All practice questions

Question 481Question →

An architect is designing a triangular solar panel frame with side lengths measuring $15$ feet, $24$ feet, and $21$ feet. What is the measure, in degrees, of the interior angle opposite the side measuring $21$ feet?

Show answer & explanation

Answer: 60

Answer

The measure of the interior angle opposite the side measuring 21 feet is 60 degrees.

Using the Law of Cosines

c^2 = a^2 + b^2 - 2ab\cos(C)

with side lengths

a = 15

,

b = 24

, and

c = 21

gives

21^2 = 15^2 + 24^2 - 2(15)(24)\cos(C)

. Simplifying the equation leads to

441 = 801 - 720\cos(C)

, which rearranges to

-360 = -720\cos(C)

or

\cos(C) = 0.5

. Evaluating

\arccos(0.5)

yields an angle measure of

60^\circ

.

Step-by-Step Solution

1

Set up the Law of Cosines with the side lengths

a = 15

,

b = 24

, and target opposite side

c = 21

.

21^2 = 15^2 + 24^2 - 2(15)(24)\cos(C)

The Law of Cosines connects three side lengths of any triangle to the cosine of the angle opposite one of those sides.

2

Simplify the numerical values in the equation.

441 = 801 - 720\cos(C)

Evaluate

21^2 = 441

,

15^2 + 24^2 = 225 + 576 = 801

, and

2(15)(24) = 720

.

3

Isolate the cosine expression.

\cos(C) = 0.5

Subtracting

801

from both sides gives

-360 = -720\cos(C)

, then dividing by

-720

yields

0.5

.

4

Find the inverse cosine of

0.5

.

C = 60^\circ

In any triangle, the angle whose cosine is

0.5

is

60^\circ

.

Key Concept

Applying the Law of Cosines to solve for an unknown angle given all three side lengths of a non-right triangle.

Question 482Question →

A retail clothing store applies successive markdowns to a coat during a seasonal clearance sale. In the first week, the original price of the coat is discounted by $30\%$ . In the second week, the price is reduced by an additional $\frac{1}{5}$ of the first-week sale price. Finally, a customer uses a promotional code at checkout to receive an extra $15\%$ off the second-week sale price. If the customer pays a final price of $\$119.00$ before tax, what was the original price of the coat, in dollars?

Show answer & explanation

Answer: 250

Answer

The original price of the coat was

250

dollars.

To find the original price of the coat, express each successive discount as a multiplier representing the portion of the remaining price. The first discount of

30\%

leaves

70\%

(

0.70

). The second discount of

\frac{1}{5}

(

20\%

) leaves

80\%

(

0.80

) of the first sale price. The final discount of

15\%

leaves

85\%

(

0.85

) of the second sale price. Combining these yields an overall price multiplier of

0.70 \times 0.80 \times 0.85 = 0.476

. Dividing the final purchase price of

\$119.00

by

0.476

gives the original price of

\$250.00

.

Step-by-Step Solution

1

Calculate the price multiplier for the first week discount.

Multiplier 1 =

0.70

A

30\%

discount leaves

100\% - 30\% = 70\%

of the original price.

2

Convert the second week fraction discount to a decimal multiplier.

Multiplier 2 =

0.80

A discount of

\frac{1}{5}

is equal to

20\%

, leaving

1 - 0.20 = 0.80

of the week-1 price.

3

Calculate the price multiplier for the promotional code.

Multiplier 3 =

0.85

A

15\%

discount leaves

100\% - 15\% = 85\%

(

0.85

) of the week-2 price.

4

Find the combined price multiplier by taking the product of all three individual multipliers.

Combined Multiplier =

0.476

Successive discounts compound multiplicatively:

0.70 \times 0.80 \times 0.85 = 0.476

.

5

Solve for the original price

P

using the equation

0.476 P = 119

.

P = 250

Dividing the final price

\$119.00

by

0.476

yields

250

.

Key Concept

Compounding successive percentage and fraction discounts and solving for the original value.

Estimated Time:2m 0s

Question 483Question →

A high-speed laser pulse travels at a constant speed of $3.0 \times 10^8$ meters per second. If the pulse takes $4.5 \times 10^{-7}$ seconds to travel the entire length of a specialized optical fiber, what is the length of the optical fiber, in meters?

Show answer & explanation

Answer: 135

Answer

The length of the optical fiber is 135 meters.

Multiplying the speed

3.0 \times 10^8

meters per second by the time

4.5 \times 10^{-7}

seconds yields

(3.0 \times 4.5) \times 10^{8 + (-7)} = 13.5 \times 10^1 = 135

meters.

Step-by-Step Solution

1

Set up the distance equation

d = (3.0 \times 10^8) \times (4.5 \times 10^{-7})

Distance equals speed multiplied by time.

2

Group coefficients and powers of 10

d = (3.0 \times 4.5) \times (10^8 \times 10^{-7})

The commutative and associative properties of multiplication allow grouping coefficients and base-10 exponential terms together.

3

Simplify the coefficients and add exponents

d = 13.5 \times 10^1 = 135

Multiplying the coefficients yields 13.5, and adding the exponents gives

8 + (-7) = 1

. Evaluating

13.5 \times 10^1

results in 135.

Key Concept

Multiplying numbers expressed in scientific notation using exponent rules.

Question 484Question →

On a standard number line, point $A$ has coordinate $-28$ and point $B$ has coordinate $12$ . Point $C$ is the midpoint of segment $AB$ . Point $D$ is positioned on the line such that point $B$ is the midpoint of segment $CD$ . What is the distance between point $C$ and point $D$ on the number line?

Show answer & explanation

Answer: 40

Answer

The distance between point

C

and point

D

on the number line is 40.

The midpoint

C

of segment

AB

is found by averaging the coordinates:

\frac{-28 + 12}{2} = -8

. Because point

B(12)

is the midpoint of segment

CD

, set up the equation

\frac{-8 + D}{2} = 12

, yielding

D = 32

. The distance between point

C(-8)

and point

D(32)

is the absolute value of their difference:

|32 - (-8)| = 40

.

Step-by-Step Solution

1

Calculate the coordinate of point

C

, which is the midpoint of segment

AB

.

The coordinate of point

C

is

-8

.

The midpoint of two points on a number line is given by their average:

\frac{-28 + 12}{2} = -8

.

2

Determine the coordinate of point

D

using the midpoint relationship for segment

CD

.

The coordinate of point

D

is

32

.

Since point

B(12)

is the midpoint of segment

CD

,

\frac{C + D}{2} = 12 \implies \frac{-8 + D}{2} = 12

, which solves to

D = 32

.

3

Find the distance between point

C

and point

D

.

The distance is

40

.

Distance on a number line is the absolute value of the difference between coordinates:

|32 - (-8)| = |40| = 40

.

Key Concept

Midpoint and Absolute Value Distance on a Number Line

Alternative Method

Alternatively, note that the distance of segment

AB

is

|12 - (-28)| = 40

. Since

C

is the midpoint of

AB

, the length of segment

CB

is

\frac{40}{2} = 20

. Since

B

is the midpoint of segment

CD

, the length of segment

BD

must equal the length of segment

CB

, which is

20

. Therefore, the total distance from

C

to

D

is

CB + BD = 20 + 20 = 40

.

Estimated Time:1m 15s

Question 485Question →

Let $a$ and $b$ be integers. On a standard number line, $a = -14$ , and the distance between $a$ and $b$ is $23$ . If $b > 0$ , what is the value of $|2b - |a||$ ?

Show answer & explanation

Answer: 4

Answer

The value of

|2b - |a||

is 4.

The distance between

a = -14

and

b

on the number line is

|b - (-14)| = |b + 14| = 23

. Solving for a positive value of

b

gives

b + 14 = 23

, so

b = 9

. The absolute value of

a

is

|-14| = 14

. Substituting these values into

|2b - |a||

yields

|2(9) - 14| = |18 - 14| = 4

.

Step-by-Step Solution

1

Determine the value of integer

b

using the distance relationship on the number line.

The distance formula

|b - (-14)| = 23

simplifies to

|b + 14| = 23

. Since

b > 0

,

b = 9

.

The distance between two points

x

and

y

on a number line is given by

|x - y|

.

2

Calculate the absolute value of

a

.

|a| = |-14| = 14

.

The absolute value of a negative number is its positive magnitude.

3

Substitute

b = 9

and

|a| = 14

into the expression

|2b - |a||

.

|2(9) - 14| = |18 - 14| = 4

.

Perform operations inside the outer absolute value first, then take the absolute value of the result.

Key Concept

Absolute Value as Distance on a Number Line

Estimated Time:1m 0s

Question 486Question →

A landscape architect is designing a triangular courtyard garden. Two adjacent edges of the garden measure $8$ meters and $15$ meters, and the angle between these two edges is $60^\circ$ . What is the length, in meters, of the third edge of the garden?

Show answer & explanation

Answer: 13

Answer

The length of the third edge of the garden is 13 meters.

Applying the Law of Cosines

c^2 = a^2 + b^2 - 2ab \cos(C)

with

a = 8

,

b = 15

, and

C = 60^\circ

yields

c^2 = 8^2 + 15^2 - 2(8)(15)\cos(60^\circ) = 64 + 225 - 240(0.5) = 169

. Taking the square root gives

c = 13

meters.

Step-by-Step Solution

1

Identify the given side lengths and included angle.

Two sides are

a = 8\text{ m}

and

b = 15\text{ m}

, and their included angle is

C = 60^\circ

.

The Law of Cosines is used when two sides and the included angle (SAS) are known.

2

Substitute the values into the Law of Cosines formula

c^2 = a^2 + b^2 - 2ab \cos(C)

.

c^2 = 8^2 + 15^2 - 2(8)(15)\cos(60^\circ) = 64 + 225 - 240(0.5) = 169

.

Evaluating the squared terms and trigonometric value

\cos(60^\circ) = 0.5

simplifies the equation to

c^2 = 169

.

3

Solve for the side length

c

by taking the square root.

c = \sqrt{169} = 13\text{ meters}

.

The physical side length of a geometric figure must be positive.

Key Concept

Law of Cosines (

c^2 = a^2 + b^2 - 2ab \cos(C)

)

Estimated Time:1m 30s

Question 487Question →

A civil engineer is designing a triangular bridge support structure with vertices $P$ , $Q$ , and $R$ . The support beam $PQ$ is $80$ feet long, the beam $QR$ is $50$ feet long, and the interior angle $\angle PQR$ measures $60^\circ$ . What is the length, in feet, of the support beam $PR$ ?

Show answer & explanation

Answer: 70

Answer

The length of the support beam

PR

is

70

feet.

Using the Law of Cosines formula

c^2 = a^2 + b^2 - 2ab\cos(C)

with side lengths

80

and

50

and included angle

60^\circ

, we calculate

PR^2 = 80^2 + 50^2 - 2(80)(50)\cos(60^\circ) = 6400 + 2500 - 4000 = 4900

. Taking the square root gives

PR = 70

feet.

Step-by-Step Solution

1

Identify the given dimensions and included angle

Side

PQ = 80

ft, side

QR = 50

ft, and included angle

\angle PQR = 60^\circ

.

The Law of Cosines applies directly when two side lengths and the included angle (SAS) are known.

2

Set up the Law of Cosines equation for the unknown side

PR

PR^2 = PQ^2 + QR^2 - 2(PQ)(QR)\cos(\angle PQR)

This formula generalizes the Pythagorean theorem to non-right triangles.

3

Substitute the known values into the equation and evaluate

PR^2 = 80^2 + 50^2 - 2(80)(50)\cos(60^\circ) = 6400 + 2500 - 4000 = 4900

Evaluating squares and using

\cos(60^\circ) = 0.5

simplifies the calculation.

4

Take the positive square root to find

PR

PR = \sqrt{4900} = 70

Side length must be a positive real number.

Key Concept

Applying the Law of Cosines to solve for an unknown side in a Side-Angle-Side (SAS) triangle.

Question 488Question →

A cafe offers a lunch special where a customer selects $1$ sandwich from $4$ available options, $1$ side dish from $3$ available options, and $1$ drink from $5$ available options. How many different total lunch combinations consisting of one sandwich, one side dish, and one drink can a customer choose?

Show answer & explanation

Answer: 60

Answer

60 combinations

According to the Fundamental Counting Principle, if one event can occur in

m

ways, a second in

n

ways, and a third in

p

ways, the total number of combinations for all three events occurring together is

m \times n \times p

. Multiplying

4

sandwiches by

3

sides by

5

drinks yields

4 \times 3 \times 5 = 60

distinct lunch combinations.

Step-by-Step Solution

1

Identify the number of options available for each independent selection.

4 sandwiches, 3 side dishes, 5 drinks

The total outcomes depend on making one selection from each distinct category.

2

Apply the Fundamental Counting Principle.

4 × 3 × 5 = 60

The total number of outcomes for independent sequential choices is the product of the number of options for each choice.

Key Concept

Fundamental Counting Principle

Estimated Time:45s

Question 489Question →

Biochemists measured the rate of glucose consumption (in $\text{mM/min}$ ) by a newly isolated bacterial strain across 3 temperature treatments ( $25^\circ\text{C}$ , $35^\circ\text{C}$ , and $45^\circ\text{C}$ ). Each temperature condition was evaluated in 3 separate trials. The results are recorded in the table below:

Temperature ( $^\circ\text{C}$ )	Trial 1 ( $\text{mM/min}$ )	Trial 2 ( $\text{mM/min}$ )	Trial 3 ( $\text{mM/min}$ )
$25$	$1.2$	$1.5$	$1.8$
$35$	$4.0$	$4.6$	$4.6$
$45$	$2.1$	$2.5$	$2.0$

Based on the data provided, what is the average rate of glucose consumption, in $\text{mM/min}$ , for the $35^\circ\text{C}$ treatment across the 3 trials?

Show answer & explanation

Answer: 4.4

Answer

The average rate of glucose consumption at

35^\circ\text{C}

across the 3 trials is

4.4\text{ mM/min}

.

To find the average glucose consumption rate at

35^\circ\text{C}

, locate the row for

35^\circ\text{C}

in the table and sum the values across the 3 trials (

4.0 + 4.6 + 4.6 = 13.2\text{ mM/min}

). Then divide by the number of trials (

3

) to get

\frac{13.2}{3} = 4.4\text{ mM/min}

.

Step-by-Step Solution

1

Extract data points for the

35^\circ\text{C}

row

Trial 1 =

4.0\text{ mM/min}

, Trial 2 =

4.6\text{ mM/min}

, Trial 3 =

4.6\text{ mM/min}

The question specifically asks for the average at

35^\circ\text{C}

.

2

Calculate the sum of the trial values

4.0 + 4.6 + 4.6 = 13.2\text{ mM/min}

Calculating an average requires finding the total sum of all trial measurements first.

3

Divide the sum by the total number of trials

\frac{13.2}{3} = 4.4\text{ mM/min}

Dividing the sum by

3

calculates the arithmetic mean across the 3 trials.

Key Concept

Calculating the average (mean) of data values from a table

Estimated Time:1m 0s

Question 490Question →

A jar contains $3$ green marbles, $5$ yellow marbles, and $12$ purple marbles. If one marble is drawn at random from the jar, what is the probability that the drawn marble is NOT purple? Express your answer as a decimal.

Show answer & explanation

Answer: 0.4

Answer

The probability that the drawn marble is NOT purple is 0.4.

To find the probability of drawing a non-purple marble, identify the number of non-purple marbles (3 green + 5 yellow = 8) and divide by the total number of marbles in the jar (3 + 5 + 12 = 20). The resulting fraction 8/20 simplifies to 0.4.

Step-by-Step Solution

1

Calculate the total number of outcomes in the sample space.

Total marbles = 3 + 5 + 12 = 20

The total sample space includes all marbles contained in the jar.

2

Determine the number of favorable outcomes.

Non-purple marbles = 3 + 5 = 8

A marble that is not purple must be either green or yellow.

3

Compute the probability.

Probability = 8 / 20 = 0.4

The probability of an event is the number of favorable outcomes divided by the total number of possible outcomes.

Key Concept

Basic Probability of Complementary Events

Question 491Question →

On a standard number line, point $P$ is located at $-17$ and point $Q$ is located at $31$ . Point $R$ lies between $P$ and $Q$ such that the ratio of the distance between $P$ and $R$ to the distance between $R$ and $Q$ is $3:5$ . What is the value of $|R - (-5)|$ ?

Show answer & explanation

Answer: 6

Answer

The value of

|R - (-5)|

is

6

.

The total distance between point

P (-17)

and point

Q (31)

is

31 - (-17) = 48

. Since point

R

divides segment

PQ

in a

3:5

ratio, the distance from

P

to

R

is

\frac{3}{8} \times 48 = 18

. Adding this distance to

-17

gives

R = 1

. Substituting

R = 1

into

|R - (-5)|

yields

|1 - (-5)| = |6| = 6

.

Step-by-Step Solution

1

Calculate the total distance between points

P

and

Q

Distance

PQ = 31 - (-17) = 48

The distance between two points on a number line is found by taking the absolute difference of their coordinates.

2

Determine the coordinate of point

R

Coordinate of

R = 1

The ratio of

PR

to

RQ

is

3:5

, meaning

PR

is

\frac{3}{3+5} = \frac{3}{8}

of the total distance

PQ

. Adding

\frac{3}{8} \times 48 = 18

to the starting coordinate

-17

gives

R = 1

.

3

Evaluate the absolute value expression

|R - (-5)|

|1 - (-5)| = |6| = 6

Substitute

R = 1

into the target expression and simplify the double negative before taking the absolute value.

Key Concept

Calculating distance on a number line, segment ratio partitioning, and absolute value evaluation.

Estimated Time:1m 30s

Question 492Question →

What is the value, in degrees, of the expression $\arcsin\left(\frac{\sqrt{3}}{2}\right) + \arccos\left(-\frac{1}{2}\right)$ ?

Show answer & explanation

Answer: 180

Answer

The value of the expression is 180 degrees.

To evaluate

\arcsin\left(\frac{\sqrt{3}}{2}\right) + \arccos\left(-\frac{1}{2}\right)

, find the principal value for each term. The principal range for

\arcsin

is

[-90^\circ, 90^\circ]

, so

\arcsin\left(\frac{\sqrt{3}}{2}\right) = 60^\circ

. The principal range for

\arccos

is

[0^\circ, 180^\circ]

, so for a negative argument, the result lies in Quadrant II, giving

\arccos\left(-\frac{1}{2}\right) = 120^\circ

. Adding these values together gives

60^\circ + 120^\circ = 180^\circ

.

Step-by-Step Solution

1

Find the principal angle for

\arcsin\left(\frac{\sqrt{3}}{2}\right)

in degrees.

\arcsin\left(\frac{\sqrt{3}}{2}\right) = 60^\circ

The inverse sine function yields outputs restricted to the range

[-90^\circ, 90^\circ]

. The angle in Quadrant I whose sine is

\frac{\sqrt{3}}{2}

is

60^\circ

.

2

Find the principal angle for

\arccos\left(-\frac{1}{2}\right)

in degrees.

\arccos\left(-\frac{1}{2}\right) = 120^\circ

The inverse cosine function yields outputs restricted to the range

[0^\circ, 180^\circ]

. For a negative input, the output must be in Quadrant II. The angle whose cosine is

-\frac{1}{2}

is

120^\circ

.

3

Sum the two evaluated angle measures.

60^\circ + 120^\circ = 180^\circ

Perform standard addition on the two principal values.

Key Concept

Evaluating inverse trigonometric functions within their standard principal value ranges.

Question 493Question →

At a local vehicle dealership, a buyer customizing a new car model can select $1$ exterior paint color from $5$ available colors, $1$ interior seat material from $3$ available materials, and $1$ transmission type from $2$ available options (manual or automatic). According to the Fundamental Counting Principle, how many different unique configurations of the car can a buyer create?

Show answer & explanation

Answer: 30

Answer

There are 30 different unique configurations possible.

According to the Fundamental Counting Principle, when making a series of independent choices, the total number of distinct outcomes is found by multiplying the number of options for each choice. Multiplying 5 exterior colors, 3 interior materials, and 2 transmission types gives

5 \times 3 \times 2 = 30

unique car configurations.

Step-by-Step Solution

1

Count the number of choices for each feature

Paint colors = 5, Interior materials = 3, Transmissions = 2

Each feature selection is an independent choice in the configuration process.

2

Multiply the number of options for all features

5 \times 3 \times 2 = 30

The Fundamental Counting Principle states that if there are

n_1

ways to make one choice,

n_2

ways to make a second, and

n_3

ways to make a third, the total number of combined outcomes is

n_1 \times n_2 \times n_3

.

Key Concept

Fundamental Counting Principle

Estimated Time:45s

Question 494Question →

On a temperature monitor scale represented as a horizontal number line, sensor $A$ is positioned at coordinate $-18$ and sensor $B$ is positioned at coordinate $14$ . A relay station $C$ is located on the line segment between sensor $A$ and sensor $B$ such that the distance from sensor $A$ to relay station $C$ is $\frac{3}{4}$ of the total distance between sensor $A$ and sensor $B$ . What is the absolute value of the coordinate of relay station $C$ ?

Show answer & explanation

Answer: 6

Answer

The absolute value of the coordinate of relay station

C

is

6

.

To find the distance between

A

(

-18

) and

B

(

14

), subtract

-18

from

14

to get

32

units. Three-fourths of

32

is

24

units. Adding

24

units to the starting coordinate

-18

yields

6

as the coordinate for

C

. The absolute value of

6

is

6

.

Step-by-Step Solution

1

Calculate the total distance between coordinates

-18

and

14

Distance =

|14 - (-18)| = |14 + 18| = 32

units

The distance between two points on a number line is given by the absolute value of the difference of their coordinates.

2

Calculate the distance from sensor

A

to relay station

C

Distance

AC = \frac{3}{4} \times 32 = 24

units

Relay station

C

is positioned

\frac{3}{4}

of the total distance away from

A

towards

B

.

3

Find the coordinate of relay station

C

Coordinate of

C = -18 + 24 = 6

Moving rightward from

-18

by

24

units gives the location of

C

.

4

Evaluate the absolute value of the coordinate of

C

|6| = 6

The problem asks for the absolute value of the coordinate of

C

.

Key Concept

Distance on a number line and absolute value evaluation

Estimated Time:1m 15s

Question 495Question →

If $\theta = \arcsin\left(-\frac{3}{5}\right)$ , what is the exact decimal value of $\cos(2\theta)$ ?

Show answer & explanation

Answer: 0.28

Answer

The exact decimal value of

\cos(2\theta)

is

0.28

.

Given

\theta = \arcsin\left(-\frac{3}{5}\right)

, the sine of

\theta

is

\sin(\theta) = -\frac{3}{5}

. Using the double-angle formula for cosine,

\cos(2\theta) = 1 - 2\sin^2(\theta)

, we substitute

\sin(\theta)

to get

\cos(2\theta) = 1 - 2\left(-\frac{3}{5}\right)^2 = 1 - \frac{18}{25} = \frac{7}{25} = 0.28

.

Step-by-Step Solution

1

Identify the value of

\sin(\theta)

from the inverse trigonometric expression

\sin(\theta) = -\frac{3}{5}

By definition of the inverse sine function, if

\theta = \arcsin\left(-\frac{3}{5}\right)

, then

\sin(\theta) = -\frac{3}{5}

where

-\frac{\pi}{2} \le \theta \le \frac{\pi}{2}

.

2

Select the double-angle identity for cosine that uses sine

\cos(2\theta) = 1 - 2\sin^2(\theta)

This form of the double-angle identity allows direct calculation without needing to calculate

\cos(\theta)

first.

3

Substitute

\sin(\theta)

and evaluate the expression

\cos(2\theta) = 0.28

Substituting

\sin(\theta) = -\frac{3}{5}

gives

1 - 2\left(-\frac{3}{5}\right)^2 = 1 - 2\left(\frac{9}{25}\right) = 1 - \frac{18}{25} = \frac{7}{25} = 0.28

.

Key Concept

Evaluating Trigonometric Functions of Inverse Trigonometric Expressions using Double-Angle Identities

Alternative Method

Alternatively, place

\theta

in Quadrant IV (since

-\frac{\pi}{2} \le \theta < 0

for a negative inverse sine input). The adjacent side is

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

, so

\cos(\theta) = \frac{4}{5}

. Then apply the alternative double-angle identity

\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = \left(\frac{4}{5}\right)^2 - \left(-\frac{3}{5}\right)^2 = \frac{16}{25} - \frac{9}{25} = \frac{7}{25} = 0.28

.

Estimated Time:1m 15s

Question 496Question →

A conference schedule allows attendees to select 1 morning workshop out of 6 options, 1 keynote session out of 3 options, and 1 afternoon panel out of 4 options. However, 2 specific morning workshops conflict with 1 specific afternoon panel and cannot be chosen together. How many different valid 3-session schedule combinations can an attendee create?

Show answer & explanation

Answer: 66

Answer

An attendee can create 66 different valid 3-session schedule combinations.

According to the Fundamental Counting Principle, the total unrestricted number of schedule combinations is

6 \times 3 \times 4 = 72

. The restriction specifies that 2 morning workshops cannot be paired with 1 specific afternoon panel. Since there are 3 keynote speaker options available for any schedule, the number of invalid combinations is

2 \times 3 \times 1 = 6

. Subtracting the invalid options from the total gives

72 - 6 = 66

valid schedule combinations.

Step-by-Step Solution

1

Calculate total unrestricted schedule combinations

72 combinations

Multiply the choices for each session:

6 \times 3 \times 4 = 72

.

2

Calculate the number of conflicting schedule combinations

6 invalid combinations

The 2 restricted morning workshops combined with 1 restricted afternoon panel can occur alongside any of the 3 keynote choices (

2 \times 3 \times 1 = 6

).

3

Subtract conflicting combinations from total combinations

66 valid combinations

Subtracting 6 invalid schedules from 72 total schedules leaves 66 allowable options.

Key Concept

Fundamental Counting Principle with Restrictions

Estimated Time:1m 30s

Question 497Question →

A juice bar allows customers to create a custom smoothie by selecting exactly $1$ fruit base out of $4$ available options and $1$ liquid base out of $3$ available options. How many different custom smoothie combinations of $1$ fruit base and $1$ liquid base are possible?

Show answer & explanation

Answer: 12

Answer

The total number of possible smoothie combinations is 12.

According to the Fundamental Counting Principle, if there are

m

ways to make a first choice and

n

ways to make a second choice, there are

m \times n

total possible pairs. Here, multiplying

4

fruit choices by

3

liquid choices yields

4 \times 3 = 12

total smoothie combinations.

Step-by-Step Solution

1

Determine the number of options for each independent choice

Fruit base options = 4; Liquid base options = 3

Each choice is independent of the other.

2

Multiply the number of options for each choice using the Fundamental Counting Principle

4 x 3 = 12

To find the total number of combinations across multiple independent categories, multiply the number of choices in each category together.

Key Concept

Fundamental Counting Principle

Estimated Time:45s

Question 498Question →

A commercial printing press operates at a constant rate throughout the day. At 10:15 a.m., a total of 180 posters have been printed since the morning shift began. By 11:45 a.m. on the same morning, a total of 540 posters have been printed. What is the printing rate of the press, in posters per hour?

Show answer & explanation

Answer: 240

Answer

The printing rate of the press is 240 posters per hour.

The slope of a linear relationship represents the constant rate of change. The total number of posters increases by

540 - 180 = 360

over an elapsed time of

1.5

hours. Dividing the change in posters by the change in time yields

\frac{360}{1.5} = 240

posters per hour.

Step-by-Step Solution

1

Determine the change in the dependent variable (number of posters printed)

540 - 180 = 360

posters

Slope represents the change in

y

divided by the change in

x

, where

y

is the total posters printed.

2

Determine the change in the independent variable (time in hours)

From 10:15 a.m. to 11:45 a.m. is 1.5 hours

Time must be expressed in hours to match the requested unit of posters per hour.

3

Compute the slope (rate of change)

\frac{360}{1.5} = 240

posters per hour

Dividing total posters printed by total hours gives the constant rate of change.

Key Concept

Interpretation of Slope as a Constant Rate of Change

Question 499Question →

In the standard $(x, y)$ coordinate plane, a circle is defined by the equation $x^2 + y^2 - 6x + 8y - 11 = 0$ . What is the distance from the center of this circle to the origin $(0, 0)$ ?

Show answer & explanation

Answer: 5

Answer

The distance from the center of the circle to the origin is 5.

By completing the square on

x^2 - 6x

and

y^2 + 8y

, we rewrite

x^2 + y^2 - 6x + 8y - 11 = 0

as

(x - 3)^2 + (y + 4)^2 = 36

. The center of the circle is

(3, -4)

. The distance from

(3, -4)

to

(0, 0)

is

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

.

Step-by-Step Solution

1

Rewrite the general equation of the circle in standard form by completing the square.

(x - 3)^2 + (y + 4)^2 = 36

Grouping

x

and

y

terms and adding

(b/2)^2

to both sides isolates the center coordinates

(h, k)

.

2

Identify the center of the circle from standard form.

Center is

(3, -4)

The standard equation of a circle is

(x - h)^2 + (y - k)^2 = r^2

, where

(h, k)

is the center.

3

Compute the distance between the center

(3, -4)

and the origin

(0, 0)

.

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

Applying the distance formula

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

yields the distance to the origin.

Key Concept

Completing the square to find the center of a circle and applying the distance formula

Estimated Time:1m 15s

Question 500Question →

What is the exact decimal value of $\tan\left(\arcsin\left(\frac{12}{13}\right)\right)$ ?

Show answer & explanation

Answer: 2.4

Answer

The exact decimal value of

\tan\left(\arcsin\left(\frac{12}{13}\right)\right)

is

2.4

.

Letting

\theta = \arcsin\left(\frac{12}{13}\right)

, we establish a right triangle in Quadrant I with opposite side length

12

and hypotenuse length

13

. The adjacent side length is calculated via the Pythagorean theorem as

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

. The tangent of this angle is the ratio of the opposite side to the adjacent side,

\frac{12}{5}

, which equals

2.4

.

Step-by-Step Solution

1

Define the angle using the inverse trigonometric expression

Let

\theta = \arcsin\left(\frac{12}{13}\right)

, which implies

\sin(\theta) = \frac{12}{13}

in Quadrant I.

The inverse sine function returns an angle whose sine is the given ratio, restricted to the principal interval

\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]

.

2

Determine

\cos(\theta)

using the fundamental trigonometric identity

\cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - \left(\frac{12}{13}\right)^2} = \frac{5}{13}

Since

\theta

is in Quadrant I,

\cos(\theta)

is positive.

3

Evaluate

\tan(\theta)

as the ratio of sine to cosine

\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{12/13}{5/13} = \frac{12}{5} = 2.4

Dividing opposite by adjacent (or sine by cosine) yields the exact decimal value

2.4

.

Key Concept

Evaluating algebraic values of composite inverse trigonometric expressions

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