All practice questions

Question 2561Question →

An angle in standard position measures $\frac{2\pi}{3}$ radians. If the angle is increased by $45^\circ$ , what is the measure of the new angle, in radians?

\frac{3\pi}{7}

\frac{14\pi}{3}

\frac{12\pi}{11}

\frac{11\pi}{12}

\frac{3\pi}{4}

Show answer & explanation

Answer: $\frac{11\pi}{12}$

Answer

The correct answer is

\frac{11\pi}{12}

To find the final angle measure, first convert the rotation angle of

45^\circ

into radians. Since

180^\circ = \pi

radians, multiplying

45^\circ

by

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

gives

\frac{\pi}{4}

radians. Next, add the initial angle and the rotation:

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

. Finding a common denominator of

12

, the sum is

\frac{8\pi}{12} + \frac{3\pi}{12} = \frac{11\pi}{12}

radians.

Step-by-Step Solution

1

Convert the rotation angle from degrees to radians.

45^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{4}

radians

To perform the addition, both angle measures must be in the same unit (radians).

2

Add the two radian measures.

\frac{2\pi}{3} + \frac{\pi}{4} = \frac{8\pi}{12} + \frac{3\pi}{12} = \frac{11\pi}{12}

radians

An increase in angle measure corresponds to counterclockwise rotation, which means adding the two angles.

Key Concept

Converting between degrees and radians and adding angles in standard position.

Estimated Time:45s

Question 2562Question →

A wheelchair ramp is constructed in two consecutive straight segments. The first segment rises at a $30^\circ$ angle relative to the flat ground and has a length of $12$ feet. The second segment starts at the end of the first segment and rises at a $45^\circ$ angle relative to the horizontal, with a length of $8\sqrt{2}$ feet. What is the total vertical rise, in feet, from the start of the first segment to the end of the second segment?

22

6 + 8\sqrt{2}

14

8 + 6\sqrt{3}

16 + 6\sqrt{3}

Show answer & explanation

Answer: $14$

Answer

The total vertical rise is

14

feet.

The total vertical rise is the sum of the vertical rises of the two individual segments. The first segment forms a

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

right triangle where the hypotenuse is

12

feet, so the vertical rise (the leg opposite the

30^\circ

angle) is half of the hypotenuse, which is

6

feet. The second segment forms a

45^\circ-45^\circ-90^\circ

right triangle where the hypotenuse is

8\sqrt{2}

feet, so the vertical rise (the leg opposite the

45^\circ

angle) is the hypotenuse divided by

\sqrt{2}

, which is

8

feet. Adding these two values gives a total vertical rise of

14

feet.

Step-by-Step Solution

1

Calculate the vertical rise of the first segment.

The first segment has a length of

12

feet and rises at a

30^\circ

angle. It forms a

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

right triangle where the vertical rise is the side opposite the

30^\circ

angle. Since the leg opposite the

30^\circ

angle is half the hypotenuse, the vertical rise is

\frac{12}{2} = 6

feet.

To find the vertical component of the first ramp segment.

2

Calculate the vertical rise of the second segment.

The second segment has a length of

8\sqrt{2}

feet and rises at a

45^\circ

angle. It forms a

45^\circ-45^\circ-90^\circ

right triangle where the leg length is the hypotenuse divided by

\sqrt{2}

. Thus, the vertical rise is

\frac{8\sqrt{2}}{\sqrt{2}} = 8

feet.

To find the vertical component of the second ramp segment.

3

Sum the vertical rises of both segments.

The total vertical rise is

6\text{ feet} + 8\text{ feet} = 14

feet.

To find the combined vertical height gained over the entire ramp.

Key Concept

Using special right triangle ratios (

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

and

45^\circ-45^\circ-90^\circ

) to find missing side lengths.

Estimated Time:1m 30s

Question 2563Question →

An equilateral triangle $ABC$ has a side length of $8$ inches. Point $D$ lies on side $BC$ such that the distance from $B$ to $D$ is $3$ inches. What is the length, in inches, of the segment $AD$ ?

Show answer & explanation

Answer: 7

Answer

The length of segment

AD

is

7

inches.

Dropping altitude

AM

from

A

to

BC

divides the equilateral triangle into two

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

right triangles. Since

M

is the midpoint of

BC

,

BM = 4

inches. In

\triangle ABM

, the hypotenuse is

8

and the shorter leg is

4

, so the altitude

AM = 4\sqrt{3}

inches. Since

BD = 3

inches, the segment

DM

has length

BM - BD = 4 - 3 = 1

inch. Applying the Pythagorean Theorem to right triangle

\triangle ADM

gives

AD^2 = AM^2 + DM^2 = (4\sqrt{3})^2 + 1^2 = 48 + 1 = 49

, which simplifies to

AD = 7

inches.

Step-by-Step Solution

1

Find the midpoint of side

BC

by dropping altitude

AM

.

BM = 4

inches

In an equilateral triangle, the altitude to a side bisects that side.

2

Calculate the length of the altitude

AM

.

AM = 4\sqrt{3}

inches

The altitude forms a

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

triangle with the hypotenuse of

8

inches, making the altitude length equal to

8 \times \frac{\sqrt{3}}{2} = 4\sqrt{3}

.

3

Determine the length of the segment

DM

.

DM = 1

inch

Since

D

is

3

inches from

B

and

M

is

4

inches from

B

, the remaining distance is

4 - 3 = 1

.

4

Apply the Pythagorean Theorem on right triangle

\triangle ADM

to find

AD

.

AD = 7

inches

The hypotenuse squared is the sum of the squares of the legs:

AD^2 = (4\sqrt{3})^2 + 1^2 = 48 + 1 = 49

, which gives

AD = 7

.

Key Concept

Using the altitude of an equilateral triangle to create special right triangles and applying the Pythagorean Theorem.

Question 2564Question →

Match each angle measure in degrees on the left to its equivalent angle measure in radians on the right.

Click a left item, then click its matching right item

Show answer & explanation

Answer

The degree measures

30^\circ

,

45^\circ

,

60^\circ

, and

90^\circ

correspond to

\frac{\pi}{6}

,

\frac{\pi}{4}

,

\frac{\pi}{3}

, and

\frac{\pi}{2}

radians, respectively.

Each degree measure matches its correct radian value by multiplying the degree measure by

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

and simplifying the fraction.

Step-by-Step Solution

1

Apply the degree-to-radian conversion formula.

Multiply each degree measure by the conversion factor

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

.

A full circle has

360^\circ

or

2\pi

radians, meaning

180^\circ = \pi

radians. Therefore, the conversion factor from degrees to radians is

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

.

2

Simplify the resulting fractions.

30^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{6}

,

45^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{4}

,

60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3}

, and

90^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{2}

.

Reducing the fractions by dividing the numerator and denominator by their greatest common factor gives the simplified radian values.

Key Concept

Converting degree measures to radian measures on the unit circle

Estimated Time:45s

Question 2565Question →

On a coordinate map of a harbor, a lighthouse is located at $L(-4, 9)$ and a dock is located at $D(8, -7)$ . A buoy is positioned at the midpoint of the straight-line segment connecting the lighthouse and the dock. If a boat is anchored at $B(-1, 5)$ , what is the distance, in coordinate units, between the boat and the buoy?

Show answer & explanation

Answer: 5

Answer

The distance between the boat and the buoy is 5 units.

To find the distance between the boat and the buoy, first determine the location of the buoy. Since the buoy is at the midpoint of the segment connecting the lighthouse at

L(-4, 9)

and the dock at

D(8, -7)

, we use the midpoint formula:

M = \left(\frac{-4 + 8}{2}, \frac{9 + (-7)}{2}\right) = (2, 1)

. Next, find the distance between the boat at

B(-1, 5)

and the buoy at

M(2, 1)

using the distance formula:

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

.

Step-by-Step Solution

1

Find the coordinates of the buoy, which is the midpoint

M(x_m, y_m)

of the segment connecting the lighthouse

L(-4, 9)

and the dock

D(8, -7)

.

M(2, 1)

The midpoint formula is given by

x_m = \frac{x_1 + x_2}{2}

and

y_m = \frac{y_1 + y_2}{2}

. Substituting the coordinates of

L

and

D

, we get

x_m = \frac{-4 + 8}{2} = 2

and

y_m = \frac{9 + (-7)}{2} = 1

.

2

Calculate the distance

d

between the boat at

B(-1, 5)

and the buoy at

M(2, 1)

.

5

units

The distance formula is given by

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

. Substituting the coordinates of

B

and

M

yields

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

.

Key Concept

Applying the midpoint and distance formulas sequentially to solve a coordinate geometry word problem.

Question 2566Question →

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 2567Question →

A parabolic arch is modeled by the equation $y = (x - 3)^2 - 5$ in the standard $(x, y)$ coordinate plane. A straight pathway is modeled by a line where the $y$ -coordinate of any point is 2 less than its $x$ -coordinate. If the pathway intersects the arch at points $A$ and $B$ , what is the area, in square units, of the triangle with vertices at $A$ , $B$ , and the origin $(0, 0)$ ?

3

5

7

8

10

Show answer & explanation

Answer: 5

Answer

The area of the triangle with vertices at the intersection points and the origin is 5 square units.

To find the area of the triangle, we first solve the system of equations. Substituting the pathway's equation y = x - 2 into the parabola's equation y = (x - 3)^2 - 5 yields x^2 - 7x + 6 = 0, which factors to (x - 6)(x - 1) = 0. This gives x = 6 and x = 1. The corresponding y-coordinates are y = 4 and y = -1, representing the intersection points (6, 4) and (1, -1). The area of the triangle with these vertices and the origin (0, 0) is calculated as 0.5 * |6(-1) - 4(1)| = 5.

Step-by-Step Solution

1

Set up the system of equations by substituting the linear equation into the quadratic equation.

(x - 3)^2 - 5 = x - 2

The pathway is described as having a y-coordinate that is 2 less than the x-coordinate, which translates to the linear equation y = x - 2. Substituting this into the parabola's equation allows us to find the intersection points.

2

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

x^2 - 7x + 6 = 0

Expanding

(x - 3)^2

gives

x^2 - 6x + 9

. Subtracting

x

and adding

2

to both sides results in standard quadratic form.

3

Factor the quadratic equation to solve for the x-coordinates.

(x - 6)(x - 1) = 0

, so

x = 6

or

x = 1

Factoring allows us to find the x-values that satisfy the intersection condition.

4

Substitute the x-values back into the linear equation to find the corresponding y-coordinates.

For

x = 6

,

y = 4

giving point

(6, 4)

. For

x = 1

,

y = -1

giving point

(1, -1)

.

The intersection points must satisfy both equations in the system.

5

Calculate the area of the triangle with vertices

(0, 0)

,

(6, 4)

, and

(1, -1)

using the coordinate area formula.

Area =

\frac{1}{2} |6(-1) - 4(1)| = \frac{1}{2} |-10| = 5

square units.

The area of a triangle with one vertex at the origin and others at

(x_1, y_1)

and

(x_2, y_2)

is

\frac{1}{2} |x_1 y_2 - y_1 x_2|

.

Key Concept

Solving systems of linear and quadratic equations and finding the area of a triangle in the coordinate plane.

Question 2568Question →

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 2569Question →

A water tank is being drained at a constant rate. After $3$ hours of draining, the tank contains $120$ gallons of water. After $5$ hours of draining, the tank contains $80$ gallons of water. If the volume of water in the tank, $y$ (in gallons), is modeled as a linear function of the time spent draining, $x$ (in hours), what is the slope of the line representing this function in the standard $(x, y)$ coordinate plane?

20

-\frac{1}{20}

-20

-11

53

Show answer & explanation

Answer: $-20$

Answer

-20

The correct answer is

-20

. The slope

m

of a line passing through two points

(x_1, y_1)

and

(x_2, y_2)

is given by the formula

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

. Identifying the points from the problem as

(3, 120)

and

(5, 80)

, we calculate the slope as

m = \frac{80 - 120}{5 - 3} = \frac{-40}{2} = -20

. This represents a constant rate of change of

-20

gallons per hour.

Step-by-Step Solution

1

Identify the coordinates of the two data points from the problem statement.

The two points on the line are

(3, 120)

and

(5, 80)

, where

x

represents the time in hours and

y

represents the volume of water in gallons.

To calculate the slope of a line, we first need to identify the coordinate points

(x_1, y_1)

and

(x_2, y_2)

that lie on the line.

2

Substitute the coordinates into the slope formula,

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

.

m = \frac{80 - 120}{5 - 3}

The slope represents the constant rate of change, which is the change in the dependent variable (

y

) divided by the change in the independent variable (

x

).

3

Simplify the expression to find the final value of the slope.

m = \frac{-40}{2} = -20

Performing the subtraction and division gives the slope of the line, showing that the volume of water decreases by

20

gallons per hour.

Key Concept

The slope of a line, representing a constant rate of change, is calculated using the formula

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

for any two points

(x_1, y_1)

and

(x_2, y_2)

on the line.

Estimated Time:1m 0s

Question 2570Question →

A convex pentagon has one interior angle that measures $108^\circ$ . The remaining four interior angles have measures in the ratio $3:4:5:6$ . What is the measure of the largest interior angle of this pentagon?

84^\circ

120^\circ

144^\circ

162^\circ

180^\circ

Show answer & explanation

Answer: $144^\circ$

Answer

The measure of the largest interior angle of the pentagon is

144^\circ

.

The correct answer is the option stating

144^\circ

. To find this, we calculate the sum of all interior angles of a pentagon using

(5-2) \times 180^\circ = 540^\circ

. Subtracting the known angle of

108^\circ

leaves

432^\circ

for the remaining four angles. The ratio

3:4:5:6

implies these angles can be represented as

3x

,

4x

,

5x

, and

6x

, summing to

18x

. Solving

18x = 432

gives

x = 24

. The largest angle is

6x = 6 \times 24^\circ = 144^\circ

.

Step-by-Step Solution

1

Calculate the sum of the interior angles of a pentagon.

The sum of the interior angles of any convex pentagon (

n = 5

) is

(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ

.

Before finding individual angles, we must determine the total sum of all interior angles in the polygon.

2

Determine the sum of the remaining four angles.

The remaining four angles sum to

540^\circ - 108^\circ = 432^\circ

.

We subtract the measure of the known angle from the total sum to find the portion allocated to the remaining angles.

3

Set up and solve an algebraic equation using the given ratio.

Let the four remaining angles be represented as

3x

,

4x

,

5x

, and

6x

. Their sum is

3x + 4x + 5x + 6x = 18x

. Setting this equal to the remaining sum gives

18x = 432

, which simplifies to

x = 24

.

Using the ratio allows us to define the relative sizes of the remaining angles in terms of a single variable, which we can solve for using their sum.

4

Find the measure of the largest interior angle.

The largest of these four angles is represented by

6x

, which is

6 \times 24^\circ = 144^\circ

. Since

144^\circ

is also larger than the given angle of

108^\circ

, it is the largest interior angle of the pentagon.

We multiply the value of

x

by the largest coefficient in the ratio and compare it to the other given angle to identify the maximum measure.

Key Concept

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

. For an irregular polygon, when some angles are known and others are in a ratio, we subtract the known angles from the total sum and distribute the remaining sum proportionally according to the ratio.

Estimated Time:1m 30s

Question 2571Question →

For the quadratic equation $0.25x^2 - 1.5x + c = 0$ , the discriminant is equal to $4$ . What is the value of the constant $c$ ?

-7.00

-6.25

-1.75

1.75

6.25

Show answer & explanation

Answer: $-1.75$

Answer

The value of the constant

c

is

-1.75

.

The correct option is

-1.75

. To find the value of

c

, substitute the coefficients

a = 0.25

,

b = -1.5

, and the discriminant

D = 4

into the formula

D = b^2 - 4ac

. This gives

4 = (-1.5)^2 - 4(0.25)c

, which simplifies to

4 = 2.25 - c

. Solving for

c

yields

c = 2.25 - 4 = -1.75

.

Step-by-Step Solution

1

Identify the values of the coefficients from the quadratic equation

0.25x^2 - 1.5x + c = 0

.

The coefficients are

a = 0.25

,

b = -1.5

, and the constant is

c

.

These coefficients are required to compute the discriminant.

2

Recall the formula for the discriminant

D

and substitute the known values, including the given discriminant

D = 4

.

4 = (-1.5)^2 - 4(0.25)c

This sets up an equation to solve for the unknown constant

c

.

3

Simplify the squared term and the multiplication of the coefficients.

4 = 2.25 - 1c

, which simplifies to

4 = 2.25 - c

.

Squaring

-1.5

yields

2.25

, and

4(0.25) = 1

.

4

Solve the linear equation for

c

.

c = -1.75

Subtracting

2.25

from both sides gives

1.75 = -c

, which means

c = -1.75

.

Key Concept

Using the discriminant formula

D = b^2 - 4ac

to solve for an unknown coefficient in a quadratic equation.

Estimated Time:1m 30s

Question 2572Question →

A circular pizza with a radius of $8\text{ inches}$ is cut into slices. If one slice has a central angle of $45^\circ$ , what is the area, in square inches, of this slice?

2\pi

8\pi

16\pi

56\pi

64\pi

Show answer & explanation

Answer: $8\pi$

Answer

The area of the slice is

8\pi

square inches.

The area of a sector is found by multiplying the circle's total area by the fraction of the circle that the sector represents. With a radius of

8

inches, the total area is

\pi \times 8^2 = 64\pi

square inches. Since the slice has a central angle of

45^\circ

, it represents

\frac{45}{360} = \frac{1}{8}

of the entire pizza. Multiplying the total area by this fraction gives

\frac{1}{8} \times 64\pi = 8\pi

square inches.

Step-by-Step Solution

1

Calculate the total area of the circular pizza using the area formula

A = \pi r^2

with

r = 8

.

The total area of the pizza is

\pi \times 8^2 = 64\pi

square inches.

To find the area of a sector, we first need to determine the area of the entire circle.

2

Find the fraction of the circle represented by the slice's central angle by dividing

45^\circ

by

360^\circ

.

The fraction is

\frac{45}{360} = \frac{1}{8}

.

A full circle has

360^\circ

, so the ratio of the central angle to

360^\circ

gives the proportion of the circle's total area that the sector occupies.

3

Multiply the total area of the pizza by the fraction representing the slice.

The area of the slice is

\frac{1}{8} \times 64\pi = 8\pi

square inches.

Multiplying the total circle area by the sector's fraction yields the sector's area.

Key Concept

The area of a sector is proportional to its central angle and can be found using the formula

A = \frac{\theta}{360} \pi r^2

when the angle is in degrees.

Question 2573Question →

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 2574Question →

In the standard $(x, y)$ coordinate plane, three vertices of a rhombus are $(1, 2)$ , $(5, 5)$ , and $(9, 2)$ . If the $y$ -coordinate of the fourth vertex is less than 2, what are the coordinates of the fourth vertex?

(5, -1)

(13, 5)

(13, -1)

(6, -2)

(-3, 5)

Show answer & explanation

Answer: $(5, -1)$

Answer

The coordinates of the fourth vertex are

(5, -1)

.

The correct answer is the coordinate pair

(5, -1)

. Since opposite sides of a rhombus are parallel and congruent, the vector from

(5, 5)

to

(1, 2)

is

(-4, -3)

. Applying this vector to

(9, 2)

gives

(9 - 4, 2 - 3) = (5, -1)

. This point satisfies the requirement that the

y

-coordinate is less than 2, and all four side lengths are exactly 5.

Step-by-Step Solution

1

Calculate the vector translation between two adjacent vertices of the rhombus.

The vector from vertex

(5, 5)

to vertex

(1, 2)

is

(1 - 5, 2 - 5) = (-4, -3)

.

In a rhombus, opposite sides must be parallel and equal in length, meaning the translation from one vertex to another on one side must equal the translation on the opposite side.

2

Apply the translation vector to the third vertex to find the fourth vertex.

Applying the vector

(-4, -3)

to the vertex

(9, 2)

yields

(9 - 4, 2 - 3) = (5, -1)

.

This determines the coordinates of the fourth vertex that completes the parallelogram structure.

3

Verify that the resulting vertex satisfies all conditions of the problem.

The

y

-coordinate of

(5, -1)

is

-1

, which is less than 2. The side lengths are all equal to 5:

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

and

\sqrt{(5-9)^2 + (-1-2)^2} = 5

.

This confirms that the figure is a rhombus and satisfies the

y

-coordinate constraint.

Key Concept

Using vector translations and distance formulas to determine coordinates of geometric figures on the coordinate plane.

Question 2575Question →

What is the complete set of real values of $x$ that satisfy the inequality $\frac{3}{4} - \frac{2x}{3} > 2$ ?

x < -\frac{15}{8}

x > -\frac{15}{8}

x < \frac{1}{2}

x > \frac{1}{2}

x > 24

Show answer & explanation

Answer: $x < -\frac{15}{8}$

Answer

x < -\frac{15}{8}

Subtracting

\frac{3}{4}

from both sides of the inequality gives

-\frac{2x}{3} > \frac{5}{4}

. Multiplying both sides by

-\frac{3}{2}

and reversing the inequality sign from greater than (

>

) to less than (

<

) yields

x < -\frac{15}{8}

.

Step-by-Step Solution

1

Subtract

\frac{3}{4}

from both sides of the inequality.

-\frac{2x}{3} > \frac{5}{4}

To isolate the term containing the variable

x

on one side of the inequality.

2

Multiply both sides of the inequality by

-\frac{3}{2}

and reverse the inequality sign.

x < -\frac{15}{8}

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

Key Concept

Solving linear inequalities by isolating the variable and applying the sign reversal rule when multiplying or dividing by a negative value.

Question 2576Question →

In the standard $(x, y)$ coordinate plane, line $L_1$ passes through the points $(k, 5)$ and $(4, -1)$ . Line $L_2$ is perpendicular to line $L_1$ and is represented by the equation $3x - 2y = 6$ . What is the value of $k$ ?

-2

0

-5

8

13

Show answer & explanation

Answer: -5

Answer

-5

The correct answer is

-5

. First, find the slope of line

L_2

by rewriting the equation

3x - 2y = 6

in slope-intercept form:

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

, which gives a slope of

\frac{3}{2}

. Because line

L_1

is perpendicular to line

L_2

, its slope must be the negative reciprocal of

\frac{3}{2}

, which is

-\frac{2}{3}

. Using the slope formula with the points

(k, 5)

and

(4, -1)

gives

\frac{-1 - 5}{4 - k} = -\frac{2}{3}

. Solving this equation gives

\frac{-6}{4 - k} = -\frac{2}{3}

, which simplifies to

18 = 8 - 2k

, leading to

k = -5

.

Step-by-Step Solution

1

Convert the equation of line

L_2

to slope-intercept form to find its slope.

3x - 2y = 6 \implies -2y = -3x + 6 \implies y = \frac{3}{2}x - 3

. Thus, the slope of

L_2

is

\frac{3}{2}

.

Converting to slope-intercept form (

y = mx + b

) isolates the slope as the coefficient of

x

.

2

Find the slope of line

L_1

using the relationship between perpendicular lines.

The slope of

L_1

is the negative reciprocal of

\frac{3}{2}

, which is

-\frac{2}{3}

.

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

3

Use the slope formula with the points

(k, 5)

and

(4, -1)

to solve for

k

.

\frac{-1 - 5}{4 - k} = -\frac{2}{3} \implies \frac{-6}{4 - k} = -\frac{2}{3} \implies 18 = 2(4 - k) \implies 18 = 8 - 2k \implies 2k = -10 \implies k = -5

.

Setting the calculated slope of

L_1

equal to the slope formula expression allows us to solve for the missing coordinate parameter.

Key Concept

Finding a coordinate parameter by using the negative reciprocal relationship between the slopes of perpendicular lines.

Question 2577Question →

A central angle of a circle measures $315^\circ$ . What is the radian measure of this angle?

\frac{4}{7\pi}

\frac{4\pi}{7}

\frac{2\pi}{3}

\frac{7\pi}{8}

\frac{7\pi}{4}

Show answer & explanation

Answer: $\frac{7\pi}{4}$

Answer

\frac{7\pi}{4}

To convert degrees to radians, multiply the degree measure by

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

. Multiplying

315^\circ

by

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

gives

\frac{315\pi}{180}

. Simplifying the fraction by dividing the numerator and the denominator by their greatest common divisor, 45, results in

\frac{7\pi}{4}

radians.

Step-by-Step Solution

1

Set up the conversion from degrees to radians.

Multiply

315^\circ

by the conversion factor

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

.

The conversion factor from degrees to radians is

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

because a straight angle of

180^\circ

is equivalent to

\pi

radians.

2

Perform the multiplication and simplify the resulting fraction.

315 \times \frac{\pi}{180} = \frac{315\pi}{180} = \frac{7\pi}{4}

.

Dividing both the numerator 315 and the denominator 180 by their greatest common divisor, 45, yields the simplified fraction

\frac{7}{4}

.

Key Concept

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

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

and simplify the resulting fraction.

Estimated Time:45s

Question 2578Question →

In the figure below, quadrilateral $ABCD$ is divided by diagonal $BD$ into two right triangles. In right triangle $ABD$ , the angle at $B$ is a right angle, $\angle ADB = 30^\circ$ , and the hypotenuse $AD = 12$ centimeters. In right triangle $BCD$ , the angle at $C$ is a right angle, and $BC = CD$ . What is the perimeter, in centimeters, of quadrilateral $ABCD$ ?

18 + 6\sqrt{3}

18 + 12\sqrt{3}

18 + 6\sqrt{6}

18 + 12\sqrt{6}

12 + 6\sqrt{3} + 6\sqrt{2}

Show answer & explanation

Answer: $18 + 6\sqrt{6}$

Answer

The perimeter of the quadrilateral is

18 + 6\sqrt{6}

centimeters.

The perimeter of quadrilateral

ABCD

is the sum of the lengths of its four outer boundary sides:

AB

,

BC

,

CD

, and

DA

. In the

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

right triangle

ABD

, the shorter leg

AB

is half the hypotenuse

AD

, so

AB = 6\text{ cm}

. The longer leg

BD

is

AB\sqrt{3} = 6\sqrt{3}\text{ cm}

. In the isosceles right triangle

BCD

, the hypotenuse is

BD = 6\sqrt{3}\text{ cm}

. The legs

BC

and

CD

are congruent, with each length equal to the hypotenuse divided by

\sqrt{2}

, which simplifies to

3\sqrt{6}\text{ cm}

. Adding the four outer side lengths (

6 + 3\sqrt{6} + 3\sqrt{6} + 12

) yields a perimeter of

18 + 6\sqrt{6}\text{ cm}

.

Step-by-Step Solution

1

Find the lengths of the legs of right triangle

ABD

using the properties of a

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

right triangle.

AB = 6\text{ cm}

and

BD = 6\sqrt{3}\text{ cm}

In a

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

triangle, the leg opposite the

30^\circ

angle is half the length of the hypotenuse (

AB = \frac{12}{2} = 6

), and the leg opposite the

60^\circ

angle is

\sqrt{3}

times the shorter leg (

BD = 6\sqrt{3}

).

2

Find the lengths of the legs of the isosceles right triangle

BCD

(

45^\circ-45^\circ-90^\circ

) using the hypotenuse

BD

.

BC = CD = 3\sqrt{6}\text{ cm}

In a

45^\circ-45^\circ-90^\circ

triangle, the length of each leg is the hypotenuse divided by

\sqrt{2}

. Thus,

BC = CD = \frac{6\sqrt{3}}{\sqrt{2}} = \frac{6\sqrt{3}\cdot\sqrt{2}}{2} = 3\sqrt{6}

.

3

Calculate the perimeter of quadrilateral

ABCD

by summing the lengths of its four outer boundary sides:

AB

,

BC

,

CD

, and

DA

.

Perimeter

= 6 + 3\sqrt{6} + 3\sqrt{6} + 12 = 18 + 6\sqrt{6}\text{ cm}

The perimeter is the sum of the outer boundary sides of the quadrilateral, which are

AB

,

BC

,

CD

, and

DA

.

Key Concept

Pythagorean Theorem and Special Right Triangles

Alternative Method

Instead of using special right triangle formulas, the Pythagorean theorem can be used with variables:

AB^2 + BD^2 = AD^2

, where

AB = \frac{1}{2}AD = 6

, so

36 + BD^2 = 144 \implies BD = \sqrt{108} = 6\sqrt{3}

. Then

BC^2 + CD^2 = BD^2 \implies 2BC^2 = 108 \implies BC = \sqrt{54} = 3\sqrt{6}

.

Estimated Time:1m 30s

Question 2579Question →

Ancient Lunar Magnetic Field Models

The Moon currently lacks a global magnetic field, but magnetized crustal rocks indicate it possessed one between $4.25$ and $3.56\text{ billion years ago}$ ( $\text{Ga}$ ). Two models are proposed to explain this ancient lunar dynamo:

*Model 1 (Thermal-Compositional Dynamo)*
This model proposes that the lunar dynamo was driven by thermal and compositional convection within a liquid metallic core. As the core cooled, solid iron crystallized at the center, releasing lighter elements (such as sulfur) into the remaining liquid outer core. The buoyant rise of these lighter elements, combined with heat loss, drove fluid convection that maintained a continuous, stable magnetic field of approximately $100\text{ }\mu\text{T}$ for nearly $700\text{ million years}$ , ending only when core crystallization was complete.

*Model 2 (Mechanical Impact Dynamo)*
This model proposes that the lunar dynamo was not continuously active but was periodically restarted by massive basin-forming impacts. A giant impactor would transfer immense angular momentum to the Moon’s mantle, temporarily altering its rotation rate relative to the liquid core. This differential rotation at the core-mantle boundary generated shear and turbulence in the liquid core, initiating a dynamo. Each dynamo event lasted only $10\text{ to }20\text{ million years}$ before friction synchronized the rotation of the mantle and core, extinguishing the magnetic field until the next major impact.

Statement: According to Model 2, a lunar crustal rock that crystallized continuously over a $50\text{-million-year}$ period of impact quiescence (a time with no large impacts) would record a strong, steadily active global magnetic field throughout its entire formation.

True

False

Show answer & explanation

Answer: False

Answer

The statement is false because Model 2 claims that the lunar dynamo is transient and only operates for

10\text{ to }20\text{ million years}

following a massive impact. During a

50\text{-million-year}

period of impact quiescence, the dynamo would be inactive for the majority of the time, so a rock forming continuously throughout this period would not record a steady magnetic field.

The correct answer is false because Model 2 clearly states that the magnetic field is a temporary phenomenon triggered by impacts, lasting at most

20\text{ million years}

. A

50\text{-million-year}

window without impacts would leave the Moon without a global magnetic field for at least the latter

30\text{ million years}

of that interval, meaning any rock forming continuously throughout this duration would not record a steady field.

Step-by-Step Solution

1

Identify the mechanism and duration of the magnetic field proposed under Model 2.

Model 2 asserts that the lunar dynamo is not continuous but is periodically restarted by impacts, with each event lasting only

10\text{ to }20\text{ million years}

before core-mantle synchronization extinguishes the field.

This establishes the temporal constraints of the dynamo activity according to the second model.

2

Compare the proposed duration of a single dynamo event with the period described in the statement.

The statement describes a rock crystallizing over a

50\text{-million-year}

period of impact quiescence. This duration (

50\text{ million years}

) is significantly longer than the maximum lifespan of a single dynamo event (

20\text{ million years}

).

This comparison determines if the magnetic field could physically persist throughout the entire rock formation process without a new impact trigger.

3

Evaluate the validity of the statement based on the comparison.

Since the quiet period exceeds the dynamo's maximum lifespan, the magnetic field would turn off before the rock finished forming, making the statement false.

This leads to the final determination of the true/false value.

Key Concept

Distinguishing between continuous and transient models to make inferences about magnetic field history.

Question 2580Question →

A circular metal plate has a radius of $6\text{ inches}$ . A sector of the plate with a central angle of $60^\circ$ is cut out to make a custom spacer. What is the area, in square inches, of the cut-out sector?

2\pi

\frac{3\pi}{2}

6\pi

\frac{36\pi}{5}

36\pi

Show answer & explanation

Answer: $6\pi$

Answer

The correct area of the sector is

6\pi

square inches.

The correct answer of

6\pi

is found by calculating the total area of the circular plate using

A = \pi r^2 = \pi (6)^2 = 36\pi

, and then multiplying this by the ratio of the central angle to the total degrees in a circle, which is

\frac{60^\circ}{360^\circ} = \frac{1}{6}

. Thus, the sector area is

\frac{1}{6} \times 36\pi = 6\pi

square inches.

Step-by-Step Solution

1

Calculate the area of the entire circular plate.

The total area of the circle is

A = \pi (6)^2 = 36\pi

square inches.

Before finding the area of a sector, we need the total area of the circle of which it is a part.

2

Determine the fraction of the circle represented by the sector.

The fraction is

\frac{60^\circ}{360^\circ} = \frac{1}{6}

.

A circle contains

360^\circ

, so a central angle of

60^\circ

corresponds to

\frac{60}{360}

of the full circle.

3

Multiply the total area of the circle by the fraction.

The sector area is

36\pi \times \frac{1}{6} = 6\pi

square inches.

The area of a sector is proportional to its central angle relative to the total angle of a circle.

Key Concept

The area of a sector with radius

r

and central angle

\theta

in degrees is given by the formula

A = \pi r^2 \left(\frac{\theta}{360^\circ}\right)

.

Alternative Method

Alternatively, since

60^\circ

is

\frac{1}{6}

of a full

360^\circ

circle, the sector area is simply one-sixth of the total area of the circle (

36\pi

), which gives

6\pi

square inches.

Estimated Time:45s

Answer

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