All practice questions

Question 261Question →

In acute triangle $ABC$ , the measure of $\angle A$ is $74^\circ$ . The altitude from vertex $B$ to side $AC$ and the altitude from vertex $C$ to side $AB$ intersect at point $H$ inside the triangle. What is the measure, in degrees, of $\angle BHC$ ?

Show answer & explanation

Answer: 106

Answer

The measure of

\angle BHC

is 106 degrees.

In quadrilateral

AEHD

, the sum of the angles is

360^\circ

. Since the angles at

E

and

D

are

90^\circ

because they are formed by altitudes, the sum of

\angle A

and

\angle EHD

must be

180^\circ

. Thus,

\angle EHD = 180^\circ - 74^\circ = 106^\circ

. Since

\angle BHC

and

\angle EHD

are vertical angles,

\angle BHC = 106^\circ

.

Step-by-Step Solution

1

Define the intersection points of the altitudes with the opposite sides.

Let the altitude from vertex

B

intersect side

AC

at point

D

, and let the altitude from vertex

C

intersect side

AB

at point

E

. Therefore,

\angle ADH = 90^\circ

and

\angle AEH = 90^\circ

.

Altitudes by definition are perpendicular to the sides they intersect, forming

90^\circ

angles.

2

Analyze the sum of interior angles in the quadrilateral

AEHD

.

The sum of the interior angles of a quadrilateral is

360^\circ

, so

\angle A + \angle AEH + \angle EHD + \angle ADH = 360^\circ

.

Any quadrilateral can be split into two triangles, making the sum of its interior angles

360^\circ

.

3

Substitute the known angle measures to find the measure of

\angle EHD

.

74^\circ + 90^\circ + \angle EHD + 90^\circ = 360^\circ \implies 254^\circ + \angle EHD = 360^\circ \implies \angle EHD = 106^\circ

.

Solving the linear equation for the unknown angle

\angle EHD

.

4

Relate

\angle EHD

to the target angle

\angle BHC

.

Since line segments

BD

and

CE

intersect at

H

, the angles

\angle BHC

and

\angle EHD

are vertical angles, so

\angle BHC = \angle EHD = 106^\circ

.

Vertical angles are equal in measure.

Key Concept

The sum of angles in quadrilaterals, the definition of altitudes, and vertical angles within triangles.

Estimated Time:2m 0s

Question 262Question →

A circle in the first quadrant of the standard $(x,y)$ coordinate plane is tangent to the $x$ -axis and is also tangent to the line $y = \frac{4}{3}x$ . If the center of the circle lies on the line with equation $y = 3x - 10$ , what is the radius of the circle?

Show answer & explanation

Answer: 2

Answer

The radius of the circle is 2.

The correct answer is obtained by determining that the center of a circle tangent to the

x

-axis in the first quadrant has the form

(h, R)

where

R

is the radius. Using the distance from this point to the line

4x - 3y = 0

gives

h = 2R

to ensure the center remains in the first quadrant. Substituting

(2R, R)

into the line

y = 3x - 10

yields

R = 3(2R) - 10

, which solves to

R = 2

.

Step-by-Step Solution

1

Determine the relation between the circle's center coordinates and its radius.

The center is

(h, R)

where

R

is the radius.

Because the circle is tangent to the

x

-axis and lies in the first quadrant, the y-coordinate of its center must equal its radius.

2

Apply the point-to-line distance formula from the center to the line

y = \frac{4}{3}x

.

The relation is

|4h - 3R| = 5R

.

The distance from the center

(h, R)

to the line

4x - 3y = 0

must equal the radius

R

.

3

Solve the absolute value equation for

h

in terms of

R

.

Since

h > 0

, we find

h = 2R

.

The positive case

4h - 3R = 5R

gives

h = 2R

, whereas the negative case

4h - 3R = -5R

gives a negative

h

which violates the first-quadrant condition.

4

Substitute the center coordinates

(2R, R)

into the line

y = 3x - 10

and solve for

R

.

R = 2

.

The center lies on this line, so its coordinates must satisfy the line's equation.

Key Concept

Equations and Graphs of Circles

Question 263Question →

A circle in the standard $(x, y)$ coordinate plane is defined by the equation $(x - 1)^2 + (y - 2)^2 = 10$ , and a line is defined by the equation $y = 3x - 1$ . The line intersects the circle at two points, $(x_1, y_1)$ and $(x_2, y_2)$ . What is the sum of the $y$ -coordinates of these two points of intersection?

Show answer & explanation

Answer: 4

Answer

The sum of the

y

-coordinates of the intersection points is 4.

Substituting

y = 3x - 1

into the circle's equation gives

(x-1)^2 + (3x-3)^2 = 10

. Factoring out 3 from the second term yields

(x-1)^2 + 9(x-1)^2 = 10

, which simplifies to

10(x-1)^2 = 10

, or

(x-1)^2 = 1

. Solving for

x

gives

x = 0

and

x = 2

. Substituting these values into the linear equation gives the corresponding

y

-coordinates:

y = -1

when

x = 0

, and

y = 5

when

x = 2

. The sum of these

y

-coordinates is

5 + (-1) = 4

.

Step-by-Step Solution

1

Substitute the expression for

y

from the linear equation into the circle's equation.

(x-1)^2 + (3x-3)^2 = 10

To reduce the system of two equations to a single equation in terms of

x

.

2

Factor out 3 from the term

(3x-3)

and simplify the equation.

10(x-1)^2 = 10

, which simplifies to

(x-1)^2 = 1

To solve for the

x

-coordinates of the intersection points.

3

Solve the simplified quadratic equation for

x

.

x_1 = 2

and

x_2 = 0

To find the

x

-coordinates of the two intersection points.

4

Substitute the

x

-values back into the linear equation

y = 3x - 1

to find the corresponding

y

-coordinates.

y_1 = 3(2) - 1 = 5

and

y_2 = 3(0) - 1 = -1

To determine the

y

-coordinates of the intersection points

(2, 5)

and

(0, -1)

.

5

Add the two

y

-coordinates together.

5 + (-1) = 4

To find the sum of the

y

-coordinates as requested by the question.

Key Concept

Solving systems of linear and circular equations by substitution

Alternative Method

Instead of finding the individual coordinates, substitute

y_1 = 3x_1 - 1

and

y_2 = 3x_2 - 1

to write the sum as

y_1 + y_2 = 3(x_1 + x_2) - 2

. Expanding the substitution equation gives

10x^2 - 20x = 0

. By Vieta's formulas, the sum of the roots

x_1 + x_2 = -(-20)/10 = 2

. Substituting this back gives

y_1 + y_2 = 3(2) - 2 = 4

.

Estimated Time:1m 30s

Question 264Question →

In the standard $(x,y)$ coordinate plane, a rectangle has vertices at $(-2, -3)$ , $(4, -3)$ , $(4, 2)$ , and $(-2, 2)$ . What is the area, in square units, of this rectangle?

Show answer & explanation

Answer: 30

Answer

The area of the rectangle is 30 square units.

The area of a rectangle is the product of its length and width. By finding the difference between the x-coordinates of the horizontal vertices (

4 - (-2) = 6

) and the difference between the y-coordinates of the vertical vertices (

2 - (-3) = 5

), we find the dimensions to be 6 and 5. Multiplying these gives

6 \times 5 = 30

.

Step-by-Step Solution

1

Determine the length of the horizontal sides of the rectangle.

The horizontal sides have a length of 6 units.

The horizontal sides connect vertices with the same y-coordinates, such as

(-2, -3)

and

(4, -3)

. The distance is the difference in their x-coordinates:

4 - (-2) = 6

.

2

Determine the length of the vertical sides of the rectangle.

The vertical sides have a length of 5 units.

The vertical sides connect vertices with the same x-coordinates, such as

(4, -3)

and

(4, 2)

. The distance is the difference in their y-coordinates:

2 - (-3) = 5

.

3

Calculate the area of the rectangle.

The area of the rectangle is 30 square units.

The area of a rectangle is found by multiplying its length by its width:

\text{Area} = 6 \times 5 = 30

.

Key Concept

Finding the area of a rectangle on the coordinate plane by calculating the lengths of its horizontal and vertical sides.

Question 265Question →

The length $L$ , in centimeters, of a copper rod at a temperature of $T$ degrees Celsius can be modeled by the linear equation $L = 150.04 + 0.012T$ . If the length of the rod is measured to be $150.40$ centimeters, what is its temperature in degrees Celsius?

Show answer & explanation

Answer: 30

Answer

The temperature of the rod is 30 degrees Celsius.

The correct answer is 30. By substituting the given length of

150.40

centimeters into the equation for

L

, we get

150.40 = 150.04 + 0.012T

. Subtracting

150.04

from both sides gives

0.36 = 0.012T

. Finally, dividing both sides by

0.012

yields the temperature

T = 30

degrees Celsius.

Step-by-Step Solution

1

Substitute the measured length

L = 150.40

into the equation.

150.40 = 150.04 + 0.012T

To set up the equation with the given value for length.

2

Subtract 150.04 from both sides of the equation.

0.36 = 0.012T

To isolate the variable term containing

T

.

3

Divide both sides of the equation by 0.012.

T = 30

To find the temperature

T

.

Key Concept

Solving a multi-step linear equation involving decimals

Question 266Question →

A straight hiking trail ascends a hill at a constant incline. On a coordinate grid where the units represent meters, the path of the trail is a straight line starting at the coordinate point $(10, 150)$ and ending at the coordinate point $(90, 190)$ . What is the slope of this trail?

Show answer & explanation

Answer: 0.5

Answer

The correct answer is

0.5

(or

\frac{1}{2}

).

The slope of a line is defined as the change in the

y

-coordinates divided by the change in the

x

-coordinates:

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

. By substituting the coordinates of the start of the trail

(10, 150)

and the end of the trail

(90, 190)

, we calculate

\frac{190 - 150}{90 - 10} = \frac{40}{80} = 0.5

.

Step-by-Step Solution

1

Identify the coordinates from the problem statement.

(x_1, y_1) = (10, 150)

and

(x_2, y_2) = (90, 190)

To calculate the slope between two points, we first need to define their coordinates.

2

Recall the formula for the slope of a line passing through two points.

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

Slope represents the vertical change (rise) divided by the horizontal change (run).

3

Substitute the coordinate values into the slope formula and simplify.

m = \frac{190 - 150}{90 - 10} = \frac{40}{80} = 0.5

Plugging the values into the formula yields the constant rate of change (slope) of the trail.

Key Concept

Slope of a Line

Question 267Question →

A triangle in the standard $(x, y)$ coordinate plane has vertices at $A(1, 1)$ , $B(10, 16)$ , and $C(5, 9)$ . A line passes through the point $P(4, 7)$ on the side $AC$ and intersects the side $AB$ at a point $Q$ . If this line divides the triangle into two regions of equal area, what is the length of the line segment $PQ$ ?

Show answer & explanation

Answer: 5

Answer

5

The total area of the triangle is 6. To divide the triangle into two equal-area regions, each region must have an area of 3. Since point

P(4,7)

lies

\frac{3}{4}

of the way from

A

to

C

, the area of the sub-triangle

\triangle PBC

is only

\frac{1}{4} \times 6 = 1.5

. Thus, the dividing line must intersect side

AB

at a point

Q

to form

\triangle APQ

with an area of 3. Using the area ratio formula,

\text{Area}(\triangle APQ) = \frac{AP}{AC} \times \frac{AQ}{AB} \times \text{Area}(\triangle ABC) \implies 3 = \frac{3}{4} \times \frac{AQ}{AB} \times 6

, which gives

\frac{AQ}{AB} = \frac{2}{3}

. Using the section formula, the coordinates of

Q

are

A + \frac{2}{3}(B - A) = (1, 1) + \frac{2}{3}(9, 15) = (7, 11)

. Finally, the length of

PQ

is

\sqrt{(7-4)^2 + (11-7)^2} = \sqrt{3^2 + 4^2} = 5

.

Step-by-Step Solution

1

Calculate the area of the entire triangle

ABC

using the Shoelace formula.

Area of

\triangle ABC = 6

.

Establishing the total area of the triangle is necessary to determine the target area of 3 for each of the two equal-area regions.

2

Determine which side of the triangle the dividing line intersects by comparing the area of

\triangle PBC

to the target area of 3.

The line must intersect side

AB

at a point

Q

.

Since

P(4,7)

lies

\frac{3}{4}

of the way along

AC

, the base

PC

is

\frac{1}{4}

of

AC

. The area of

\triangle PBC

is

\frac{1}{4} \times 6 = 1.5

. Since this is less than 3, the dividing line cannot intersect side

BC

and must intersect side

AB

instead.

3

Set up the area ratio equation for

\triangle APQ

to find the ratio

\frac{AQ}{AB}

.

\frac{AQ}{AB} = \frac{2}{3}

.

The area of

\triangle APQ

is given by

\text{Area}(\triangle APQ) = \frac{AP}{AC} \times \frac{AQ}{AB} \times \text{Area}(\triangle ABC) \implies 3 = \frac{3}{4} \times \frac{AQ}{AB} \times 6 \implies \frac{AQ}{AB} = \frac{2}{3}

.

4

Find the coordinates of

Q

using the section formula along segment

AB

from

A(1,1)

to

B(10,16)

.

Q(7, 11)

.

Applying

Q = A + \frac{2}{3}(B - A) = (1, 1) + \frac{2}{3}(9, 15) = (7, 11)

yields the exact coordinates of

Q

.

5

Calculate the length of segment

PQ

using the distance formula between

P(4,7)

and

Q(7,11)

.

PQ = 5

.

The question asks for the length of the segment

PQ

, which is the distance between these two points.

Key Concept

Using coordinate geometry formulas and area ratios to solve problems involving geometric figures on the coordinate plane.

Question 268Question →

In the standard $(x, y)$ coordinate plane, the point $M$ is the midpoint of the line segment with endpoints $P(-4, -2)$ and $Q(4, 2)$ . A second line segment is drawn from $M$ to a point $R(x, y)$ such that the length of the segment $MR$ is $8$ . If the midpoint of the segment $MR$ lies on the line $3x - 4y + 12 = 0$ , what is the smallest possible value of $x$ ?

Show answer & explanation

Answer: -8

Answer

The smallest possible value of

x

is

-8

.

By finding the midpoint

M(0,0)

of

PQ

and writing the midpoint of

MR

as

\left(\frac{x}{2}, \frac{y}{2}\right)

, we substitute this into the line equation to find

y = \frac{3}{4}x + 6

. We then substitute this into the distance formula equation

x^2 + y^2 = 64

to get the quadratic equation

25x^2 + 144x - 448 = 0

, which yields the solutions

x = -8

and

x = 2.24

. The smallest possible value is

-8

.

Step-by-Step Solution

1

Calculate the coordinates of the midpoint

M

of segment

PQ

.

M = (0, 0)

The midpoint formula states that the midpoint of a segment with endpoints

(x_1, y_1)

and

(x_2, y_2)

is

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

.

2

Express the midpoint

N

of segment

MR

in terms of

R(x, y)

.

N = \left(\frac{x}{2}, \frac{y}{2}\right)

The midpoint of

M(0, 0)

and

R(x, y)

is found by averaging their coordinates.

3

Substitute the coordinates of

N

into the equation of the line

3x - 4y + 12 = 0

.

y = \frac{3}{4}x + 6

Since the midpoint

N

lies on the line, its coordinates must satisfy the line's equation, which gives a linear relationship between

x

and

y

.

4

Set up the equation for the distance

MR = 8

.

x^2 + y^2 = 64

The distance formula between

M(0, 0)

and

R(x, y)

is

d = \sqrt{x^2 + y^2}

, and squaring both sides gives

x^2 + y^2 = d^2

.

5

Substitute

y = \frac{3}{4}x + 6

into the distance equation and solve the quadratic equation.

x = -8

and

x = 2.24

Substituting the linear relationship into the quadratic circle equation gives a single quadratic equation in terms of

x

, which can be solved using the quadratic formula.

6

Determine the smallest value of

x

from the two possible solutions.

-8

Comparing the two solutions,

-8

is smaller than

2.24

.

Key Concept

Distance and Midpoint Formulas

Alternative Method

Instead of solving algebraically, one can scale the line

3x - 4y + 12 = 0

by a factor of 2 centered at the origin

M(0,0)

to directly obtain the line equation on which

R

lies:

3x - 4y + 24 = 0

. Then, find the intersection of this line with the circle

x^2 + y^2 = 64

using substitution.

Estimated Time:2m 30s

Question 269Question →

A line has a slope of $-3$ . If the sum of its $x$ -intercept and its $y$ -intercept is $12$ , what is the $y$ -intercept of the line?

Show answer & explanation

Answer: 9

Answer

The y-intercept of the line is 9.

The equation of the line is

y = -3x + b

, where

b

is the

y

-intercept. The

x

-intercept is found by setting

y=0

, which yields

x = \frac{b}{3}

. Setting their sum to

12

gives

\frac{b}{3} + b = 12

, which simplifies to

4b = 36

and results in

b = 9

.

Step-by-Step Solution

1

Express the equation of the line using the slope-intercept form.

The equation of the line is

y = -3x + b

, where

b

is the

y

-intercept.

We are given that the slope of the line is

-3

.

2

Find the

x

-intercept of the line in terms of

b

.

Setting

y = 0

gives

0 = -3x + b \implies 3x = b \implies x = \frac{b}{3}

. Thus, the

x

-intercept is

\frac{b}{3}

.

The

x

-intercept of a line is the

x

-coordinate where the line crosses the

x

-axis, which occurs when

y = 0

.

3

Set up an equation using the given sum of the intercepts.

\frac{b}{3} + b = 12

The problem states that the sum of the

x

-intercept and the

y

-intercept is

12

.

4

Solve the linear equation for the

y

-intercept

b

.

Multiply the entire equation by

3

to clear the denominator:

b + 3b = 36 \implies 4b = 36 \implies b = 9

.

Solving this equation yields the value of the

y

-intercept.

Key Concept

Linear equations, slope-intercept form, and finding intercepts.

Alternative Method

Alternatively, you can write the equation of the line in intercept form:

\frac{x}{a} + \frac{y}{b} = 1

, where

a

is the

x

-intercept and

b

is the

y

-intercept. The slope of this line is

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

, which means

b = 3a

. Since the sum of the intercepts is

12

, we write

a + b = 12

. Substituting

b = 3a

into this sum gives

a + 3a = 12 \implies 4a = 12 \implies a = 3

. Therefore, the

y

-intercept

b

is

3(3) = 9

.

Estimated Time:1m 30s

Question 270Question →

In the standard $(x, y)$ coordinate plane, line $p$ is perpendicular to the line represented by the equation $y = -0.2x + 9$ . What is the slope of line $p$ ?

Show answer & explanation

Answer: 5

Answer

The slope of line

p

is

5

.

The given line is in slope-intercept form

y = mx + b

with a slope of

-0.2

, which can be written as

-\frac{1}{5}

. The slope of any line perpendicular to this line is the negative reciprocal of

-\frac{1}{5}

, which is

5

.

Step-by-Step Solution

1

Identify the slope of the given line from its equation.

The slope of the line

y = -0.2x + 9

is

-0.2

(or

-\frac{1}{5}

).

The equation is in slope-intercept form

y = mx + b

, where the coefficient of

x

represents the slope

m

.

2

Calculate the slope of the perpendicular line.

The negative reciprocal of

-\frac{1}{5}

is

5

.

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

m_1 \cdot m_2 = -1

).

Key Concept

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.

Estimated Time:45s

Question 271Question →

If $x$ is a real number such that $\log_4(x - 3) = \frac{1}{2} + \log_4(2)$ , what is the value of $x$ ?

Show answer & explanation

Answer: 7

Answer

The value of

x

is 7.

By moving the logarithmic terms to the same side and applying the quotient rule, the equation simplifies to

\log_4\left(\frac{x-3}{2}\right) = \frac{1}{2}

. Converting this to exponential form yields

\frac{x-3}{2} = 4^{1/2}

. Since

4^{1/2} = 2

, we have

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

, which gives

x - 3 = 4

, or

x = 7

. Substituting

x = 7

back into the original equation shows it is a valid solution.

Step-by-Step Solution

1

Group logarithmic terms on one side of the equation.

\log_4(x - 3) - \log_4(2) = \frac{1}{2}

Grouping the logarithms allows them to be combined using logarithmic properties.

2

Apply the quotient property of logarithms.

\log_4\left(\frac{x - 3}{2}\right) = \frac{1}{2}

The quotient property states that

\log_b(a) - \log_b(c) = \log_b(\frac{a}{c})

.

3

Convert the equation from logarithmic form to exponential form.

\frac{x - 3}{2} = 4^{1/2}

A logarithmic equation

\log_b(y) = z

is equivalent to

b^z = y

.

4

Evaluate the fractional exponent and solve the linear equation for

x

.

x = 7

Since

4^{1/2} = 2

, the equation becomes

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

. Multiplying both sides by 2 gives

x - 3 = 4

, so adding 3 to both sides yields

x = 7

.

Key Concept

Solving logarithmic equations using properties of logarithms

Estimated Time:1m 30s

Question 272Question →

In $\triangle PQR$ , the measure of interior angle $\angle P$ is $40^\circ$ . If the measure of $\angle Q$ is twice the measure of $\angle P$ , what is the measure, in degrees, of the interior angle $\angle R$ ?

Show answer & explanation

Answer: 60

Answer

The measure of interior angle

\angle R

is 60 degrees.

By definition, the interior angles of any triangle sum to 180 degrees. Given that angle P is 40 degrees, and angle Q is twice angle P (80 degrees), the sum of angles P and Q is 120 degrees. Subtracting this from 180 degrees gives a remaining measure of 60 degrees for angle R.

Step-by-Step Solution

1

Calculate the measure of angle Q

m\angle Q = 80^\circ

The problem states that the measure of angle Q is twice the measure of angle P, which is 40 degrees.

2

Apply the Triangle Angle Sum Theorem

m\angle P + m\angle Q + m\angle R = 180^\circ

The sum of the measures of the interior angles of any triangle is always 180 degrees.

3

Solve for the measure of angle R

m\angle R = 60^\circ

Substitute the values of angles P and Q into the equation: 40 + 80 + m\angle R = 180, which simplifies to 120 + m\angle R = 180, and solving for m\angle R gives 60.

Key Concept

The sum of the measures of the interior angles of a triangle is always 180 degrees.

Question 273Question →

A landscaping company charges a one-time equipment mobilization fee of $45$ plus an hourly labor rate of $32.50$ per worker. A homeowner hires a crew of $3$ workers to clear their yard. If the total bill for the job is $435$ dollars, for how many hours did the crew work?

Show answer & explanation

Answer: 4

Answer

4

The total cost of

435

dollars is the sum of the one-time

45

dollar mobilization fee and the hourly labor cost of

32.50

dollars per worker for

3

workers over

h

hours. This translates to the equation

45 + 3(32.50)h = 435

, which simplifies to

45 + 97.50h = 435

. Subtracting

45

from both sides gives

97.50h = 390

. Dividing by

97.50

yields

h = 4

hours.

Step-by-Step Solution

1

Set up the algebraic expression representing the total cost based on the number of hours worked,

h

. The total cost consists of a fixed mobilization fee of

45

dollars and a variable labor cost of

32.50

dollars per worker per hour.

45 + 3(32.50)h

To represent the relation between the hours worked and the total charge.

2

Equate the expression for the total cost to the actual total bill of

435

dollars and simplify the labor rate coefficient.

45 + 97.50h = 435

To form a solvable linear equation in terms of the unknown number of hours,

h

.

3

Subtract the fixed mobilization fee of

45

from both sides of the equation.

97.50h = 390

To isolate the term containing the variable

h

.

4

Divide both sides of the equation by

97.50

to solve for

h

.

h = 4

To determine the number of hours the crew worked.

Key Concept

Formulating and solving linear equations from real-world contexts

Estimated Time:1m 30s

Question 274Question →

In the standard $(x, y)$ coordinate plane, a line $L_1$ is perpendicular to the line that contains the points $(3, 5)$ and $(-1, 8)$ . If $L_1$ is also parallel to the line defined by the equation $ax + 6y = 15$ , what is the value of the constant $a$ ?

Show answer & explanation

Answer: -8

Answer

The value of the constant

a

is

-8

.

The slope of the line containing

(3, 5)

and

(-1, 8)

is

-\frac{3}{4}

. The slope of a line perpendicular to it is the negative reciprocal, which is

\frac{4}{3}

. Because line

L_1

is parallel to the line

ax + 6y = 15

, they must have equal slopes. The slope of

ax + 6y = 15

is

-\frac{a}{6}

. Setting the two slopes equal gives

-\frac{a}{6} = \frac{4}{3}

, which yields

a = -8

.

Step-by-Step Solution

1

Calculate the slope of the line containing the points

(3, 5)

and

(-1, 8)

.

The slope is

-\frac{3}{4}

.

Using the slope formula

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

for the points

(3, 5)

and

(-1, 8)

, we get

m = \frac{8 - 5}{-1 - 3} = \frac{3}{-4} = -\frac{3}{4}

.

2

Determine the slope of line

L_1

using the perpendicular relationship.

The slope of

L_1

is

\frac{4}{3}

.

Since line

L_1

is perpendicular to the line with slope

-\frac{3}{4}

, its slope must be the negative reciprocal, which is

-\frac{1}{-3/4} = \frac{4}{3}

.

3

Express the slope of the line

ax + 6y = 15

in terms of

a

.

The slope is

-\frac{a}{6}

.

Rewriting the equation

ax + 6y = 15

in slope-intercept form (

y = mx + b

) gives

6y = -ax + 15

, which simplifies to

y = -\frac{a}{6}x + \frac{5}{2}

. The slope is the coefficient of

x

, which is

-\frac{a}{6}

.

4

Set the slope of

L_1

equal to the slope of the parallel line to solve for

a

.

a = -8

Because line

L_1

is parallel to the line

ax + 6y = 15

, their slopes are equal:

-\frac{a}{6} = \frac{4}{3}

. Multiplying both sides by

-6

gives

a = -8

.

Key Concept

Parallel lines have equal slopes, and perpendicular lines have slopes that are negative reciprocals of each other.

Estimated Time:1m 30s

Question 275Question →

If $u$ and $v$ are positive real numbers such that $\frac{(u^{-2} v^3)^3}{u^k v^{-1}} = \frac{v^{10}}{u^{10}}$ , what is the value of the exponent $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of the exponent is 4.

By applying the exponent rules systematically, the expression on the left simplifies to

u^{-6-k} v^{10}

, and the expression on the right is

u^{-10} v^{10}

. Equating the exponents of

u

gives

-6-k = -10

, which solves to

k = 4

.

Step-by-Step Solution

1

Apply the power of a product and power of a power properties to the numerator

(u^{-2} v^3)^3

.

u^{-6} v^9

According to the power of a product rule,

(xy)^a = x^a y^a

, and the power of a power rule,

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

.

2

Use the quotient of powers property to simplify the left side of the equation.

u^{-6-k} v^{10}

The quotient of powers rule states that

\frac{x^a}{x^b} = x^{a-b}

, so the exponents of like bases are subtracted:

-6 - k

for

u

and

9 - (-1) = 10

for

v

.

3

Rewrite the right side of the equation,

\frac{v^{10}}{u^{10}}

, using a negative exponent.

u^{-10} v^{10}

Applying the negative exponent rule,

\frac{1}{x^a} = x^{-a}

.

4

Set the simplified expressions equal and solve for

k

.

k = 4

Since

u^{-6-k} v^{10} = u^{-10} v^{10}

, the exponents of the base

u

must be equal. Therefore,

-6 - k = -10

, which simplifies to

k = 4

.

Key Concept

Properties of Exponents in Algebraic Expressions

Estimated Time:1m 30s

Question 276Question →

What value of the base $b$ satisfies the equation $\log_b (3b + 10) = 2$ ?

Show answer & explanation

Answer: 5

Answer

The correct answer is 5.

Applying the definition of logarithms to the equation

\log_b(3b + 10) = 2

converts it to the quadratic equation

b^2 = 3b + 10

. Subtracting

3b

and

10

from both sides results in

b^2 - 3b - 10 = 0

. Factoring the quadratic yields

(b - 5)(b + 2) = 0

, giving the potential solutions

b = 5

and

b = -2

. Because the base of a logarithm must be positive (

b > 0

), the negative solution is extraneous. This leaves

5

as the only valid base.

Step-by-Step Solution

1

Convert the logarithm to exponential form.

b^2 = 3b + 10

By the definition of logarithms,

\log_b(x) = y

is equivalent to

b^y = x

.

2

Rearrange into standard quadratic form.

b^2 - 3b - 10 = 0

Subtracting

3b

and

10

from both sides sets the quadratic expression equal to zero.

3

Factor the quadratic equation.

(b - 5)(b + 2) = 0

Finding two integers that multiply to

-10

and add to

-3

gives

-5

and

2

.

4

Solve for the variable and apply base constraints.

b = 5

Solving the factored equation yields

b = 5

or

b = -2

. Since a logarithmic base must be strictly positive (

b > 0

), we reject

b = -2

as extraneous, leaving

b = 5

.

Key Concept

Converting logarithmic equations to exponential form and verifying base restrictions.

Estimated Time:1m 30s

Question 277Question →

In a triangle, two of the sides have lengths $13$ and $20$ . The third side has a length of $s$ , where $s$ is an integer. If the side of length $20$ is the longest side of the triangle, and the triangle is obtuse, what is the number of possible values for $s$ ?

Show answer & explanation

Answer: 8

Answer

There are 8 possible integer values for

s

.

To find the number of possible integer values for

s

, we combine the Triangle Inequality Theorem (

13 + s > 20 \implies s > 7

) and the condition for an obtuse triangle with

20

as the longest side (

20^2 > 13^2 + s^2 \implies s^2 < 231 \implies s \leq 15

). This limits

s

to integers in the range

[8, 15]

, which contains exactly

8

values.

Step-by-Step Solution

1

Apply the Triangle Inequality Theorem to find the lower bound for

s

.

s > 7

, so the minimum integer value is

8

.

The sum of the two shorter sides of a triangle must be strictly greater than the longest side.

2

Set up the obtuse triangle inequality with

20

as the longest side.

20^2 > 13^2 + s^2

In any obtuse triangle with longest side

c

, the inequality

c^2 > a^2 + b^2

must hold.

3

Solve the inequality

20^2 > 13^2 + s^2

for

s

.

s^2 < 231 \implies s \leq 15

Simplifying the inequality gives

400 > 169 + s^2 \implies s^2 < 231

. The largest integer whose square is less than

231

is

15

.

4

Determine the number of integers in the range

[8, 15]

.

8 possible values

The integers satisfying both conditions are

\{8, 9, 10, 11, 12, 13, 14, 15\}

, which count to

8

.

Key Concept

Triangle Inequality Theorem and obtuse triangle classification using side lengths

Alternative Method

List the perfect squares and verify which ones satisfy both

s^2 < 231

and the Triangle Inequality Theorem

s > 7

.

Estimated Time:2m 0s

Question 278Question →

The functions $f$ and $g$ are defined for all permissible real numbers by $f(x) = \frac{x + 3}{x - 1}$ and $g(x) = 2x - 5$ . If $(f \circ g)(x) = 3$ , what is the value of $x$ ?

Show answer & explanation

Answer: 4

Answer

The value of

x

that satisfies the equation

(f \circ g)(x) = 3

is

4

.

To solve for

x

in

(f \circ g)(x) = 3

, we find the composite function

f(g(x))

by substituting

g(x) = 2x - 5

into

f(x)

. This yields

f(g(x)) = \frac{(2x-5)+3}{(2x-5)-1} = \frac{2x-2}{2x-6}

. Setting this equal to

3

gives

\frac{2x-2}{2x-6} = 3

. Multiplying by

2x-6

yields

2x-2 = 6x-18

. Rearranging terms to isolate

x

gives

4x = 16

, which results in

x = 4

.

Step-by-Step Solution

1

Substitute the expression for

g(x)

into

f(x)

to obtain the composite function

(f \circ g)(x)

.

(f \circ g)(x) = \frac{2x - 2}{2x - 6}

By definition of function composition,

(f \circ g)(x) = f(g(x))

. Substituting

g(x) = 2x - 5

into

f(x) = \frac{x+3}{x-1}

yields

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

, which simplifies to

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

.

2

Set the composite function expression equal to

3

and solve for

x

.

x = 4

We set

\frac{2x-2}{2x-6} = 3

. Multiplying both sides by the denominator

2x-6

gives

2x-2 = 3(2x-6)

. Expanding the right side gives

2x-2 = 6x-18

. Subtracting

2x

from both sides and adding

18

to both sides results in

16 = 4x

. Dividing by

4

gives

x = 4

.

Key Concept

Function Composition and Evaluation

Estimated Time:1m 30s

Question 279Question →

A closed cardboard box has a height of $x$ inches, a width of $3x - 2$ inches, and a length of $2x + 5$ inches. When the volume of the box, in cubic inches, is written as a polynomial in standard form, what is the coefficient of the $x^2$ term?

Show answer & explanation

Answer: 11

Answer

The coefficient of the

x^2

term is 11.

Expanding the volume expression

V(x) = x(3x - 2)(2x + 5)

yields

6x^3 + 11x^2 - 10x

. The coefficient of the

x^2

term is the numerical value associated with

x^2

, which is 11.

Step-by-Step Solution

1

Set up the polynomial expression for the volume.

V(x) = x(3x - 2)(2x + 5)

The volume of a rectangular prism is the product of its length, width, and height.

2

Multiply the binomials

(3x - 2)

and

(2x + 5)

by distributing terms.

(3x - 2)(2x + 5) = 6x^2 + 15x - 4x - 10 = 6x^2 + 11x - 10

To find the product of two binomials, multiply each term of the first binomial by each term of the second binomial and combine like terms.

3

Distribute the monomial

x

to each term in the simplified trinomial.

x(6x^2 + 11x - 10) = 6x^3 + 11x^2 - 10x

The height

x

must scale the entire base area polynomial.

4

Identify the coefficient of the quadratic term

x^2

.

11

The coefficient of a term is the numerical factor multiplied by the variable part.

Key Concept

Multiplying polynomials and identifying coefficients of specific terms in the resulting standard form polynomial.

Question 280Question →

If $x$ is a real number that satisfies the equation $\frac{3}{5}(2x - 7) + 0.4 = 0.2(x + 3)$ , what is the value of $5x - 2$ ?

Show answer & explanation

Answer: 20

Answer

The value of the expression

5x - 2

is

20

.

First, convert the fraction

\frac{3}{5}

to the decimal

0.6

. The equation becomes

0.6(2x - 7) + 0.4 = 0.2(x + 3)

. Distribute on both sides to get

1.2x - 4.2 + 0.4 = 0.2x + 0.6

. Combine constant terms on the left side to get

1.2x - 3.8 = 0.2x + 0.6

. Subtract

0.2x

and add

3.8

to both sides to isolate the variable, resulting in

x = 4.4

. Finally, substitute

x = 4.4

into the expression

5x - 2

to get

5(4.4) - 2 = 22 - 2 = 20

.

Step-by-Step Solution

1

Convert the fraction and distribute the coefficients

1.2x - 4.2 + 0.4 = 0.2x + 0.6

To clear parentheses and align terms using decimals.

2

Combine constants on the left side

1.2x - 3.8 = 0.2x + 0.6

To simplify the left-hand side of the linear equation.

3

Isolate the variable

x

x = 4.4

To determine the value of the unknown variable.

4

Evaluate the target expression

20

To compute the final value of the expression

5x - 2

.

Key Concept

Solving linear equations with fractional and decimal coefficients

Answer

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