All practice questions

Question 381Question →

The path of a particle in the standard $(x, y)$ coordinate plane is described by the linear equation $3x - y = 2$ , and the path of another particle is described by the quadratic equation $y = x^2 - x - 7$ . If the two paths intersect at two locations, what is the sum of the $y$ -coordinates of these intersection points?

Show answer & explanation

Answer: 8

Answer

The sum of the

y

-coordinates of the intersection points is 8.

The correct answer is 8. Solving the system by setting

x^2 - x - 7 = 3x - 2

results in the quadratic equation

x^2 - 4x - 5 = 0

. Factoring gives

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

, which yields intersection

x

-coordinates of

x = 5

and

x = -1

. Substituting these back into the linear equation gives

y

-coordinates of

13

and

-5

. Adding these values together yields

13 + (-5) = 8

.

Step-by-Step Solution

1

Express the linear equation in terms of

y

.

y = 3x - 2

This allows for direct substitution into the quadratic equation.

2

Equate the linear and quadratic expressions to solve for the

x

-coordinates of the intersection points.

x^2 - x - 7 = 3x - 2

Intersection points share the same coordinates for both equations.

3

Set the quadratic equation to zero.

x^2 - 4x - 5 = 0

This puts the equation in standard form so it can be solved by factoring.

4

Factor the quadratic equation.

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

, so

x = 5

or

x = -1

Factoring determines the

x

-coordinates of the intersection points.

5

Substitute the

x

-values into the linear equation to determine the

y

-coordinates.

For

x = 5

,

y = 13

. For

x = -1

,

y = -5

.

Finding the

y

-coordinates is necessary to compute their sum.

6

Add the

y

-coordinates together.

13 + (-5) = 8

The question asks for the sum of the

y

-coordinates of the intersection points.

Key Concept

Solving systems of linear and quadratic equations by substitution and factoring

Alternative Method

We can use Vieta's formulas to find the sum of the

y

-coordinates without calculating each individual coordinate. The sum of the

y

-coordinates is

y_1 + y_2 = (3x_1 - 2) + (3x_2 - 2) = 3(x_1 + x_2) - 4

. Since

x_1

and

x_2

are the roots of

x^2 - 4x - 5 = 0

, Vieta's formulas state that the sum of the roots is

x_1 + x_2 = -\frac{-4}{1} = 4

. Substituting this value into our sum expression yields

3(4) - 4 = 12 - 4 = 8

.

Estimated Time:1m 30s

Question 382Question →

A rectangular park $ABCD$ has a length of $20$ meters and a width of $15$ meters. A straight walking path is built from corner $A$ to a point $P$ on the diagonal path $BD$ such that the path $AP$ is perpendicular to $BD$ . What is the length, in meters, of the path $AP$ ?

Show answer & explanation

Answer: 12

Answer

The length of the path

AP

is

12

meters.

The diagonal

BD

divides the rectangular park into two congruent right triangles. For right triangle

ABD

, the legs are

AB = 20

and

AD = 15

. Using the Pythagorean theorem, the hypotenuse

BD = \sqrt{20^2 + 15^2} = 25

meters. The area of triangle

ABD

is

\frac{1}{2} \times 20 \times 15 = 150

square meters. Since

AP

is perpendicular to

BD

,

AP

is the altitude to base

BD

, so the area can also be written as

\frac{1}{2} \times 25 \times AP

. Equating the two areas,

12.5 \times AP = 150

, which simplifies to

AP = 12

meters.

Step-by-Step Solution

1

Calculate the length of the diagonal

BD

using the Pythagorean theorem on right triangle

ABD

.

BD = \sqrt{20^2 + 15^2} = 25

meters.

The diagonal forms the hypotenuse of the right triangle

ABD

, which is needed to calculate the altitude

AP

.

2

Express the area of right triangle

ABD

using the two perpendicular legs,

AB

and

AD

.

\text{Area} = \frac{1}{2} \times 20 \times 15 = 150

square meters.

This establishes the total area of the triangle.

3

Express the area of the same triangle using the diagonal

BD

as the base and the perpendicular path

AP

as the height.

\text{Area} = \frac{1}{2} \times 25 \times AP

.

This sets up an equation using the unknown path length

AP

.

4

Equate the two area expressions and solve for

AP

.

12.5 \times AP = 150 \implies AP = 12

meters.

Since both expressions represent the area of the same triangle, they must be equal.

Key Concept

Using the Pythagorean theorem to find the hypotenuse of a right triangle, and then using the area formula to find the altitude to the hypotenuse.

Alternative Method

Alternatively, you can use similar right triangles. Since triangle

ABP

is similar to triangle

DBA

, the ratio of their corresponding sides is equal:

\frac{AP}{AD} = \frac{AB}{BD}

. Substituting the known values gives

\frac{AP}{15} = \frac{20}{25}

, which simplifies to

AP = 15 \times \frac{4}{5} = 12

meters.

Estimated Time:1m 30s

Question 383Question →

A warehouse is cooling a refrigerated storage room. The initial temperature of the room is $22^\circ\text{C}$ . A cooling system lowers the temperature by $1.8^\circ\text{C}$ per hour. To store a specific vaccine, the temperature of the room must be kept strictly below $-5^\circ\text{C}$ . What is the minimum number of whole hours the cooling system must run to reach a safe storage temperature?

Show answer & explanation

Answer: 16

Answer

The minimum number of whole hours the cooling system must run is 16.

To find the minimum number of whole hours, we set up the inequality

22 - 1.8h < -5

. Subtracting 22 from both sides yields

-1.8h < -27

. Dividing by the negative rate

-1.8

requires reversing the inequality sign, which gives

h > 15

. The smallest integer greater than 15 is 16.

Step-by-Step Solution

1

Set up the inequality representing the temperature condition.

22 - 1.8h < -5

The initial temperature is

22^\circ\text{C}

, and it decreases by

1.8^\circ\text{C}

per hour

h

. The temperature must be strictly below

-5^\circ\text{C}

.

2

Isolate the variable term by subtracting 22 from both sides.

-1.8h < -27

Subtracting 22 simplifies the left side and groups the constant terms on the right side.

3

Divide both sides by -1.8 and reverse the inequality sign.

h > 15

Dividing both sides of an inequality by a negative number reverses the direction of the inequality sign.

4

Determine the smallest integer value for h that satisfies the inequality.

16

Since the time must be strictly greater than 15 hours, the smallest whole number of hours that satisfies this is 16.

Key Concept

Solving linear inequalities by isolating the variable and reversing the inequality sign when dividing by a negative number.

Question 384Question →

The quadratic equation $1.5x^2 - kx + 6 = 0$ , where $k$ is a positive constant, has exactly one real solution. What is the value of $k$ ?

Show answer & explanation

Answer: 6

Answer

6

For the quadratic equation

1.5x^2 - kx + 6 = 0

to have exactly one real solution, the discriminant

b^2 - 4ac

must equal

0

. Substituting

a = 1.5

,

b = -k

, and

c = 6

gives

(-k)^2 - 4(1.5)(6) = k^2 - 36 = 0

, which yields

k^2 = 36

. Since

k

must be a positive constant,

k

must be

6

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation

1.5x^2 - kx + 6 = 0

.

a = 1.5

,

b = -k

, and

c = 6

To use the discriminant formula, we need to know the values of

a

,

b

, and

c

from the standard form

ax^2 + bx + c = 0

.

2

Set the discriminant equal to zero.

b^2 - 4ac = 0

A quadratic equation has exactly one real solution if and only if its discriminant is equal to zero.

3

Substitute the coefficients into the discriminant formula and simplify.

k^2 - 36 = 0

Substituting

a = 1.5

,

b = -k

, and

c = 6

into the formula gives

(-k)^2 - 4(1.5)(6) = k^2 - 36 = 0

.

4

Solve the equation for the positive constant

k

.

k = 6

Solving

k^2 = 36

gives

k = 6

or

k = -6

. Since the problem states that

k

is a positive constant, we choose

k = 6

.

Key Concept

Determining the number of real solutions of a quadratic equation using the discriminant

Question 385Question →

In trapezoid $ABCD$ , the bases $AB$ and $CD$ are parallel. The diagonals $AC$ and $BD$ intersect at point $E$ . If the area of $\triangle ABE$ is $16\text{ cm}^2$ and the area of $\triangle CDE$ is $36\text{ cm}^2$ , what is the total area, in square centimeters, of trapezoid $ABCD$ ?

Show answer & explanation

Answer: 100

Answer

The total area of trapezoid

ABCD

is

100\text{ cm}^2

.

Triangles

ABE

and

CDE

are similar with an area ratio of

16:36

, yielding a side length ratio of

2:3

. The adjacent triangles

\triangle ADE

and

\triangle BCE

each share an altitude with the base triangles, making their areas equal to the geometric mean of the base areas:

\sqrt{16 \times 36} = 24\text{ cm}^2

each. Summing the four regions gives

16 + 36 + 24 + 24 = 100\text{ cm}^2

.

Step-by-Step Solution

1

Establish the similarity of triangles

ABE

and

CDE

.

\triangle ABE \sim \triangle CDE

Since bases

AB

and

CD

are parallel, alternate interior angles are congruent (

\angle EAB \cong \angle ECD

and

\angle EBA \cong \angle EDC

). By AA Similarity, the triangles are similar.

2

Calculate the linear scale factor between the similar triangles.

\frac{AE}{EC} = \frac{BE}{ED} = \sqrt{\frac{16}{36}} = \frac{2}{3}

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding side lengths.

3

Find the areas of the adjacent triangles

\triangle ADE

and

\triangle BCE

.

\text{Area}(\triangle ADE) = 24\text{ cm}^2

and

\text{Area}(\triangle BCE) = 24\text{ cm}^2

Triangles

ADE

and

CDE

share the same altitude from vertex

D

to diagonal

AC

, meaning the ratio of their areas is equal to the ratio of their bases:

\frac{\text{Area}(\triangle ADE)}{\text{Area}(\triangle CDE)} = \frac{AE}{EC} = \frac{2}{3} \implies \text{Area}(\triangle ADE) = \frac{2}{3} \times 36 = 24

. By similar logic,

\frac{\text{Area}(\triangle BCE)}{\text{Area}(\triangle ABE)} = \frac{EC}{AE} = \frac{3}{2} \implies \text{Area}(\triangle BCE) = \frac{3}{2} \times 16 = 24

.

4

Sum the areas of the four individual triangles.

\text{Area}(ABCD) = 16 + 36 + 24 + 24 = 100\text{ cm}^2

The total area of the trapezoid is the sum of the areas of the four non-overlapping triangles formed by its diagonals.

Key Concept

For any trapezoid with diagonals intersecting at

E

and parallel bases forming triangles of areas

A_1

and

A_2

, the other two triangles each have an area equal to

\sqrt{A_1 A_2}

, and the total area of the trapezoid is given by

(\sqrt{A_1} + \sqrt{A_2})^2

.

Question 386Question →

An angle in standard position has a measure of $\frac{5\pi}{6}$ radians. What is the degree measure of this angle?

Show answer & explanation

Answer: 150

Answer

The degree measure of the angle is 150.

To convert an angle from radians to degrees, multiply the radian measure by

\frac{180^\circ}{\pi}

. In this case, multiplying

\frac{5\pi}{6}

by

\frac{180^\circ}{\pi}

simplifies to

150^\circ

because the

\pi

terms cancel and

180

divided by

6

is

30

, which is then multiplied by

5

.

Step-by-Step Solution

1

Multiply the given radian measure by the conversion factor

\frac{180^\circ}{\pi}

to convert from radians to degrees.

\frac{5\pi}{6} \times \frac{180^\circ}{\pi}

One full rotation is

360^\circ

, which is equal to

2\pi

radians. Therefore,

180^\circ = \pi

radians, yielding the conversion factor

\frac{180^\circ}{\pi}

.

2

Simplify the expression by canceling out common terms.

150^\circ

The common term

\pi

cancels out from the numerator and denominator, leaving

\frac{5 \times 180^\circ}{6} = 5 \times 30^\circ = 150^\circ

.

Key Concept

Converting radian measures to degree measures

Question 387Question →

In the standard $(x, y)$ coordinate plane, a square has two opposite vertices at $(1, -3)$ and $(7, 5)$ . What is the area of the square, in square units?

Show answer & explanation

Answer: 50

Answer

The area of the square is 50 square units.

The length of the diagonal of the square is found using the distance formula between the two opposite vertices:

d = \sqrt{(7 - 1)^2 + (5 - (-3))^2} = \sqrt{6^2 + 8^2} = 10

. The area of a square can be calculated using its diagonal length

d

with the formula

\text{Area} = \frac{d^2}{2}

. Substituting

d = 10

gives

\text{Area} = \frac{10^2}{2} = 50

square units.

Step-by-Step Solution

1

Calculate the length of the diagonal of the square using the distance formula.

The diagonal length is

10

.

The distance between opposite vertices of a square represents the length of its diagonal.

2

Determine the area of the square using the diagonal length.

The area is

50

.

The area of a square with diagonal

d

is given by

\frac{d^2}{2}

.

Key Concept

Finding the area of a square on the coordinate plane using its diagonal.

Estimated Time:1m 30s

Question 388Question →

Two lines in a coordinate plane, $T_1$ and $T_2$ , are perpendicular to each other. Line $T_1$ has the equation $3x - 5y = 15$ , and line $T_2$ has the equation $ax + 9y = 20$ , where $a$ is a constant. What is the value of $a$ ?

Show answer & explanation

Answer: 15

Answer

The value of the constant

a

is 15.

The correct answer is 15. The slope of the line

3x - 5y = 15

is

\frac{3}{5}

. Since the two lines are perpendicular, the slope of the second line must be the negative reciprocal of

\frac{3}{5}

, which is

-\frac{5}{3}

. The slope of the line

ax + 9y = 20

is

-\frac{a}{9}

. Equating the two slopes yields

-\frac{a}{9} = -\frac{5}{3}

, which simplifies to

a = 15

.

Step-by-Step Solution

1

Find the slope of line

T_1

The slope of line

T_1

is

\frac{3}{5}

.

Writing the equation

3x - 5y = 15

in slope-intercept form

y = \frac{3}{5}x - 3

isolates the slope coefficient.

2

Find the perpendicular slope

The perpendicular slope is

-\frac{5}{3}

.

Perpendicular lines have slopes that are negative reciprocals of each other, so we invert the fraction and change the sign.

3

Determine the slope of line

T_2

in terms of

a

The slope of line

T_2

is

-\frac{a}{9}

.

Rewriting the equation

ax + 9y = 20

in slope-intercept form

y = -\frac{a}{9}x + \frac{20}{9}

isolates the slope coefficient.

4

Solve for

a

a = 15

Setting the slope of

T_2

equal to the perpendicular slope yields the equation

-\frac{a}{9} = -\frac{5}{3}

, which simplifies to

a = 15

.

Key Concept

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

Question 389Question →

A convex hexagon has interior angles with measures of $2x^\circ$ , $(3x - 15)^\circ$ , $(2x + 25)^\circ$ , $(3x + 10)^\circ$ , $(4x - 50)^\circ$ , and $(x + 30)^\circ$ . What is the degree measure of the smallest interior angle of this hexagon?

Show answer & explanation

Answer: 78

Answer

The degree measure of the smallest interior angle is 78.

The sum of the interior angles of a hexagon is calculated as

(6 - 2) \times 180^\circ = 720^\circ

. Adding the algebraic expressions for the six angles yields

15x = 720

, which gives

x = 48

. Substituting

x = 48

back into the expressions gives the angle measures of

96^\circ

,

129^\circ

,

121^\circ

,

154^\circ

,

142^\circ

, and

78^\circ

. The smallest angle is

78^\circ

.

Step-by-Step Solution

1

Calculate the sum of the interior angles of the hexagon.

Sum of interior angles = 720 degrees

The sum of the interior angles of any convex polygon with

n

sides is given by

(n - 2) \times 180^\circ

. For a hexagon (

n = 6

), the sum is

(6 - 2) \times 180^\circ = 720^\circ

.

2

Set up an algebraic equation by summing all the given angle expressions and equating them to 720.

2x + (3x - 15) + (2x + 25) + (3x + 10) + (4x - 50) + (x + 30) = 720

The sum of the actual measures of the interior angles must equal the calculated sum of 720 degrees.

3

Combine like terms to simplify the equation.

15x = 720

Grouping the

x

terms (

2x + 3x + 2x + 3x + 4x + x = 15x

) and the constant terms (

-15 + 25 + 10 - 50 + 30 = 0

) simplifies the expression.

4

Solve for

x

.

x = 48

Dividing both sides of the equation

15x = 720

by 15 isolates the variable

x

.

5

Substitute the value of

x

back into the angle expressions to identify the smallest angle.

The angles are

96^\circ

,

129^\circ

,

121^\circ

,

154^\circ

,

142^\circ

, and

78^\circ

. The smallest measure is

78^\circ

.

Evaluating each expression at

x = 48

determines the actual angle measures, from which the smallest can be chosen. Evaluating

(x + 30)^\circ

gives

48 + 30 = 78^\circ

, which is the minimum value.

Key Concept

The sum of the interior angles of a convex

n

-sided polygon is

(n - 2) \times 180^\circ

. Setting up and solving linear algebraic equations is required to determine unknown angle measures.

Estimated Time:1m 30s

Question 390Question →

What is the greatest integer value of $x$ that satisfies the inequality $\frac{7 - 5x}{3} - 4 \ge 2(x + 6)$ ?

Show answer & explanation

Answer: -4

Answer

The greatest integer value of

x

that satisfies the inequality is

-4

.

To find the greatest integer value of

x

that satisfies the inequality, solve the inequality algebraically. First, clear the fraction by multiplying all terms by 3:

7 - 5x - 12 \ge 6(x + 6)

. Simplify the left side to

-5x - 5

and distribute the right side to get

6x + 36

. Move variables to the left side by subtracting

6x

to get

-11x - 5 \ge 36

. Add 5 to both sides to get

-11x \ge 41

. Finally, divide by

-11

and reverse the inequality sign, yielding

x \le -\frac{41}{11}

, which is approximately

x \le -3.73

. The greatest integer less than or equal to

-3.73

is

-4

.

Step-by-Step Solution

1

Multiply the entire inequality by 3 to clear the fraction

7 - 5x - 12 \ge 6(x + 6)

Multiplying by the common denominator eliminates the fraction and simplifies further algebraic steps.

2

Simplify the left side and distribute the right side

-5x - 5 \ge 6x + 36

Combines constant terms on the left side and expands the parentheses on the right side.

3

Subtract

6x

from both sides

-11x - 5 \ge 36

Gathers all variable terms on the left side of the inequality.

4

Add 5 to both sides

-11x \ge 41

Isolates the variable term by moving the constant to the right side.

5

Divide by -11 and reverse the inequality sign

x \le -\frac{41}{11}

Isolates the variable

x

. Reversing the inequality sign is required when multiplying or dividing both sides by a negative number.

6

Determine the greatest integer satisfying the inequality

x \le -3.73

, so the greatest integer is

-4

Since

-\frac{41}{11} \approx -3.73

, the largest integer that is less than or equal to this value is

-4

.

Key Concept

Solving linear inequalities, applying the sign-flip rule when dividing by a negative number, and identifying integer boundaries.

Question 391Question →

A hiker starts at a trailhead, point $A$ , and walks $9$ miles due east, then $12$ miles due north to reach a campsite, point $C$ . A lookout tower, point $T$ , is located due west of the campsite $C$ . The straight-line distance from the starting point $A$ to the tower $T$ is $13$ miles. If the tower $T$ is located east of the north-south line passing through point $A$ , what is the distance, in miles, between the campsite $C$ and the tower $T$ ?

Show answer & explanation

Answer: 4

Answer

The distance between the campsite and the tower is 4 miles.

The correct answer is 4 miles. By modeling the hiker's path, the vertical height of both the campsite and the tower is 12 miles north of the starting point's east-west line. A right triangle is formed by the trailhead, the projection of the tower onto the east-west axis, and the tower itself. The hypotenuse is 13 miles and the vertical leg is 12 miles. By the Pythagorean theorem, the horizontal leg is 5 miles. Since the tower is east of the trailhead's north-south line, it is 5 miles east of the trailhead. The campsite is 9 miles east of the trailhead. The horizontal distance between the campsite and the tower is the difference: 9 - 5 = 4 miles.

Step-by-Step Solution

1

Determine the vertical height of the campsite and the tower.

The campsite

C

is

12

miles north of the trailhead

A

's east-west line. Since the tower

T

is located due west of

C

, it lies on the same horizontal line. Therefore, the vertical distance from the east-west line to the tower

T

is also

12

miles.

Points on the same due east-west line share the same vertical offset (latitude) from the reference axis.

2

Use the Pythagorean theorem to find the horizontal distance from the trailhead to the tower.

Let

D

be the point on the east-west line directly below the tower

T

. A right triangle

\triangle ADT

is formed where the vertical leg

DT = 12

miles, the hypotenuse

AT = 13

miles, and the horizontal leg is

AD

. Using the Pythagorean theorem:

AD^2 + DT^2 = AT^2 \implies AD^2 + 12^2 = 13^2 \implies AD^2 + 144 = 169 \implies AD^2 = 25 \implies AD = 5

miles.

Calculating the horizontal offset of the tower from the trailhead's north-south line.

3

Calculate the horizontal distance between the campsite and the tower.

The campsite

C

is

9

miles east of the trailhead

A

's north-south line, and the tower

T

is

5

miles east of it. The horizontal distance between them is the difference:

9 - 5 = 4

miles.

Since both points have the same vertical coordinate, the direct distance is simply the difference in their horizontal coordinates.

Key Concept

Using the Pythagorean theorem to solve multi-step geometry problems on a coordinate-like plane.

Estimated Time:1m 30s

Question 392Question →

A line with a slope of $2.5$ passes through the point $(3, 4)$ in the coordinate plane. What is the y-coordinate of the point on this line whose x-coordinate is $-1$ ?

Show answer & explanation

Answer: -6

Answer

-6

By applying the slope formula

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

with the slope equal to

2.5

, the known point at

(3, 4)

, and the target point at

(-1, y)

, we obtain the linear equation

2.5 = \frac{y - 4}{-1 - 3}

. Simplifying the denominator gives

2.5 = \frac{y - 4}{-4}

. Multiplying both sides by

-4

yields

-10 = y - 4

. Finally, adding

4

to both sides gives

y = -6

.

Step-by-Step Solution

1

Set up the slope formula relation

2.5 = \frac{y - 4}{-1 - 3}

The slope

m

of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is defined as the change in y-values divided by the change in x-values.

2

Solve for the unknown y-coordinate

y = -6

Simplifying the denominator to

-4

, multiplying both sides by

-4

gives

-10 = y - 4

, and adding

4

to both sides isolates

y

.

Key Concept

Finding a missing coordinate on a line given its slope and another coordinate point.

Question 393Question →

A kite has vertices $K(2, 8)$ , $I(6, 3)$ , $T(2, -4)$ , and $E(-2, 3)$ in the standard $(x, y)$ coordinate plane. What is the area of kite $KITE$ , in square units?

Show answer & explanation

Answer: 48

Answer

The area of kite

KITE

is 48 square units.

The area of kite

KITE

is 48 because the horizontal diagonal

IE

has a length of 8 units, the vertical diagonal

KT

has a length of 12 units, and the area of a kite is calculated as half the product of its diagonal lengths:

\frac{1}{2} \times 8 \times 12 = 48

.

Step-by-Step Solution

1

Calculate the lengths of the vertical diagonal

KT

and the horizontal diagonal

IE

.

The length of

KT

is 12 and the length of

IE

is 8.

The vertices

K(2, 8)

and

T(2, -4)

share the same

x

-coordinate, so the diagonal is vertical with length

8 - (-4) = 12

. The vertices

I(6, 3)

and

E(-2, 3)

share the same

y

-coordinate, so the diagonal is horizontal with length

6 - (-2) = 8

.

2

Apply the area formula for a kite:

\text{Area} = \frac{1}{2} d_1 d_2

.

Area = 48

Since the diagonals of a kite are perpendicular, the area is half the product of the lengths of the diagonals:

\frac{1}{2} \times 12 \times 8 = 48

.

Key Concept

Finding the area of a geometric figure on the coordinate plane by using diagonal lengths.

Question 394Question →

For an angle $\theta$ such that $\pi < \theta < \frac{3\pi}{2}$ , the value of $\cos\theta = -0.8$ . What is the value of $10\sin\theta$ ?

Show answer & explanation

Answer: -6

Answer

The value of

10\sin\theta

is

-6

.

The correct answer is

-6

. Substituting

\cos\theta = -0.8

into the fundamental Pythagorean identity

\sin^2\theta + \cos^2\theta = 1

yields

\sin^2\theta + 0.64 = 1

, which simplifies to

\sin^2\theta = 0.36

. Since the angle

\theta

lies in the interval

\pi < \theta < \frac{3\pi}{2}

(Quadrant III), its sine value must be negative. Thus,

\sin\theta = -0.6

. Multiplying this value by 10 gives

10\sin\theta = -6

.

Step-by-Step Solution

1

Use the Pythagorean identity

\sin^2\theta + \cos^2\theta = 1

with the given cosine value

\cos\theta = -0.8

.

\sin^2\theta + (-0.8)^2 = 1 \Rightarrow \sin^2\theta + 0.64 = 1 \Rightarrow \sin^2\theta = 0.36

The Pythagorean identity relates the sine and cosine values of any angle.

2

Take the square root of both sides, selecting the correct sign based on the quadrant constraint

\pi < \theta < \frac{3\pi}{2}

.

Since the angle lies in Quadrant III, the sine function must be negative. Thus,

\sin\theta = -\sqrt{0.36} = -0.6

.

In the third quadrant, y-coordinates on the unit circle are negative, meaning

\sin\theta

must be negative.

3

Multiply

\sin\theta

by 10 to get the final requested value.

10\sin\theta = 10 \times (-0.6) = -6

This calculation yields the final answer requested by the problem.

Key Concept

Applying the Pythagorean identity

\sin^2\theta + \cos^2\theta = 1

and choosing the correct sign based on the angle's quadrant.

Question 395Question →

An angle $\theta$ lies in the second quadrant and satisfies $\sin\theta = \frac{5}{13}$ . What is the value of $12\tan\theta$ ?

Show answer & explanation

Answer: -5

Answer

The value of

12\tan\theta

is -5.

Applying the Pythagorean identity

\sin^2\theta + \cos^2\theta = 1

with

\sin\theta = \frac{5}{13}

yields

\cos^2\theta = \frac{144}{169}

. Since the angle

\theta

lies in the second quadrant, its cosine is negative, meaning

\cos\theta = -\frac{12}{13}

. Using the quotient identity

\tan\theta = \frac{\sin\theta}{\cos\theta}

, we find

\tan\theta = -\frac{5}{12}

. Multiplying by 12 gives the correct value of -5.

Step-by-Step Solution

1

Use the Pythagorean identity to find the magnitude of

\cos\theta

.

\cos^2\theta = \frac{144}{169}

The Pythagorean identity states that

\sin^2\theta + \cos^2\theta = 1

. Substituting

\sin\theta = \frac{5}{13}

gives

\cos^2\theta = 1 - \left(\frac{5}{13}\right)^2

.

2

Determine the value of

\cos\theta

by applying the quadrant sign rule.

\cos\theta = -\frac{12}{13}

Since

\theta

is in the second quadrant, the cosine of

\theta

must be negative.

3

Calculate

\tan\theta

using the quotient identity.

\tan\theta = -\frac{5}{12}

The quotient identity is

\tan\theta = \frac{\sin\theta}{\cos\theta}

.

4

Multiply

\tan\theta

by 12 to find the required expression's value.

-5

Multiplying the value of

\tan\theta

(which is

-\frac{5}{12}

) by 12 yields

-5

.

Key Concept

Fundamental Trigonometric Identities

Estimated Time:1m 0s

Question 396Question →

A convex polygon has $n$ sides. The sum of the measures of $n-1$ of its interior angles is $2030^\circ$ . What is the measure, in degrees, of the remaining interior angle?

Show answer & explanation

Answer: 130

Answer

The measure of the remaining interior angle is 130 degrees.

The sum of the interior angles of a convex polygon must be a multiple of

180^\circ

. The multiple of

180^\circ

immediately greater than

2030^\circ

is

2160^\circ

(which is

12 \times 180^\circ

). The difference between the total sum and the sum of the

n-1

angles is

2160^\circ - 2030^\circ = 130^\circ

. Since

130^\circ

is less than

180^\circ

, this is a valid interior angle for a convex polygon.

Step-by-Step Solution

1

Write the formula for the total sum of the interior angles of a convex polygon.

The sum of the interior angles of a polygon with

n

sides is

(n-2) \times 180^\circ

.

This formula relates the number of sides to the total sum of the interior angles.

2

Apply the properties of convex polygons to set up an inequality for the total sum.

Since the remaining interior angle must be greater than

0^\circ

and less than

180^\circ

, the total sum of all

n

angles must satisfy

2030^\circ < (n-2) \times 180^\circ < 2030^\circ + 180^\circ

, which simplifies to

2030^\circ < (n-2) \times 180^\circ < 2210^\circ

.

By definition, every interior angle of a convex polygon is strictly less than

180^\circ

.

3

Solve the inequality to find the integer value of

n-2

.

Dividing the entire inequality by

180^\circ

gives

11.28 < n-2 < 12.28

. Since the number of sides

n

must be an integer,

n-2

must be the integer

12

.

A polygon must have a whole number of sides.

4

Calculate the total sum of all interior angles and find the remaining angle.

The total sum is

12 \times 180^\circ = 2160^\circ

. Subtracting the sum of the other

n-1

angles gives the remaining angle:

2160^\circ - 2030^\circ = 130^\circ

.

The difference between the total sum of all interior angles and the sum of the

n-1

angles is the measure of the final angle.

Key Concept

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, where each interior angle is strictly less than

180^\circ

.

Question 397Question →

An angle $\theta$ in standard position is coterminal with an angle of $-\frac{17\pi}{4}$ radians. If $0 \le \theta < 2\pi$ , what is the value of $\theta$ expressed as a decimal multiple of $\pi$ ? (For example, if the angle were $\frac{3\pi}{2} = 1.5\pi$ , the answer would be 1.5.)

Show answer & explanation

Answer: 1.75

Answer

1.75

To find a coterminal angle in the interval

[0, 2\pi)

for

-\frac{17\pi}{4}

, we add multiples of

2\pi

. Since

-\frac{17\pi}{4} = -4.25\pi

, we add

6\pi

(three full rotations) to get

-4.25\pi + 6\pi = 1.75\pi

. The multiple of

\pi

is therefore 1.75.

Step-by-Step Solution

1

Convert the coefficient of the given angle from a fraction to a decimal.

-\frac{17}{4} = -4.25

Converting the fraction to a decimal makes it easier to work with the addition of full rotations.

2

Add multiples of

2\pi

(which corresponds to adding 2 to the coefficient of

\pi

) to find a coterminal angle in the interval

[0, 2\pi)

.

-4.25 + 2 = -2.25

;

-2.25 + 2 = -0.25

;

-0.25 + 2 = 1.75

Adding

2\pi

representing full counterclockwise rotations on the unit circle results in a coterminal angle. We repeat this process until the coefficient lies in the interval

[0, 2)

.

3

Identify the coefficient of

\pi

for the coterminal angle.

1.75

The question asks for the angle as a decimal multiple of

\pi

, which is the coefficient of

\pi

in the expression

1.75\pi

.

Key Concept

Coterminal angles are angles in standard position that share the same terminal side. They can be found by adding or subtracting multiples of

2\pi

radians.

Estimated Time:1m 30s

Question 398Question →

In the standard $(x, y)$ coordinate plane, a line segment has endpoints at $P(2, 1)$ and $Q(8, 5)$ . Line $L$ is parallel to segment $PQ$ and passes through the point $(3, -2)$ . If the point $(9, y)$ also lies on line $L$ , what is the value of $y$ ?

Show answer & explanation

Answer: 2

Answer

The value of

y

is

2

.

The slope of segment

PQ

is

\frac{5 - 1}{8 - 2} = \frac{2}{3}

. Since line

L

is parallel to segment

PQ

, its slope is also

\frac{2}{3}

. The slope of line

L

through

(3, -2)

and

(9, y)

is given by

\frac{y - (-2)}{9 - 3} = \frac{y + 2}{6}

. Equating the two slopes yields

\frac{y + 2}{6} = \frac{2}{3}

, which simplifies to

y + 2 = 4

, so

y = 2

.

Step-by-Step Solution

1

Calculate the slope of segment

PQ

.

Slope of

PQ = \frac{2}{3}

Since parallel lines have equal slopes, finding the slope of the reference segment

PQ

is the first step in determining the slope of line

L

.

2

Set up the slope equation for line

L

using its parallel relationship to segment

PQ

.

\frac{y - (-2)}{9 - 3} = \frac{2}{3}

Because line

L

is parallel to segment

PQ

, its slope must also be

\frac{2}{3}

. The slope of line

L

is calculated using the points

(3, -2)

and

(9, y)

.

3

Solve the equation for

y

.

y = 2

Simplifying the numerator gives

y + 2

and the denominator gives

6

. Multiplying both sides of

\frac{y + 2}{6} = \frac{2}{3}

by

6

yields

y + 2 = 4

, which solves to

y = 2

.

Key Concept

Parallel lines have equal slopes in the coordinate plane.

Question 399Question →

A vertical flagpole is secured by two straight guide wires anchored to the flat ground on opposite sides of the pole. The first guide wire is $13$ feet long and its anchor is $5$ feet from the base of the pole. The second guide wire is anchored such that it makes a $30^\circ$ angle of elevation with the ground. If both guide wires are attached to the flagpole at the same height, what is the length, in feet, of the second guide wire?

Show answer & explanation

Answer: 24

Answer

The length of the second guide wire is

24

feet.

The correct answer is

24

. First, the height of the attachment point is found using the Pythagorean theorem:

h = \sqrt{13^2 - 5^2} = 12

feet. Since the second wire makes a

30^\circ

angle of elevation with the ground, it forms a 30-60-90 right triangle where the flagpole height of

12

feet is the leg opposite the

30^\circ

angle. The length of the wire is the hypotenuse of this triangle, which is twice the length of the opposite leg:

2 \times 12 = 24

feet.

Step-by-Step Solution

1

Use the Pythagorean theorem to calculate the height of the flagpole where the guide wires are attached.

The flagpole height is

12

feet.

The first guide wire, the flagpole, and the ground form a right triangle with a hypotenuse of

13

feet and a horizontal leg of

5

feet. Thus,

h = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = 12

.

2

Apply the properties of a 30-60-90 special right triangle to find the length of the second guide wire.

The length of the second guide wire is

24

feet.

The second wire forms a 30-60-90 right triangle with the flagpole and the ground. The angle of elevation is

30^\circ

, which means the side opposite this angle is the vertical height of the flagpole (

12

feet). In a 30-60-90 triangle, the hypotenuse (the wire length) is twice the length of the shorter leg (opposite the

30^\circ

angle), so the length is

2 \times 12 = 24

.

Key Concept

Applying the Pythagorean Theorem and the ratio properties of 30-60-90 special right triangles to solve multi-step geometry problems.

Question 400Question →

A projectile is launched vertically upward from an initial height of $5$ meters. Its height, $h(t)$ in meters, $t$ seconds after launch is given by the function $h(t) = -4.9t^2 + 19.6t + 5$ . To the nearest tenth of a second, how many seconds after launch does the projectile reach a height of $15$ meters on its way down?

Show answer & explanation

Answer: 3.4

Answer

To the nearest tenth of a second, the projectile reaches a height of

15

meters on its way down at

3.4

seconds.

The correct answer is

3.4

seconds. Setting the height equation

h(t) = 15

yields

-4.9t^2 + 19.6t - 10 = 0

. Solving this quadratic equation via the quadratic formula gives two solutions:

t \approx 0.6

seconds and

t \approx 3.4

seconds. The projectile travels upward first, passing the

15

-meter mark at

0.6

seconds, and then descends, passing the

15

-meter mark again at

3.4

seconds.

Step-by-Step Solution

1

Set up the quadratic equation by setting the height function

h(t)

equal to

15

.

-4.9t^2 + 19.6t + 5 = 15

This allows us to find the specific values of time

t

when the height of the projectile is exactly

15

meters.

2

Rearrange the quadratic equation into the standard form

at^2 + bt + c = 0

by subtracting

15

from both sides.

-4.9t^2 + 19.6t - 10 = 0

Writing the equation in standard form is necessary before applying the quadratic formula.

3

Substitute the coefficients

a = -4.9

,

b = 19.6

, and

c = -10

into the quadratic formula.

t = \frac{-19.6 \pm \sqrt{(19.6)^2 - 4(-4.9)(-10)}}{2(-4.9)}

Since the quadratic equation has non-integer decimal coefficients, using the quadratic formula is the most reliable method to solve for the roots.

4

Simplify the discriminant and calculate the two values of

t

.

t \approx 0.6

and

t \approx 3.4

The discriminant is

19.6^2 - 196 = 188.16

. Taking the square root gives

\sqrt{188.16} \approx 13.72

, resulting in two real roots.

5

Determine which root corresponds to the projectile's motion on the way down.

t \approx 3.4

seconds

The smaller root (

0.6

seconds) represents the first time the projectile reaches

15

meters while ascending. The larger root (

3.4

seconds) represents the time the projectile passes

15

meters while descending.

Key Concept

Solving quadratic equations with decimal coefficients using the quadratic formula and interpreting the physical context of the roots.

Answer

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