5 = -2

.

Horizontal translations to the left decrease the x-value.

2

Determine the effect of translating

2

units up on the y-coordinate.

Add

2

to the y-coordinate:

-4 + 2 = -2

.

Vertical translations upward increase the y-value.

3

Combine the new coordinates to find the image point

P'

.

P' = (-2, -2)

The translation maps the original point

P(3, -4)

to the new coordinates

P'(-2, -2)

.

Key Concept

Translating a point

(x, y)

horizontally by

h

units and vertically by

k

units yields the image point

(x + h, y + k)

. Shifts to the left and down correspond to negative values for

h

and

k

, while shifts to the right and up correspond to positive values.

Question 2148Question →

In the standard $(x, y)$ coordinate plane, what is the distance, in coordinate units, between the points $(1, 2)$ and $(4, 6)$ ?

\frac{4}{3}

5

\sqrt{7}

7

25

Show answer & explanation

Answer: $5$

Answer

The distance is

5

coordinate units.

The correct answer is

5

because applying the distance formula

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

to the points

(1, 2)

and

(4, 6)

yields

\sqrt{(4-1)^2 + (6-2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = 5

.

Step-by-Step Solution

1

Identify the coordinates of the two given points.

Let

(x_1, y_1) = (1, 2)

and

(x_2, y_2) = (4, 6)

.

This establishes the coordinate values for the distance formula.

2

Calculate the difference between the

x

-coordinates and the difference between the

y

-coordinates.

x_2 - x_1 = 4 - 1 = 3

and

y_2 - y_1 = 6 - 2 = 4

.

These differences represent the horizontal and vertical side lengths of the right triangle formed by the two points.

3

Apply the distance formula

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

.

d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

.

Squaring the coordinate differences, adding them, and taking the square root yields the straight-line distance.

Key Concept

Using the distance formula to find the straight-line distance between two points in the coordinate plane.

Question 2149Question →

A model rocket is launched vertically upward from a platform. Its height, $h$ , in meters, $t$ seconds after launch is modeled by the equation $h = -4.9t^2 + v_0 t + 2$ , where $v_0$ is the constant initial upward velocity in meters per second. If the rocket never reaches a height of $12$ meters, which of the following inequalities represents all possible values of $v_0$ ?

0 < v_0 < 7

0 < v_0 < 9.8

0 < v_0 < 14

0 < v_0 < 19.6

0 < v_0 < 28

Show answer & explanation

Answer: $0 < v_0 < 14$

Answer

0 < v_0 < 14

The correct answer is the range of values where the velocity is between 0 and 14. Setting the rocket's height equal to 12 meters gives the equation

-4.9t^2 + v_0 t + 2 = 12

. Subtracting 12 from both sides results in

-4.9t^2 + v_0 t - 10 = 0

. Since the rocket never reaches 12 meters, this quadratic equation has no real solutions, meaning its discriminant must be negative. Calculating the discriminant yields

v_0^2 - 4(-4.9)(-10) < 0

, which simplifies to

v_0^2 - 196 < 0

. Solving for a positive velocity gives

0 < v_0 < 14

.

Step-by-Step Solution

1

Set up the equation for the rocket reaching a height of 12 meters.

-4.9t^2 + v_0 t + 2 = 12

To find when the rocket reaches 12 meters, we set the height equation equal to 12.

2

Rearrange the equation into standard quadratic form.

-4.9t^2 + v_0 t - 10 = 0

Subtracting 12 from both sides of the equation sets it to 0, which is the standard form

at^2 + bt + c = 0

.

3

Apply the condition for no real solutions using the discriminant.

\Delta = v_0^2 - 4(-4.9)(-10) < 0

For the rocket to never reach 12 meters, the quadratic equation must have no real solutions. This requires the discriminant

\Delta = b^2 - 4ac

to be strictly negative.

4

Solve the inequality for the positive velocity

v_0

.

v_0^2 - 196 < 0 \implies v_0^2 < 196 \implies v_0 < 14

Multiplying the constants yields

4 \times 4.9 \times 10 = 196

. Taking the square root gives

v_0 < 14

. Since initial upward velocity must be positive,

0 < v_0 < 14

.

Key Concept

Using the discriminant of a quadratic equation to determine the number of real solutions in a physical context.

Question 2150Question →

A triangular framework is being constructed using three metal rods. Two of the rods have lengths of $14$ inches and $25$ inches. The third rod must have an integer length of $x$ inches. What is the number of possible integer values for $x$ ?

14

28

27

29

39

Show answer & explanation

Answer: 27

Answer

27

The correct answer is the value representing 27 possible integers. By the Triangle Inequality Theorem, the third side

x

must satisfy the inequality

25 - 14 < x < 25 + 14

, which simplifies to

11 < x < 39

. Since

x

must be an integer, it can take any value from

12

to

38

, inclusive. Counting these integers yields

38 - 12 + 1 = 27

possible values.

Step-by-Step Solution

1

Apply the Triangle Inequality Theorem to set up the inequality for the third side

x

.

25 - 14 < x < 25 + 14

The Triangle Inequality Theorem states that the length of any side of a triangle must be strictly greater than the difference of the other two sides and strictly less than their sum.

2

Simplify the compound inequality to find the bounds for

x

.

11 < x < 39

Subtracting and adding the side lengths gives the range of possible values for the third side.

3

Identify the set of integers that satisfy the simplified inequality.

x \in \{12, 13, 14, \dots, 37, 38\}

Since

x

must be an integer and the inequalities are strict, the minimum integer value is

12

and the maximum is

38

.

4

Calculate the count of integers in the range from

12

to

38

, inclusive.

38 - 12 + 1 = 27

To find the number of integers in an inclusive range

[a, b]

, use the formula

b - a + 1

.

Key Concept

Triangle Inequality Theorem

Alternative Method

To find the number of integers strictly between two integers

a

and

b

(where

a < b

), you can use the formula

(b - a) - 1

. For the bounds

11 < x < 39

, the calculation is

(39 - 11) - 1 = 28 - 1 = 27

.

Estimated Time:1m 15s

Question 2151Question →

If $w$ is a real number such that $|3 - 2w| \le 9$ , what is the complete range of possible values for $w$ ?

w \ge -3

w \le -3

or

w \ge 6

-3 \le w \le 6

-9 \le w \le 9

-6 \le w \le 3

Show answer & explanation

Answer: $-3 \le w \le 6$

Answer

-3 \le w \le 6

The inequality

|3 - 2w| \le 9

is equivalent to the compound inequality

-9 \le 3 - 2w \le 9

. Subtracting

3

from all parts yields

-12 \le -2w \le 6

. Dividing all parts by

-2

and reversing the inequality signs results in

6 \ge w \ge -3

, which can be rewritten as

-3 \le w \le 6

.

Step-by-Step Solution

1

Set up the compound inequality representing the absolute value inequality.

-9 \le 3 - 2w \le 9

An absolute value inequality of the form

|f(x)| \le c

is equivalent to

-c \le f(x) \le c

.

2

Subtract

3

from all three parts of the inequality to isolate the term containing

w

.

-12 \le -2w \le 6

To solve for

w

, we must isolate the variable term by performing inverse operations on all parts of the inequality.

3

Divide all three parts of the inequality by

-2

and reverse the inequality signs.

6 \ge w \ge -3

, which is equivalent to

-3 \le w \le 6

Dividing an inequality by a negative number requires reversing the direction of the inequality signs to preserve the truth of the statement.

Key Concept

Solving absolute value inequalities of the form

|ax + b| \le c

by converting them into compound inequalities and solving for the variable while reversing the inequality signs when dividing by a negative number.

Estimated Time:1m 30s

Question 2152Question →

In $\triangle XYZ$ , the measure of $\angle X$ is $50^\circ$ , and the measure of $\angle Y$ is twice the measure of $\angle X$ . What is the measure of $\angle Z$ ?

80^\circ

100^\circ

30^\circ

105^\circ

150^\circ

Show answer & explanation

Answer: $30^\circ$

Answer

The correct answer is

30^\circ

.

Since the measure of

\angle X

is

50^\circ

and the measure of

\angle Y

is twice that of

\angle X

, the measure of

\angle Y

is

2 \times 50^\circ = 100^\circ

. Because the sum of the interior angles of any triangle is

180^\circ

, we can find the measure of

\angle Z

by subtracting the sum of the measures of

\angle X

and

\angle Y

from

180^\circ

:

180^\circ - (50^\circ + 100^\circ) = 30^\circ

. Thus, the correct option is the one stating

30^\circ

.

Step-by-Step Solution

1

Calculate the measure of

\angle Y

using the given relationship with

\angle X

.

\angle Y = 2 \times 50^\circ = 100^\circ

The problem states that the measure of

\angle Y

is twice the measure of

\angle X

, which is given as

50^\circ

.

2

Calculate the measure of

\angle Z

using the Triangle Angle Sum Theorem.

\angle Z = 180^\circ - (50^\circ + 100^\circ) = 30^\circ

The sum of the interior angles of any triangle is always

180^\circ

.

Key Concept

Triangle Angle Sum Theorem

Estimated Time:45s

Question 2153Question →

Two vertices of an equilateral triangle are located at the points $(\frac{1}{2}, 2)$ and $(\frac{5}{2}, 2)$ in the standard $(x,y)$ coordinate plane. If the third vertex is located above the given line segment, what are its coordinates?

(\frac{3}{4}, 2 + \sqrt{3})

(\frac{3}{2}, 2 + \sqrt{3})

(\frac{7}{4}, 2 + \sqrt{3})

(\frac{3}{2}, \sqrt{7})

(\frac{3}{2}, 4)

Show answer & explanation

Answer: $(\frac{3}{2}, 2 + \sqrt{3})$

Answer

(\frac{3}{2}, 2 + \sqrt{3})

The midpoint of the base segment is at

(\frac{3}{2}, 2)

and the vertical height of the equilateral triangle is

\sqrt{3}

. Since the vertex lies above the segment, we add the height to the

y

-coordinate of the midpoint, giving the coordinates

(\frac{3}{2}, 2 + \sqrt{3})

.

Step-by-Step Solution

1

Find the midpoint and length of the segment connecting the two given vertices

(\frac{1}{2}, 2)

and

(\frac{5}{2}, 2)

.

Since both points lie on the horizontal line

y = 2

, the distance between them is

\frac{5}{2} - \frac{1}{2} = 2

. The

x

-coordinate of the midpoint is

\frac{\frac{1}{2} + \frac{5}{2}}{2} = \frac{3}{2}

, so the midpoint is at

(\frac{3}{2}, 2)

.

The third vertex of an equilateral triangle lies on the perpendicular bisector of the opposite side, which passes through its midpoint.

2

Calculate the height of the equilateral triangle using the Pythagorean theorem.

The side length of the triangle is

2

. The distance from a vertex to the midpoint of the opposite side is

1

. The height

h

satisfies

1^2 + h^2 = 2^2 \Rightarrow 1 + h^2 = 4 \Rightarrow h = \sqrt{3}

.

The height of an equilateral triangle divides it into two

30^\circ\text{-}60^\circ\text{-}90^\circ

right triangles.

3

Determine the coordinates of the third vertex by applying the height to the midpoint.

Since the base is horizontal, the altitude is vertical. Thus, the third vertex has the same

x

-coordinate as the midpoint,

\frac{3}{2}

. The

y

-coordinate is the

y

-coordinate of the midpoint plus the height, which is

2 + \sqrt{3}

.

The vertex must be located above the segment, so we add the height to the

y

-coordinate of the midpoint.

Key Concept

Distance and Midpoint Formulas

Question 2154Question →

A circle in the standard $(x, y)$ coordinate plane has its center at $(2, -1)$ and passes through the point $(5, 3)$ . What is the diameter, in coordinate units, of the circle?

5

10

14

25

50

Show answer & explanation

Answer: 10

Answer

10

The distance between the center of the circle at

(2, -1)

and the point on the circle

(5, 3)

represents the radius (

r

). Applying the distance formula:

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

. Since the diameter of a circle is twice its radius, the diameter is

2 \times 5 = 10

.

Step-by-Step Solution

1

Use the distance formula to find the radius of the circle, which is the distance between the center

(2, -1)

and the point

(5, 3)

.

r = \sqrt{(5 - 2)^2 + (3 - (-1))^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

The radius of a circle is the distance between its center and any point on its boundary.

2

Multiply the radius by 2 to find the diameter of the circle.

d = 2r = 2(5) = 10

The diameter of a circle is always twice the length of its radius.

Key Concept

Calculating the radius of a circle using the distance formula and doubling it to find the diameter.

Question 2155Question →

Given the functions $f(x) = \sqrt{3x + 4}$ and $g(x) = x^2 - 5$ , what is the value of $g(f(7))$ ?

Show answer & explanation

Answer: 20

Answer

20

Evaluating the inner function first gives

f(7) = \sqrt{3(7) + 4} = 5

. Substituting this value into the outer function yields

g(5) = 5^2 - 5 = 20

.

Step-by-Step Solution

1

Evaluate the inner function

f(x)

at

x = 7

.

f(7) = 5

To evaluate a composite function of the form

g(f(x))

, we first calculate the value of the inner function

f(x)

at the given input.

2

Substitute the output of the inner function as the input for the outer function

g(x)

.

g(5) = 20

Since

f(7) = 5

, evaluating

g(f(7))

is equivalent to evaluating

g(5)

.

Key Concept

Function composition involves evaluating an inner function and then using that result as the input for an outer function.

Question 2156Question →

If the expression $\frac{(a^{-3} b^2)^{-2}}{a^4 (b^{-1})^3}$ is simplified to the form $a^x b^y$ , where $a$ and $b$ are positive real numbers, what is the value of the product $xy$ ?

-14

-2

2

10

18

Show answer & explanation

Answer: $-2$

Answer

-2

The correct answer is

-2

. Simplifying the expression step-by-step: first, apply the power of a product and power of a power rules to the numerator to get

a^6 b^{-4}

. Next, simplify the denominator to get

a^4 b^{-3}

. Then, apply the quotient rule to simplify the fraction to

a^{6-4} b^{-4-(-3)} = a^2 b^{-1}

. This gives

x=2

and

y=-1

. The product

xy

is

2 \cdot (-1) = -2

.

Step-by-Step Solution

1

Simplify the numerator using the power of a power rule

(x^p)^q = x^{pq}

and the power of a product rule

(xy)^p = x^p y^p

.

(a^{-3} b^2)^{-2} = a^{(-3)(-2)} b^{(2)(-2)} = a^6 b^{-4}

To raise a product to a power, raise each factor to that power by multiplying their exponents.

2

Simplify the denominator using the power of a power rule.

a^4 (b^{-1})^3 = a^4 b^{(-1)(3)} = a^4 b^{-3}

Multiply the exponents of

b

to simplify the term

(b^{-1})^3

.

3

Divide the simplified numerator by the simplified denominator using the quotient rule

\frac{x^p}{x^q} = x^{p-q}

.

\frac{a^6 b^{-4}}{a^4 b^{-3}} = a^{6-4} b^{-4 - (-3)} = a^2 b^{-1}

Subtract the exponent of the denominator from the exponent of the numerator for each base.

4

Identify the values of

x

and

y

from the simplified form

a^x b^y

and calculate the product

xy

.

x = 2

,

y = -1

, so

xy = 2 \cdot (-1) = -2

Match the simplified exponents with the variables

x

and

y

, then multiply them to find the final value.

Key Concept

Properties of Exponents in Algebraic Expressions

Alternative Method

Alternatively, you can rewrite the negative exponents as positive exponents by moving them across the fraction bar first. The numerator

(a^{-3} b^2)^{-2}

becomes

\frac{1}{(a^{-3} b^2)^2} = \frac{1}{a^{-6} b^4} = \frac{a^6}{b^4}

. The denominator

a^4 (b^{-1})^3

becomes

\frac{a^4}{b^3}

. Dividing these gives

\frac{a^6}{b^4} \cdot \frac{b^3}{a^4} = \frac{a^2}{b^1} = a^2 b^{-1}

. This yields

x = 2

and

y = -1

, leading to the product

xy = -2

.

Estimated Time:1m 30s

Question 2157Question →

An ellipse in the standard $(x, y)$ coordinate plane is defined by the equation $25x^2 + 9y^2 - 100x + 54y - 44 = 0$ . Which of the following points is a focus of this ellipse?

(-2, 7)

(2, -1)

(2, 1)

(6, -3)

(2, 19)

Show answer & explanation

Answer: $(2, 1)$

Answer

(2, 1)

The correct answer is the point

(2, 1)

because rewriting the general equation of the ellipse in standard form gives

\frac{(x-2)^2}{9} + \frac{(y+3)^2}{25} = 1

. The center is

(2, -3)

and the major axis is vertical with a focal distance of

c = \sqrt{25 - 9} = 4

. Adding this distance to the

y

-coordinate of the center yields the focus

(2, -3 + 4) = (2, 1)

.

Step-by-Step Solution

1

Group the

x

and

y

terms and factor out the coefficients.

25(x^2 - 4x) + 9(y^2 + 6y) = 44

This prepares the algebraic equation for completing the square.

2

Complete the square for both the

x

and

y

groups.

25(x-2)^2 + 9(y+3)^2 = 225

Adding

25 \times 4 = 100

and

9 \times 9 = 81

to both sides maintains equality while converting the quadratic expressions into perfect square trinomials.

3

Divide both sides by

225

to write the equation in standard form.

\frac{(x-2)^2}{9} + \frac{(y+3)^2}{25} = 1

The standard form of an ellipse equation is

\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1

(for a vertical major axis), which directly reveals the center

(h, k)

and the axis parameters.

4

Identify the center, axis lengths, and calculate the focal distance

c

.

Center is

(2, -3)

,

a^2 = 25

,

b^2 = 9

. Thus,

c = \sqrt{25 - 9} = 4

.

The focal distance

c

for an ellipse is determined by the relation

c = \sqrt{a^2 - b^2}

.

5

Determine the coordinates of the foci.

Foci are

(2, -3 \pm 4)

, which simplifies to

(2, 1)

and

(2, -7)

.

Since

a^2 = 25

is under the

y

-term, the ellipse is vertically oriented, meaning the foci lie on the vertical line passing through the center.

Key Concept

Rewriting the general equation of an ellipse to standard form and finding its foci.

Estimated Time:2m 30s

Question 2158Question →

On a coordinate plane, triangle $PQR$ has vertices $P(2, 3)$ , $Q(5, 3)$ , and $R(2, 7)$ . The triangle undergoes a sequence of three transformations:

1. A dilation centered at the point $C(1, 1)$ with a scale factor of $2$ .
2. A reflection across the line $y = -x$ .
3. A translation of $3$ units to the left and $4$ units up.

What are the coordinates of the final image of vertex $Q$ after this sequence of transformations?

(-9, -6)

(2, 13)

(-2, -13)

(-8, -5)

(-7, -4)

Show answer & explanation

Answer: $(-8, -5)$

Answer

The final image of vertex

Q

is located at the coordinates

(-8, -5)

.

Applying the transformations sequentially yields the correct result. First, the vector from the dilation center at

(1, 1)

to

(5, 3)

is

(4, 2)

. Scaling this vector by

2

gives

(8, 4)

, which when added back to

(1, 1)

places the intermediate image at

(9, 5)

. Second, reflecting

(9, 5)

across the line

y = -x

yields

(-5, -9)

. Third, translating this point

3

units left and

4

units up results in

(-5 - 3, -9 + 4) = (-8, -5)

.

Step-by-Step Solution

1

Find the vector from the center of dilation

C(1, 1)

to the point

Q(5, 3)

.

Vector

\vec{CQ} = (5 - 1, 3 - 1) = (4, 2)

.

To perform a dilation centered at a point other than the origin, we must first find the displacement of the target point relative to that center.

2

Multiply the vector by the scale factor of

2

and add it back to the coordinates of the center

C(1, 1)

.

First intermediate point

Q' = (1, 1) + 2(4, 2) = (1 + 8, 1 + 4) = (9, 5)

.

This scales the distance from the center of dilation by

2

and finds the coordinate of the image point

Q'

.

3

Apply the reflection rule for the line

y = -x

to the point

Q'(9, 5)

.

Second intermediate point

Q'' = (-5, -9)

.

Reflecting a coordinate

(x, y)

across the line

y = -x

swaps and negates both coordinates, mapping

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

.

4

Translate the point

Q''(-5, -9)

by subtracting

3

from the x-coordinate and adding

4

to the y-coordinate.

Final point

Q''' = (-5 - 3, -9 + 4) = (-8, -5)

.

Translating left reduces the x-value, and translating up increases the y-value.

Key Concept

Composite transformations in the coordinate plane including non-origin dilations, reflections, and translations.

Question 2159Question →

The measures of the three interior angles of a triangle are in the ratio $2:3:5$ . What is the measure, in degrees, of the largest angle of the triangle?

18^\circ

36^\circ

50^\circ

54^\circ

90^\circ

Show answer & explanation

Answer: $90^\circ$

Answer

90 degrees

The sum of the measures of the interior angles of a triangle is

180^\circ

. Given the ratio

2:3:5

, the angles can be expressed as

2x

,

3x

, and

5x

. Adding these yields

2x + 3x + 5x = 180

, which simplifies to

10x = 180

. Solving for

x

gives

x = 18

. The largest angle is represented by

5x

. Substituting

18

for

x

gives

5 \times 18 = 90

. Therefore, the measure of the largest angle is

90^\circ

.

Step-by-Step Solution

1

Set up an equation representing the sum of the angles in a triangle.

2x + 3x + 5x = 180

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

180^\circ

. We can represent the angles as

2x

,

3x

, and

5x

using the given ratio.

2

Solve for the scale factor

x

.

10x = 180 \Rightarrow x = 18

Combine like terms to find the total number of parts, then divide

180

by

10

to find the value of one part.

3

Calculate the measure of the largest angle.

5 \times 18 = 90

The largest angle corresponds to the largest term in the ratio, which is

5x

. Multiplying the scale factor

18

by

5

gives the measure of the largest angle.

Key Concept

The interior angles of a triangle sum to

180^\circ

. Ratios can be solved by defining a common multiplier for each part and setting their sum equal to the total.

Estimated Time:45s

Question 2160Question →

A local courier service offers two delivery options inside the city limits: Standard and Express. A Standard delivery costs a flat fee of $15.00 plus$ 0.75 per mile. An Express delivery costs a flat fee of $25.00 plus$ 0.50 per mile. If the cost of a Standard delivery is exactly $5.00 less than the cost of an Express delivery, what is the total cost of the Express delivery?

$30.00

$35.00

$45.00

$55.00

$60.00

Show answer & explanation

Answer: $35.00

Answer

The total cost of the Express delivery is $35.00.

The correct answer is

35.00. First, represent the costs as algebraic functions of the distance

m

:

15 + 0.75m

for Standard and

25 + 0.50m

for Express. Set up the equation reflecting that the Standard cost is

5.00 less than the Express cost:

15 + 0.75m = (25 + 0.50m) - 5

. Simplifying this equation gives

15 + 0.75m = 20 + 0.50m

. Subtracting

0.50m

and

15

from both sides results in

0.25m = 5

, which yields

m = 20

. Finally, plug

m = 20

into the Express cost function:

25 + 0.50(20) = 35.00

dollars.

Step-by-Step Solution

1

Define the variables and write the cost functions for both options.

Let

m

represent the number of miles. The Standard delivery cost is

S = 15 + 0.75m

, and the Express delivery cost is

E = 25 + 0.50m

.

Translating the verbal descriptions into algebraic expressions establishes the basis for solving the problem.

2

Set up the equation using the given relationship that Standard is 5.00 dollars less than Express.

15 + 0.75m = (25 + 0.50m) - 5

The phrase 'Standard cost is 5.00 dollars less than Express cost' translates to

S = E - 5

.

3

Solve the equation for

m

.

15 + 0.75m = 20 + 0.50m \implies 0.25m = 5 \implies m = 20

miles.

Solving for

m

determines the delivery distance that satisfies the cost relationship.

4

Calculate the total cost of the Express delivery by substituting the value of

m

into the Express cost function.

E = 25 + 0.50(20) = 25 + 10 = 35

dollars.

The question specifically asks for the total cost of the Express delivery, not the distance or the Standard delivery cost.

Key Concept

Translating verbal descriptions of costs and differences into algebraic equations and solving them.

Alternative Method

You can test the choices. For example, if the Express cost is

35.00, then the distance

m

is found by solving

25 + 0.50m = 35 \implies 0.50m = 10 \implies m = 20

. At

20

miles, the Standard cost is

15 + 0.75(20) = 30

. Since

30

is indeed exactly

5

less than

35$, this choice is correct.

Estimated Time:1m 30s

Answer

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