All practice questions

Question 241Question →

In the standard $(x, y)$ coordinate plane, the triangular region $R$ is bounded by the lines $y = 2x$ , $y = -x + 9$ , and the $x$ -axis. A vertical line $x = k$ (where $0 < k < 9$ ) divides region $R$ into two sub-regions. If the area of the sub-region to the right of the line $x = k$ is exactly $8$ , what is the value of $k$ ?

Show answer & explanation

Answer: 5

Answer

5

The boundary lines intersect to form a triangle with vertices at

(0, 0)

,

(9, 0)

, and

(3, 6)

. Since the area to the right of

x = k

is

8

, and the total area of the triangle is

27

,

k

must be greater than

3

. The region to the right of

x = k

is a right triangle with a base of

9 - k

and a height of

9 - k

. Setting its area

\frac{1}{2}(9 - k)^2

equal to

8

yields

(9 - k)^2 = 16

, which gives

9 - k = 4

(since

k < 9

), and thus

k = 5

.

Step-by-Step Solution

1

Find the vertices of the triangular region

R

by finding the intersection points of the boundary lines

y = 2x

,

y = -x + 9

, and

y = 0

.

The vertices of the triangle are

A(0, 0)

,

B(9, 0)

, and

C(3, 6)

.

This establishes the boundaries and shape of the triangular region.

2

Determine which side of the peak

x = 3

the vertical line

x = k

must lie. Calculate the total area and the area of the left portion.

The total area of the triangle is

27

. The area to the left of the peak

x = 3

is

9

. Since the area of the region to the right of

x = k

is

8

, which is less than

18

,

k

must be greater than or equal to

3

.

This determines the geometric shape of the sub-region to the right of

x = k

as a right triangle.

3

Set up the area formula for the right-hand triangle with vertices

(k, 0)

,

(9, 0)

, and

(k, -k + 9)

, and set it equal to

8

.

The area is

\frac{1}{2}(9 - k)^2 = 8

, which simplifies to

(9 - k)^2 = 16

.

This relates the given area to the unknown coordinate

k

.

4

Solve the equation

(9 - k)^2 = 16

for

k

, keeping in mind that

k < 9

.

Taking the square root gives

9 - k = 4 \implies k = 5

.

This yields the final value of

k

.

Key Concept

Finding the area of a region defined by linear boundary equations and dividing it with a vertical line.

Alternative Method

Using similar triangles: The right-hand triangle formed by the line

x = 3

, the line

y = -x + 9

, and the

x

-axis has vertices at

(3,0)

,

(9,0)

, and

(3,6)

, with an area of

\frac{1}{2} \times 6 \times 6 = 18

. The smaller triangle to the right of

x = k

has an area of

8

and is similar to the larger triangle. The ratio of their areas is

\frac{8}{18} = \frac{4}{9}

, which means the ratio of their linear dimensions is

\sqrt{\frac{4}{9}} = \frac{2}{3}

. The base of the larger triangle is

9 - 3 = 6

, so the base of the smaller triangle must be

6 \times \frac{2}{3} = 4

. This gives

9 - k = 4 \implies k = 5

.

Estimated Time:2m 30s

Question 242Question →

In the standard $(x, y)$ coordinate plane, line $j$ has the equation $y = -\frac{1}{4}x + 7$ . If line $k$ is perpendicular to line $j$ , what is the slope of line $k$ ?

Show answer & explanation

Answer: 4

Answer

The slope of line

k

is 4.

The slope of the perpendicular line is

4

because the negative reciprocal of the given slope,

-\frac{1}{4}

, is

4

.

Step-by-Step Solution

1

Identify the slope of line

j

.

The slope of line

j

is

-\frac{1}{4}

.

The equation of line

j

is given in slope-intercept form,

y = mx + b

, where the coefficient of

x

represents the slope.

2

Calculate the negative reciprocal of the slope of line

j

to find the slope of perpendicular line

k

.

The slope of line

k

is

4

.

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

-\frac{1}{4}

is

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

.

Key Concept

The slopes of perpendicular lines are negative reciprocals of each other.

Estimated Time:45s

Question 243Question →

In $\triangle ABC$ , point $D$ lies on side $AC$ . If $AB = BD$ , the measure of $\angle A$ is $70^\circ$ , and the measure of $\angle C$ is $25^\circ$ , what is the measure, in degrees, of $\angle DBC$ ?

Show answer & explanation

Answer: 45

Answer

The measure of

\angle DBC

is 45 degrees.

In

\triangle ABD

, since

AB = BD

, the triangle is isosceles and the base angles opposite these sides are equal. Therefore, the measure of

\angle ADB

is equal to the measure of

\angle A

, which is

70^\circ

. Because points

A

,

D

, and

C

form a straight line, the angles

\angle ADB

and

\angle BDC

are supplementary, meaning the measure of

\angle BDC = 180^\circ - 70^\circ = 110^\circ

. The sum of the interior angles in

\triangle BCD

must be

180^\circ

, so the measure of

\angle DBC = 180^\circ - 110^\circ - 25^\circ = 45^\circ

.

Step-by-Step Solution

1

Determine the measure of

\angle ADB

using the properties of isosceles triangle

\triangle ABD

.

The measure of

\angle ADB

is

70^\circ

.

Since

AB = BD

, the angles opposite those sides,

\angle A

and

\angle ADB

, are equal.

2

Calculate the measure of the supplementary angle

\angle BDC

.

The measure of

\angle BDC

is

110^\circ

.

Points

A

,

D

, and

C

are collinear, meaning

\angle ADB

and

\angle BDC

form a linear pair and sum to

180^\circ

.

3

Determine the measure of

\angle DBC

using the triangle angle sum theorem on

\triangle BCD

.

The measure of

\angle DBC

is

45^\circ

.

The sum of the interior angles of

\triangle BCD

is

180^\circ

, so the measure of

\angle DBC

is

180^\circ - (110^\circ + 25^\circ) = 45^\circ

.

Key Concept

Using properties of isosceles triangles, linear pairs, and the triangle angle sum theorem to trace unknown angles.

Question 244Question →

An architect is designing a triangular window with side lengths, in feet, represented by $x$ , $3x - 2$ , and $18$ . If the value of $x$ must be an integer, what is the sum of all possible values of $x$ ?

Show answer & explanation

Answer: 30

Answer

The sum of all possible integer values of

x

is 30.

By applying the Triangle Inequality Theorem, the sum of any two sides of a triangle must be strictly greater than the third side. This yields the inequalities

x + (3x - 2) > 18

(which simplifies to

x > 5

),

x + 18 > 3x - 2

(which simplifies to

x < 10

), and

(3x - 2) + 18 > x

(which simplifies to

x > -8

). The intersection of these inequalities is

5 < x < 10

. Since

x

must be an integer, the possible values are

6

,

7

,

8

, and

9

. The sum of these values is

6 + 7 + 8 + 9 = 30

.

Step-by-Step Solution

1

Set up the first triangle inequality constraint where the sum of the two variable sides is greater than the constant side.

x + (3x - 2) > 18 \implies 4x > 20 \implies x > 5

The Triangle Inequality Theorem states that the sum of any two sides of a triangle must be strictly greater than the third side.

2

Set up the second triangle inequality constraint where the sum of

x

and the constant side is greater than the other variable side.

x + 18 > 3x - 2 \implies 20 > 2x \implies x < 10

To satisfy the Triangle Inequality Theorem for all combinations of sides.

3

Set up the third triangle inequality constraint where the sum of the second variable side and the constant side is greater than the first variable side.

(3x - 2) + 18 > x \implies 2x > -16 \implies x > -8

To ensure the third side combination is mathematically valid.

4

Find the intersection of all three inequalities to determine the valid range for

x

.

5 < x < 10

The value of

x

must satisfy all three inequalities simultaneously.

5

Identify the integer values of

x

within the open interval

(5, 10)

and calculate their sum.

The integers are

6, 7, 8,

and

9

. Sum =

6 + 7 + 8 + 9 = 30

.

The problem specifies that

x

must be an integer, so we sum only the integers strictly between 5 and 10.

Key Concept

Triangle Inequality Theorem

Estimated Time:2m 0s

Question 245Question →

A technician uses a linear model to estimate the time, $t$ hours, required to complete a project. The estimate satisfies the equation:

\frac{2}{3}(t - 12) + \frac{5}{6}(t + 6) = 18

What is the value of $t$ ?

Show answer & explanation

Answer: 14

Answer

The value of

t

is

14

.

To solve the linear equation, we first multiply all terms by the least common multiple of the denominators, which is

6

. This simplifies the equation to

4(t - 12) + 5(t + 6) = 108

. Distributing the terms gives

4t - 48 + 5t + 30 = 108

. Combining like terms yields

9t - 18 = 108

. Adding

18

to both sides gives

9t = 126

. Finally, dividing by

9

gives

t = 14

.

Step-by-Step Solution

1

Multiply both sides of the equation by

6

to eliminate the denominators.

4(t - 12) + 5(t + 6) = 108

Multiplying by the least common multiple of the denominators clears the fractions, making the linear equation easier to solve.

2

Apply the distributive property to expand the terms.

4t - 48 + 5t + 30 = 108

Distributing the constants allows us to group like terms.

3

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

9t - 18 = 108

Simplifying the expression is a necessary step before isolating the variable.

4

Add

18

to both sides of the equation.

9t = 126

Adding the constant to both sides isolates the variable term on the left side.

5

Divide both sides of the equation by

9

to solve for

t

.

t = 14

Dividing by the coefficient of the variable yields the final solution.

Key Concept

Solving linear equations with fractional coefficients by clearing denominators and isolating the variable.

Question 246Question →

In $\triangle XYZ$ , the measure of $\angle X$ is $40^\circ$ . If the measure of $\angle Y$ is three times the measure of $\angle X$ , what is the measure, in degrees, of $\angle Z$ ?

Show answer & explanation

Answer: 20

Answer

The measure of

\angle Z

is

20

degrees.

To find the measure of

\angle Z

, we first determine the measure of

\angle Y

. Since

\angle Y

is three times the measure of

\angle X

(

40^\circ

), we calculate

3 \times 40^\circ = 120^\circ

. Because the interior angles of a triangle must sum to

180^\circ

, the remaining angle

\angle Z

is found by subtracting the measures of

\angle X

and

\angle Y

from

180^\circ

:

180^\circ - 40^\circ - 120^\circ = 20^\circ

.

Step-by-Step Solution

1

Calculate the measure of

\angle Y

.

The measure of

\angle Y

is

120^\circ

.

The measure of

\angle Y

is specified to be three times the measure of

\angle X

, which is given as

40^\circ

. Multiplying

40^\circ

by

3

gives

120^\circ

.

2

Calculate the measure of

\angle Z

.

The measure of

\angle Z

is

20^\circ

.

The interior angles of any triangle sum to

180^\circ

. Subtracting the sum of the measures of

\angle X

(

40^\circ

) and

\angle Y

(

120^\circ

) from

180^\circ

yields the measure of

\angle Z

.

Key Concept

The sum of the interior angles of a triangle is always

180^\circ

.

Estimated Time:45s

Question 247Question →

If $a > 1.5$ is a real number that satisfies the equation $4^a - 3(2^{a+1}) + 8 = 0$ , what is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

Substituting

u = 2^a

turns the equation into

u^2 - 6u + 8 = 0

. Factoring gives

(u-2)(u-4)=0

, meaning

u=2

or

u=4

. Reversing the substitution gives

2^a = 2 \implies a=1

and

2^a = 4 \implies a=2

. Since

a > 1.5

, the correct value is 2.

Step-by-Step Solution

1

Express the equation in terms of base 2.

(2^a)^2 - 6(2^a) + 8 = 0

Since 4 is

2^2

and

2^{a+1} = 2 \cdot 2^a

, expressing all terms in base 2 allows for algebraic substitution.

2

Substitute

u = 2^a

to form a quadratic equation.

u^2 - 6u + 8 = 0

Substitution simplifies the exponential equation into a standard quadratic form.

3

Factor the quadratic equation.

(u-2)(u-4) = 0

Factoring allows us to find the roots of the quadratic equation.

4

Solve for the variable

a

.

a = 1

or

a = 2

Solving

2^a = 2

yields

a = 1

, and solving

2^a = 4

yields

a = 2

.

5

Apply the given constraint on

a

.

a = 2

The problem states that

a > 1.5

, so

a = 1

is discarded and

a = 2

is the correct value.

Key Concept

Solving exponential equations using quadratic substitution

Alternative Method

Instead of using substitution, test values for

a

. Since

a > 1.5

, testing small integer values starting with

a=2

shows

4^2 - 3(2^3) + 8 = 16 - 24 + 8 = 0

, validating that

a=2

is the solution.

Estimated Time:1m 30s

Question 248Question →

If $w$ and $z$ are positive real numbers and $k$ is a constant such that the expression $\frac{(w^{3/2} z^{-1})^4}{(w z^k)^2}$ is equivalent to $w^4 z^6$ , what is the value of $k$ ?

Show answer & explanation

Answer: -5

Answer

The value of

k

is

-5

.

Applying the exponent rules, the expression simplifies to

w^4 z^{-4-2k}

. Equating the exponent of

z

to the exponent in the target expression

w^4 z^6

gives

-4-2k = 6

, which solves to

k = -5

.

Step-by-Step Solution

1

Apply the power of a product rule to the expression in the numerator

(w^{3/2} z^{-1})^4 = w^6 z^{-4}

When raising a product to a power, multiply the exponent of each factor by the outer exponent:

(x^a y^b)^c = x^{ac} y^{bc}

.

2

Apply the power of a product rule to the expression in the denominator

(w z^k)^2 = w^2 z^{2k}

Multiply the exponent of each factor in the denominator by

2

.

3

Divide the numerator by the denominator using the quotient rule for exponents

\frac{w^6 z^{-4}}{w^2 z^{2k}} = w^4 z^{-4-2k}

When dividing terms with the same base, subtract the exponent in the denominator from the exponent in the numerator:

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

.

4

Equate the exponent of

z

in the simplified expression to the exponent of

z

in the target expression

-4 - 2k = 6

Since the simplified expression is equivalent to

w^4 z^6

, the exponents of the corresponding variable bases must be equal.

5

Solve the linear equation for

k

k = -5

Add

4

to both sides of the equation to get

-2k = 10

, then divide both sides by

-2

.

Key Concept

Properties of Exponents in Algebraic Expressions

Question 249Question →

Two volunteer teams, Team A and Team B, are packing food boxes for a local shelter. Team A packs at a constant rate of $15$ boxes per hour and begins packing at 8:00 a.m. Team B packs at a constant rate of $20$ boxes per hour and begins packing at 9:30 a.m. If both teams pack continuously at their respective rates, how many hours after Team A begins will both teams have packed the same total number of boxes?

Show answer & explanation

Answer: 6

Answer

Both teams will have packed the same total number of boxes

6

hours after Team A begins.

The correct answer is

6

. Let

x

be the number of hours Team A packs. Team B begins 1.5 hours later, so Team B packs for

x - 1.5

hours. Setting their total boxes packed equal gives

15x = 20(x - 1.5)

. Distributing yields

15x = 20x - 30

. Subtracting

20x

from both sides gives

-5x = -30

, which simplifies to

x = 6

.

Step-by-Step Solution

1

Define the variable for time and identify the time difference.

Let

x

be the number of hours Team A packs. Since Team B starts 1 hour and 30 minutes (which is

1.5

hours) later, Team B's time is represented as

x - 1.5

hours.

Establishing correct algebraic representations for time is necessary to set up the equation.

2

Set up an equation equating the total boxes packed by both teams.

The equation is

15x = 20(x - 1.5)

.

Since both teams pack a constant number of boxes per hour, multiplying their rate by their active time gives the total boxes packed.

3

Solve the linear equation for

x

.

Distribute the 20:

15x = 20x - 30

. Subtract

20x

from both sides:

-5x = -30

. Divide by

-5

:

x = 6

.

Solving the equation gives the number of hours after Team A starts when their packed boxes are equal.

Key Concept

Translating real-world rates and time shifts into linear equations and solving them.

Question 250Question →

In the standard $(x, y)$ coordinate plane, a line segment has midpoint $M(4, 1)$ and one endpoint $P(x, 5)$ . If the total length of the line segment is $10$ units, and $x < 4$ , what is the value of $x$ ?

Show answer & explanation

Answer: 1

Answer

1

The midpoint

M

divides the line segment into two equal segments, meaning the distance from endpoint

P

to midpoint

M

is half of the total length of the segment,

\frac{10}{2} = 5

units. Applying the distance formula between

P(x, 5)

and

M(4, 1)

gives

\sqrt{(x - 4)^2 + (5 - 1)^2} = 5

. Squaring both sides yields

(x - 4)^2 + 16 = 25

, which simplifies to

(x - 4)^2 = 9

. Solving for

x

gives

x - 4 = 3

or

x - 4 = -3

, meaning

x = 7

or

x = 1

. The problem specifies that

x < 4

, so the only valid value is

1

.

Step-by-Step Solution

1

Find the distance from the endpoint to the midpoint.

The distance

PM

is

5

units.

The midpoint divides the line segment into two equal parts, so the distance from any endpoint to the midpoint is half of the total length of the segment:

\frac{10}{2} = 5

.

2

Set up the distance formula equation for the segment

PM

.

\sqrt{(x - 4)^2 + (5 - 1)^2} = 5

Using the coordinates of

P(x, 5)

and

M(4, 1)

with the distance formula

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

.

3

Simplify the equation and square both sides.

(x - 4)^2 + 16 = 25

Calculating

(5 - 1)^2 = 16

and squaring both sides of the equation to eliminate the radical.

4

Isolate the squared term and solve for

x

.

x = 7

or

x = 1

Subtracting

16

from both sides gives

(x - 4)^2 = 9

. Taking the square root gives

x - 4 = 3

or

x - 4 = -3

, yielding

x = 7

or

x = 1

.

5

Apply the constraint

x < 4

.

x = 1

Since the question states that

x < 4

, we choose

x = 1

instead of

x = 7

.

Key Concept

The midpoint divides a segment into two segments of equal length, and the distance between two points

(x_1, y_1)

and

(x_2, y_2)

is given by

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

.

Question 251Question →

In the standard $(x, y)$ coordinate plane, a line passes through the point $(3, 1)$ and has a slope of $-\frac{2}{3}$ . If the line intersects the $x$ -axis at the point $(p, 0)$ , what is the value of $p$ ?

Show answer & explanation

Answer: 4.5

Answer

The value of

p

is

4.5

.

The line equation is

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

. By substituting

y = 0

for the

x

-intercept, the equation becomes

-1 = -\frac{2}{3}(p - 3)

. Multiplying both sides by

-3

gives

3 = 2(p - 3)

, which simplifies to

2p = 9

, resulting in

p = 4.5

.

Step-by-Step Solution

1

Write the point-slope equation of the line.

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

We use the point-slope form

y - y_1 = m(x - x_1)

with the point

(3, 1)

and slope

-\frac{2}{3}

.

2

Substitute the point

(p, 0)

into the line's equation.

-1 = -\frac{2}{3}(p - 3)

The

x

-intercept

(p, 0)

lies on the line, so its coordinates must satisfy the equation of the line.

3

Solve for the variable

p

.

p = 4.5

Multiply by

-3

to get

3 = 2(p - 3)

, add

6

to both sides to get

2p = 9

, and divide by

2

.

Key Concept

Linear equations and graphing, specifically point-slope form and x-intercept calculation.

Question 252Question →

An ellipse in the standard $(x, y)$ coordinate plane is defined by the equation $\frac{(x-4)^2}{169} + \frac{(y+3)^2}{144} = 1$ . What is the distance between the two foci of this ellipse?

Show answer & explanation

Answer: 10

Answer

The distance between the two foci of the ellipse is 10.

By comparing the given equation to the standard form of an ellipse, we find

a^2 = 169

and

b^2 = 144

. The distance from the center to each focus,

c

, is given by

c = \sqrt{a^2 - b^2} = \sqrt{169 - 144} = \sqrt{25} = 5

. The total distance between the two foci is

2c = 2(5) = 10

.

Step-by-Step Solution

1

Identify

a^2

and

b^2

from the given equation of the ellipse.

a^2 = 169

and

b^2 = 144

The standard equation of a horizontal ellipse is

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

, where

a^2

is the larger denominator.

2

Calculate the value of

c

, the distance from the center to a focus.

c = 5

For an ellipse, the focal distance

c

is related to the semi-major axis

a

and semi-minor axis

b

by the equation

c^2 = a^2 - b^2

.

3

Calculate the distance between the two foci, which is

2c

.

10

The distance between the two foci of an ellipse is twice the distance from the center to each focus (

2c

).

Key Concept

Focal distance of an ellipse

Estimated Time:1m 15s

Question 253Question →

In the standard $(x, y)$ coordinate plane, a line with a non-zero slope $m$ passes through the point $(8, 2)$ . If the product of the line's $x$ -intercept and its $y$ -intercept is $64$ , what is the value of $m$ ?

Show answer & explanation

Answer: -0.25

Answer

The value of the slope

m

is

-0.25

(or

-\frac{1}{4}

).

By writing the line as

y = mx + b

, the given point

(8, 2)

establishes that

b = 2 - 8m

. Since the

x

-intercept is

-\frac{b}{m}

, the product of the intercepts is

-\frac{b^2}{m}

. Setting this product to

64

gives

-\frac{(2-8m)^2}{m} = 64

, which simplifies to the quadratic equation

16m^2 + 8m + 1 = 0

. Factoring this perfect square trinomial gives

(4m + 1)^2 = 0

, which yields the unique solution

m = -0.25

.

Step-by-Step Solution

1

Express the

y

-intercept in terms of

m

.

b = 2 - 8m

The line equation is

y = mx + b

and it passes through

(8, 2)

, so

2 = 8(m) + b

.

2

Express the

x

-intercept in terms of

m

.

x = -\frac{b}{m}

The

x

-intercept is the value of

x

when

y = 0

in the line equation

y = mx + b

.

3

Write the product of the intercepts and set it equal to

64

.

b^2 = -64m

The product of the intercepts is

b \cdot \left(-\frac{b}{m}\right) = -\frac{b^2}{m} = 64

.

4

Substitute

b = 2 - 8m

into the product equation and simplify to a quadratic equation.

64m^2 + 32m + 4 = 0

Substituting

b

gives

(2 - 8m)^2 = -64m

. Expanding and combining like terms yields the quadratic equation.

5

Solve the quadratic equation for

m

.

m = -0.25

Dividing by

4

gives

16m^2 + 8m + 1 = (4m + 1)^2 = 0

, which has the single real solution

m = -0.25

.

Key Concept

Using linear equation forms, point-slope relationships, and intercept properties to solve coordinate geometry problems.

Question 254Question →

An exterior angle of a triangle measures $135^\circ$ . The two nonadjacent interior angles of the triangle have measures in the ratio $2:3$ . What is the measure, in degrees, of the largest interior angle of the triangle?

Show answer & explanation

Answer: 81

Answer

The measure of the largest interior angle of the triangle is 81 degrees.

According to the Exterior Angle Theorem, the sum of the two nonadjacent interior angles equals the measure of the exterior angle:

2x + 3x = 135

, which simplifies to

5x = 135

and gives

x = 27

. The nonadjacent interior angles are

2(27) = 54^\circ

and

3(27) = 81^\circ

. The adjacent interior angle is

180^\circ - 135^\circ = 45^\circ

. Comparing the three angles (

45^\circ

,

54^\circ

,

81^\circ

), the largest is

81^\circ

.

Step-by-Step Solution

1

Use the Exterior Angle Theorem to relate the two nonadjacent interior angles to the given exterior angle.

Equation:

2x + 3x = 135

The theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two nonadjacent interior angles.

2

Solve the linear equation for

x

.

x = 27

Combine like terms to get

5x = 135

, then divide both sides by 5.

3

Calculate the measures of the two nonadjacent interior angles.

First angle:

2(27) = 54^\circ

; Second angle:

3(27) = 81^\circ

Multiply each part of the ratio by the scale factor

x = 27

.

4

Calculate the measure of the remaining interior angle.

Adjacent angle:

180^\circ - 135^\circ = 45^\circ

An interior angle and its adjacent exterior angle lie on a straight line and are supplementary (they sum to

180^\circ

).

5

Compare the three interior angle measures to find the largest.

The largest angle is

81^\circ

.

Comparing

45^\circ

,

54^\circ

, and

81^\circ

shows that

81^\circ

is the maximum value.

Key Concept

Exterior Angle Theorem and interior angle relationships in a triangle

Alternative Method

Find the adjacent interior angle first:

180^\circ - 135^\circ = 45^\circ

. Since the sum of all interior angles in a triangle is

180^\circ

, the sum of the remaining two interior angles must be

180^\circ - 45^\circ = 135^\circ

. Set up the ratio equation

2x + 3x = 135

and solve for

x

to find the other two angles.

Estimated Time:1m 15s

Question 255Question →

A pet care service, Canine Care, charges a flat monthly registration fee of $25 plus$ 15 per dog walk. A competing service, Paws & Claws, charges a flat monthly registration fee of $45 plus$ 10 per dog walk. If a client's total monthly cost with Canine Care is $30 more than their total monthly cost with Paws & Claws for the same number of walks, how many dog walks did the client's dog receive during that month?

Show answer & explanation

Answer: 10

Answer

The client's dog received 10 dog walks during the month.

The correct answer is 10. By translating the verbal descriptions into algebraic expressions, we find that Canine Care's monthly cost is

15w + 25

and Paws & Claws' monthly cost is

10w + 45

. Setting up the equation where Canine Care is

30

more than Paws & Claws gives

15w + 25 = 10w + 45 + 30

. Simplifying the right side results in

15w + 25 = 10w + 75

. Subtracting

10w

from both sides gives

5w + 25 = 75

. Subtracting

25

from both sides gives

5w = 50

, and dividing by

5

yields

w = 10

.

Step-by-Step Solution

1

Define the variable and write the cost expression for Canine Care.

15w + 25

, where

w

is the number of walks.

To represent the total monthly cost of Canine Care algebraically based on the flat fee and per-walk rate.

2

Write the cost expression for Paws & Claws.

10w + 45

, where

w

is the number of walks.

To represent the total monthly cost of Paws & Claws algebraically based on the flat fee and per-walk rate.

3

Set up the equation relating the two costs.

15w + 25 = 10w + 45 + 30

The problem states the cost of Canine Care is $30 more than the cost of Paws & Claws.

4

Simplify the equation and solve for

w

.

5w = 50 \Rightarrow w = 10

Combine like terms and isolate the variable

w

to find the number of dog walks.

Key Concept

Translating and Solving Algebraic Word Problems

Question 256Question →

In the standard $(x, y)$ coordinate plane, line $L_1$ has a negative slope and passes through the point $(1, 4)$ . Line $L_2$ has a positive slope and passes through the point $(7, 3)$ . Both lines intersect the $x$ -axis at the same point $P$ . If the product of the slopes of $L_1$ and $L_2$ is $-\frac{4}{3}$ , and line $L_3$ is perpendicular to $L_1$ and passes through the point $(2, -2)$ , what is the $y$ -intercept of $L_3$ ?

Show answer & explanation

Answer: -3.5

Answer

The

y

-intercept of line

L_3

is

-3.5

.

By writing the slope of the first line as

\frac{-4}{x_p - 1}

and the slope of the second line as

\frac{-3}{x_p - 7}

, their product is set to

-\frac{4}{3}

. Solving the resulting quadratic equation

(x_p - 4)^2 = 0

yields

x_p = 4

. Substituting this back gives a slope of

-\frac{4}{3}

for the first line. The perpendicular line must have a slope of

\frac{3}{4}

. Using the point-slope formula with point

(2, -2)

and slope

\frac{3}{4}

gives the line

y = \frac{3}{4}x - 3.5

, which crosses the

y

-axis at

-3.5

.

Step-by-Step Solution

1

Express the slopes of

L_1

and

L_2

in terms of the unknown

x

-coordinate of point

P

.

m_1 = \frac{0 - 4}{x_p - 1} = \frac{-4}{x_p - 1}

and

m_2 = \frac{0 - 3}{x_p - 7} = \frac{-3}{x_p - 7}

Since

P

lies on the

x

-axis, its coordinates are

(x_p, 0)

. We apply the slope formula

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

to the points on each line.

2

Set up the equation for the product of the slopes.

\left(\frac{-4}{x_p - 1}\right)\left(\frac{-3}{x_p - 7}\right) = -\frac{4}{3} \implies \frac{12}{(x_p - 1)(x_p - 7)} = -\frac{4}{3}

The problem states that the product of the slopes of

L_1

and

L_2

is

-\frac{4}{3}

.

3

Solve the equation for

x_p

.

x_p = 4

Cross-multiplying gives

36 = -4(x_p^2 - 8x_p + 7)

, which simplifies to

-9 = x_p^2 - 8x_p + 7 \implies x_p^2 - 8x_p + 16 = 0 \implies (x_p - 4)^2 = 0

.

4

Find the slope of

L_1

.

m_1 = \frac{-4}{4 - 1} = -\frac{4}{3}

Substituting

x_p = 4

back into the expression for

m_1

yields the slope of

L_1

.

5

Find the slope of

L_3

.

m_3 = \frac{3}{4}

Since

L_3

is perpendicular to

L_1

, its slope is the negative reciprocal of

m_1

.

6

Write the linear equation for

L_3

and determine its

y

-intercept.

y = \frac{3}{4}x - 3.5

, so the

y

-intercept is

-3.5

.

Using the point-slope form with point

(2, -2)

and slope

m_3 = \frac{3}{4}

, we get

y - (-2) = \frac{3}{4}(x - 2) \implies y + 2 = \frac{3}{4}x - 1.5 \implies y = \frac{3}{4}x - 3.5

.

Key Concept

The slope of a line is defined as

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

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

m_1 \cdot m_2 = -1

. The

y

-intercept of a line is the value of

y

when

x = 0

.

Question 257Question →

In a coordinate plane, two perpendicular lines, $L_1$ and $L_2$ , intersect at a point on the positive $y$ -axis. The line $L_1$ passes through the point $(-4, 3)$ , and the line $L_2$ has an $x$ -intercept at $(2.5, 0)$ . What is the $y$ -coordinate of the intersection point of $L_1$ and $L_2$ ?

Show answer & explanation

Answer: 5

Answer

The y-coordinate of the intersection point is 5.

The intersection point on the positive

y

-axis has coordinates

(0, 5)

. The slope of the line passing through

(-4, 3)

and

(0, 5)

is

0.5

. The slope of the line passing through

(2.5, 0)

and

(0, 5)

is

-2

. Since the product of these slopes is

0.5 \times (-2) = -1

, the lines are perpendicular.

Step-by-Step Solution

1

Identify the coordinates of the intersection point.

The intersection point of

L_1

and

L_2

is

(0, b)

, where

b > 0

.

Since the intersection point lies on the positive

y

-axis, its

x

-coordinate is

0

and its

y

-coordinate

b

must be positive.

2

Calculate the slope of

L_1

.

The slope of

L_1

is

m_1 = \frac{b - 3}{0 - (-4)} = \frac{b - 3}{4}

.

The slope formula

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

is applied to the points

(-4, 3)

and

(0, b)

.

3

Calculate the slope of

L_2

.

The slope of

L_2

is

m_2 = \frac{b - 0}{0 - 2.5} = -\frac{b}{2.5}

.

The line

L_2

passes through the

y

-intercept

(0, b)

and its

x

-intercept

(2.5, 0)

.

4

Apply the perpendicular slope condition.

m_1 \cdot m_2 = -1 \implies \left(\frac{b - 3}{4}\right)\left(-\frac{b}{2.5}\right) = -1 \implies b(b - 3) = 10

.

Perpendicular lines have slopes that are negative reciprocals of each other, meaning their product is

-1

.

5

Solve the quadratic equation for

b

.

Expanding the equation gives

b^2 - 3b - 10 = 0

, which factors into

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

. Since

b > 0

, the only valid solution is

b = 5

.

Solving the factored equation yields

b = 5

or

b = -2

. The constraint that the intersection lies on the positive

y

-axis rules out the negative value.

Key Concept

Perpendicular line slopes and coordinate intercepts

Estimated Time:2m 30s

Question 258Question →

In $\triangle ABC$ , the measure of interior angle $\angle A$ is $45^\circ$ , and the exterior angle at vertex $B$ measures $125^\circ$ . What is the measure, in degrees, of interior angle $\angle C$ ?

Show answer & explanation

Answer: 80

Answer

The measure of interior angle

\angle C

is

80

degrees.

According to the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. In this case, the exterior angle at vertex

B

measures

125^\circ

, and one of its remote interior angles,

\angle A

, measures

45^\circ

. The other remote interior angle is

\angle C

. Setting up the equation:

125^\circ = 45^\circ + \text{m}\angle C

. Solving for the measure of

\angle C

gives

125^\circ - 45^\circ = 80^\circ

.

Step-by-Step Solution

1

Use the Exterior Angle Theorem to relate the given angles.

The exterior angle at vertex

B

(

125^\circ

) is equal to the sum of the remote interior angles,

\angle A

and

\angle C

. This gives the equation:

125^\circ = 45^\circ + \text{m}\angle C

.

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

2

Solve the equation for the measure of

\angle C

.

\text{m}\angle C = 125^\circ - 45^\circ = 80^\circ

Subtract

45^\circ

from both sides of the equation to isolate the measure of

\angle C

.

Key Concept

Exterior Angle Theorem

Question 259Question →

In the standard $(x, y)$ coordinate plane, line $L_1$ passes through the points $(-1, 2)$ and $(3, 14)$ . If line $L_2$ is parallel to line $L_1$ , what is the slope of line $L_2$ ?

Show answer & explanation

Answer: 3

Answer

The slope of line

L_2

is

3

.

To find the slope of line

L_2

, we calculate the slope of line

L_1

using the formula

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

. Substituting the coordinates

(-1, 2)

and

(3, 14)

yields

m = \frac{14 - 2}{3 - (-1)} = \frac{12}{4} = 3

. Because parallel lines have the same slope, the slope of line

L_2

is also

3

.

Step-by-Step Solution

1

Calculate the slope of line

L_1

using the coordinates of the two given points,

(-1, 2)

and

(3, 14)

.

m = 3

The slope formula is

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

, which evaluates to

m = \frac{14 - 2}{3 - (-1)} = \frac{12}{4} = 3

.

2

Determine the slope of line

L_2

based on its relationship to line

L_1

.

The slope of line

L_2

is

3

.

Parallel lines always have identical slopes.

Key Concept

Parallel lines have the same slope.

Question 260Question →

If $\log_2(x + 5) + \log_2(x - 1) = 4$ , what is the value of $x$ ?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

The correct answer is 3. Combining the logarithmic terms using the product property gives

\log_2((x + 5)(x - 1)) = 4

. Converting this to exponential form yields

(x + 5)(x - 1) = 2^4 = 16

. Expanding and rewriting in standard form gives

x^2 + 4x - 21 = 0

, which factors as

(x + 7)(x - 3) = 0

. This gives potential solutions of

x = 3

and

x = -7

. However,

x = -7

results in negative arguments for the logarithms in the original equation, making it extraneous. Thus, the only valid solution is 3.

Step-by-Step Solution

1

Apply the product property of logarithms to combine the terms on the left side of the equation.

\log_2((x + 5)(x - 1)) = 4

The sum of logarithms with the same base is equal to the logarithm of their product:

\log_b(A) + \log_b(B) = \log_b(AB)

.

2

Convert the logarithmic equation to its equivalent exponential form.

(x + 5)(x - 1) = 2^4

By definition,

\log_b(Y) = C

is equivalent to

b^C = Y

.

3

Expand the product and simplify the equation into standard quadratic form.

x^2 + 4x - 21 = 0

Expanding

(x + 5)(x - 1)

yields

x^2 + 4x - 5

. Subtracting 16 from both sides gives the standard quadratic form

Ax^2 + Bx + C = 0

.

4

Factor the quadratic equation to find the potential values of

x

.

(x + 7)(x - 3) = 0

, so

x = -7

or

x = 3

We need two numbers that multiply to

-21

and add to

4

, which are

7

and

-3

.

5

Check the potential solutions against the domain of the original logarithmic equation.

x = 3

The arguments of the logarithms,

x + 5

and

x - 1

, must be strictly positive. For

x = -7

, the arguments are negative, so

x = -7

is an extraneous solution. For

x = 3

, both arguments are positive, so

x = 3

is the only valid solution.

Key Concept

Solving logarithmic equations by applying logarithmic properties and converting to exponential form, while checking for extraneous solutions.

Answer

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