Tüm alıştırma soruları

Soru 461Soru →

A mobile water purification system can process $5,400$ gallons of water in $1.5$ hours. What is the unit rate of water processed, in gallons per minute?

36

60

600

3,600

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

The correct unit rate is 60 gallons per minute.

The correct answer is found by first converting

1.5

hours to minutes (

1.5 \times 60 = 90

minutes). Then, divide the total gallons by the total minutes to find the rate:

5,400 \div 90 = 60

gallons per minute.

Adım Adım Çözüm

1

Convert the given time from hours to minutes.

1.5 \text{ hours} \times 60 \text{ minutes/hour} = 90 \text{ minutes}

.

The question specifically asks for the unit rate in gallons per minute, so the time measurement must match.

2

Divide the total amount of water by the total number of minutes to find the unit rate.

5,400 \text{ gallons} \div 90 \text{ minutes} = 60 \text{ gallons per minute}

.

A unit rate represents the quantity of one variable for every single unit of another variable.

Anahtar Kavram

Calculating unit rates involving time conversions

Tahmini Süre:1m 0s

Soru 462Soru →

Match each geometric angle scenario to its final calculated angle measure.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The geometric relationships determine the equations: the supplementary pair resolves to

140^\circ

, the complementary pair resolves to

60^\circ

, the alternate interior angles evaluate to

95^\circ

, and the vertical angles evaluate to

90^\circ

.

Each correct match correctly identifies the angle relationship, translates it into an accurate algebraic equation, solves for the variable, and completes the final substitution to find the requested angle measure.

Adım Adım Çözüm

1

Identify the core geometric relationship for each scenario.

Determine if the angles sum to

180^\circ

(supplementary), sum to

90^\circ

(complementary), or are mathematically equivalent (alternate interior, vertical).

Recognizing the geometric property is necessary to set up the correct algebraic equation.

2

Translate the text into algebraic equations and solve for the unknown variable.

The variables are calculated as

x = 40

,

x = 60

,

x = 20

, and

y = 20

for the four respective scenarios.

Isolating the variable is an intermediate step required to find the specific angle measurements.

3

Calculate the final requested angle measure using the solved variable.

Substitute the variables back into the expressions or apply the logic to find the specific 'larger angle' requested.

The question asks for the final angle measure, not just the isolated variable.

Anahtar Kavram

Lines and Angles

Soru 463Soru →

In a convex pentagon, two of the interior angles are equal in measure. The other three interior angles are also equal to each other, and each of these three is $40^\circ$ larger than the measure of one of the first two angles. What is the measure of the largest exterior angle of this pentagon?

132^\circ

96^\circ

80^\circ

48^\circ

Cevabı ve açıklamayı göster

Cevap: $96^\circ$

Cevap

The largest exterior angle is

96^\circ

.

The correct answer is

96^\circ

because setting up the equation for a pentagon's sum of interior angles,

2x + 3(x + 40) = 540

, gives

x = 84^\circ

as the smallest interior angle. The exterior angle is supplementary to the interior angle, so

180^\circ - 84^\circ = 96^\circ

.

Adım Adım Çözüm

1

Determine the sum of the interior angles of a pentagon.

Using the formula

(n - 2) \times 180^\circ

, for

n = 5

, the sum is

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

.

You must know the total sum of the interior angles before you can solve for any individual unknown angle.

2

Set up an algebraic equation to represent the interior angles.

Let the two smaller equal angles be

x

. The three larger equal angles are

(x + 40^\circ)

. The equation is

2x + 3(x + 40^\circ) = 540^\circ

.

Translating the word problem into algebra allows us to solve for the specific angle measures.

3

Solve the equation for

x

.

Distribute the 3 to get

2x + 3x + 120^\circ = 540^\circ

. Combine like terms:

5x + 120^\circ = 540^\circ

. Subtract 120 from both sides:

5x = 420^\circ

. Divide by 5:

x = 84^\circ

.

Solving for

x

provides the measure of the smallest interior angles.

4

Determine the largest exterior angle.

An exterior angle is supplementary to its adjacent interior angle. The largest exterior angle corresponds to the smallest interior angle. The largest exterior angle is

180^\circ - 84^\circ = 96^\circ

.

Because exterior and interior angles on a straight line add up to 180 degrees, the smallest interior angle creates the largest exterior angle.

Anahtar Kavram

Quadrilaterals and Polygons

Soru 464Soru →

A reservist earns a base pay of $\$115.50$ per drill period. During a weekend drill, she completes $4$ drill periods. Additionally, she receives a travel allowance of $\$0.65$ per mile for her $42$ -mile round trip. What is her total compensation for the weekend in dollars?

Cevabı ve açıklamayı göster

Cevap: 489.3

Cevap

489.30

The total compensation is the sum of the drill pay (

4 \times \$115.50 = \$462.00

) and the travel allowance (

42 \times \$0.65 = \$27.30

), which results in a final total of

\$489.30

.

Adım Adım Çözüm

1

Calculate the total pay for the drill periods.

\$462.00

The reservist gets paid

\$115.50

for each of the

4

drill periods.

2

Calculate the total travel allowance.

\$27.30

The travel allowance is calculated at a rate of

\$0.65

per mile for a total of

42

miles.

3

Add the drill pay and travel allowance together.

\$489.30

The total compensation is the sum of all earnings for the weekend.

Anahtar Kavram

Decimal operations and currency word problems involving rates and total quantities.

Soru 465Soru →

Find the ordered pair $(x, y)$ that satisfies both equations below:

y = 3x - 2

y = x + 6

Which of the following is the correct solution?

(4, 10)

(2, 8)

(8, 14)

(4, 14)

Cevabı ve açıklamayı göster

Cevap: $(4, 10)$

Cevap

The correct solution is the ordered pair

(4, 10)

.

By setting the two expressions for y equal to each other (

3x - 2 = x + 6

), we can isolate x to find

x = 4

. Substituting

4

back into either original equation gives

y = 10

. Therefore, the solution is

(4, 10)

.

Adım Adım Çözüm

1

Set the right sides of both equations equal to each other.

3x - 2 = x + 6

Since both expressions are equal to y, they must be equal to each other.

2

Subtract

x

from both sides to group the variables.

2x - 2 = 6

This begins the process of isolating the variable x.

3

Add

2

to both sides of the equation.

2x = 8

This isolates the term with the x variable on one side and the constant on the other.

4

Divide both sides by

2

.

x = 4

This fully isolates x to find its numerical value.

5

Substitute

x = 4

into the second equation to find

y

.

y = 4 + 6 = 10

Substituting the known x value back into either original equation yields the corresponding y value.

Anahtar Kavram

Solving systems of linear equations using the substitution method.

Soru 466Soru →

Match each polynomial expression with its completely factored algebraic form.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The expressions correctly match their completely factored forms by applying the rules for difference of squares, perfect trinomials, extracting a greatest common factor, and factoring by grouping.

Each polynomial must be analyzed to find its unique factors. Recognizing perfect squares, extracting common factors first, and finding the correct binomial number pairs based on sum and product rules lead to the correct matches.

Adım Adım Çözüm

1

Identify the structure of each polynomial to determine the correct factoring technique.

Categorize the expressions into difference of squares, monic trinomials, GCF + trinomial, and non-monic trinomials.

Applying the correct initial technique is essential to fully factoring the polynomial.

2

Apply the corresponding factoring method to break down the polynomial into its most basic prime factors.

Each expression is rewritten as a product of binomials and, if applicable, a constant scalar.

This verifies that the polynomial is factored completely, revealing the correct match.

Anahtar Kavram

Factoring Polynomials

Soru 467Soru →

An armored vehicle traverses an $18$ -mile desert route. It travels the first $10$ miles at a constant speed of $20$ miles per hour. If the vehicle completes the entire route in exactly $45$ minutes, what is its speed in miles per hour for the remaining distance?

Cevabı ve açıklamayı göster

Cevap: 32

Cevap

The speed for the remaining distance is

32

miles per hour.

To find the unknown speed, you must isolate the distance and time for the final segment. First, the time for the

10

-mile stretch is calculated as

0.5

hours, which is equivalent to

30

minutes. Subtracting this from the total

45

minutes leaves

15

minutes (or

0.25

hours) to travel the remaining

8

miles. Dividing the

8

miles by the

0.25

hours of remaining time yields a final speed of

32

miles per hour.

Adım Adım Çözüm

1

Calculate the time for the first segment.

10 \text{ miles} \div 20 \text{ mph} = 0.5 \text{ hours}

You need to know how much time was spent on the first segment to determine the time left for the remainder of the route.

2

Convert the first segment's time into minutes.

0.5 \text{ hours} \times 60 \text{ minutes/hour} = 30 \text{ minutes}

The total time is provided in minutes (

45

minutes), so making the units consistent allows you to easily subtract and find the remaining time.

3

Find the remaining time available.

45 \text{ minutes} - 30 \text{ minutes} = 15 \text{ minutes}

This step determines exactly how much time the vehicle has to complete the rest of the route.

4

Convert the remaining time into hours.

15 \text{ minutes} \div 60 \text{ minutes/hour} = 0.25 \text{ hours}

Since the requested final speed must be in miles per hour, the time value needs to be expressed in hours.

5

Determine the remaining distance.

18 \text{ miles} - 10 \text{ miles} = 8 \text{ miles}

This is the physical distance the vehicle must cover during the final

15

minutes.

6

Calculate the speed for the final segment.

8 \text{ miles} \div 0.25 \text{ hours} = 32 \text{ mph}

Dividing the remaining distance by the remaining time gives the required speed.

Anahtar Kavram

Calculating unknown rates in multi-part trips by finding intermediate distances and times, and converting units correctly.

Soru 468Soru →

A rectangular floor is $18$ feet long and $12$ feet wide. A square rug with a side length of $9$ feet is placed on the floor. What is the area of the floor that is NOT covered by the rug, in square feet?

180

24

135

297

Cevabı ve açıklamayı göster

Cevap: $135$

Cevap

The area of the floor not covered by the rug is

135

square feet.

The correct answer is found by calculating the total area of the floor (

18 \times 12 = 216

sq ft) and subtracting the area of the square rug (

9 \times 9 = 81

sq ft). This gives an uncovered area of

216 - 81 = 135

square feet.

Adım Adım Çözüm

1

Calculate the total area of the rectangular floor.

\text{Area of floor} = 18 \times 12 = 216 \text{ square feet}

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

2

Calculate the area of the square rug.

\text{Area of rug} = 9 \times 9 = 81 \text{ square feet}

The area of a square is found by squaring its side length.

3

Subtract the area of the rug from the total area of the floor.

216 - 81 = 135 \text{ square feet}

The uncovered area is the difference between the total floor area and the area covered by the rug.

Anahtar Kavram

Calculating and subtracting areas of 2D figures

Tahmini Süre:1m 0s

Soru 469Soru →

A small business owner takes out a loan of $\$15,000$ to repair commercial equipment. After exactly 8 months, the loan is paid off with a single payment of $\$15,900$ . Assuming simple interest is applied, what is the annual interest rate on the loan?

6\%

9\%

4\%

7.5\%

Cevabı ve açıklamayı göster

Cevap: $9\%$

Cevap

The annual simple interest rate on the loan is

9\%

.

To find the annual interest rate, we first determine the total interest paid by subtracting the principal from the total payment (

\$15,900 - \$15,000 = \$900

). Next, we convert the 8-month term into years by dividing by 12, which gives

2/3

of a year. Plugging these into the simple interest formula,

I = P \times R \times T

, gives

900 = 15,000 \times R \times (2/3)

. Simplifying the right side yields

900 = 10,000 \times R

. Dividing both sides by 10,000 results in

R = 0.09

, or

9\%

.

Adım Adım Çözüm

1

Calculate the total amount of interest paid.

\$15,900 - \$15,000 = \$900

The simple interest formula requires the interest amount, which is the difference between the total payoff amount and the original principal.

2

Convert the loan duration from months into a fraction of a year.

8 \text{ months} = \frac{8}{12} \text{ year} = \frac{2}{3} \text{ year}

Annual simple interest rates require the time variable (

T

) to be expressed in years.

3

Substitute the known values into the simple interest formula (

I = P \times R \times T

) and solve for the rate (

R

).

\$900 = \$15,000 \times R \times \frac{2}{3}

\$900 = \$10,000 \times R

R = \frac{900}{10,000} = 0.09 = 9\%

Isolating

R

provides the annual interest rate in decimal form, which is then converted to a percentage.

Anahtar Kavram

Calculating the annual simple interest rate when given the principal, the total maturity amount, and a fractional time period in months.

Soru 470Soru →

A graphic designer creates a rectangular product label that is $5.5$ centimeters long and $4.0$ centimeters wide. What is the area of the label, in square centimeters?

19

9.5

22

220

Cevabı ve açıklamayı göster

Cevap: $22$

Cevap

The correct area of the label is

22

square centimeters.

The area of a rectangle is found by multiplying its length by its width. The length of the label is

5.5

cm and the width is

4.0

cm. Multiplying these together gives

5.5 \times 4.0 = 22

square centimeters.

Adım Adım Çözüm

1

Identify the shape and the goal from the word problem.

The shape is a rectangle and the goal is to find its area.

Identifying the shape ensures the correct mathematical formula is applied.

2

Set up the area formula for a rectangle.

\text{Area} = \text{length} \times \text{width}

This is the standard geometric formula for the area of a rectangle.

3

Substitute the given values into the formula and multiply.

\text{Area} = 5.5 \times 4.0 = 22

Multiplying the given dimensions provides the final area in square centimeters.

Anahtar Kavram

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

A = l \times w

).

Soru 471Soru →

Two motorboats leave a marina at the same time, traveling in opposite directions on a straight path. One boat travels at a constant speed of $18$ mph, and the other travels at a constant speed of $22$ mph. How many minutes will it take for the two boats to be exactly $120$ miles apart?

3

180

360

1800

Cevabı ve açıklamayı göster

Cevap: 180

Cevap

180

Because the boats are moving in opposite directions, their relative speed of separation is the sum of their individual speeds: 18 mph + 22 mph = 40 mph. To find how long it takes to separate by 120 miles, divide the distance by the relative rate: 120 / 40 = 3 hours. Finally, convert 3 hours into minutes by multiplying by 60, resulting in 180 minutes.

Adım Adım Çözüm

1

Calculate the combined relative speed of the two boats.

18 + 22 = 40

mph

Because the boats are traveling in opposite directions, their speeds add together to create the rate at which they are separating.

2

Calculate the time in hours it takes to reach the target distance.

120 \div 40 = 3

hours

Using the formula Time = Distance / Rate, dividing the total distance by the relative speed gives the time in hours.

3

Convert the time from hours to minutes.

3 \times 60 = 180

minutes

The question explicitly asks for the final answer in minutes, not hours.

Anahtar Kavram

Relative Motion and Catch-Up Scenarios

Soru 472Soru →

A logistics command is tracking the movement of four supply convoys traveling along different routes. Match each convoy's travel data to its correct average unit rate in miles per hour (mph).

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

Alpha Convoy travels at 45 mph, Bravo Convoy at 48 mph, Charlie Convoy at 52 mph, and Delta Convoy at 50 mph.

Each convoy's speed is found by dividing the total miles traveled by the total hours elapsed, carefully converting any minute values into fractional or decimal hours first.

Adım Adım Çözüm

1

Calculate the unit rate for Alpha Convoy.

45

mph

The time is given in minutes, so it must be converted to hours.

20 \text{ minutes} = \frac{20}{60} \text{ hours} = \frac{1}{3} \text{ hours}

. Then,

15 \div \frac{1}{3} = 45

mph.

2

Calculate the unit rate for Bravo Convoy.

48

mph

The time is already in hours. Simply divide distance by time:

96 \div 2 = 48

mph.

3

Calculate the unit rate for Charlie Convoy.

52

mph

The time is in hours as a decimal. Divide distance by time:

130 \div 2.5 = 52

mph.

4

Calculate the unit rate for Delta Convoy.

50

mph

Convert minutes to hours:

42 \text{ minutes} = \frac{42}{60} \text{ hours} = 0.7 \text{ hours}

. Divide distance by time:

35 \div 0.7 = 50

mph.

Anahtar Kavram

Calculating unit rates (speed) by dividing distance by time, requiring unit conversion when time is given in minutes.

Tahmini Süre:2m 0s

Soru 473Soru →

Match each statistical request with its correct numerical value based on the provided data sets.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

Mode of 2, 4, 4, 6, 9 is 4. Median of 1, 3, 5, 10, 15 is 5. Mean of 2, 6, 10 is 6. Median of 3, 5, 9, 11 is 7.

Each statistical request is paired with its exact calculated value. The mode identifies the most common value, the median finds the central value of an ordered list (averaging the middle two if the count is even), and the mean calculates the mathematical average.

Adım Adım Çözüm

1

Calculate the mode for the first set:

2, 4, 4, 6, 9

The mode is

4

.

The number

4

is the only value that appears more than once, making it the most frequent value.

2

Determine the median for the second set:

1, 3, 5, 10, 15

The median is

5

.

Since the list is ordered and has an odd number of elements (

5

), the middle value is the third element.

3

Calculate the mean for the third set:

2, 6, 10

The mean is

6

.

Add the numbers (

2 + 6 + 10 = 18

) and divide by the total count (

3

).

18 \div 3 = 6

.

4

Determine the median for the fourth set:

3, 5, 9, 11

The median is

7

.

Since the list has an even number of elements (

4

), find the average of the two middle numbers (

5

and

9

).

\frac{5 + 9}{2} = \frac{14}{2} = 7

.

Anahtar Kavram

Measures of Central Tendency

Soru 474Soru →

Match each angle relationship term with its correct geometric definition.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

Complementary Angles match with the sum of

90^\circ

, Supplementary Angles match with the sum of

180^\circ

, Vertical Angles match with opposite angles of intersecting lines, and Adjacent Angles match with angles sharing a common vertex and side.

The correct matches directly apply the standard mathematical definitions for these fundamental angle relationships. Correctly distinguishing between these terms is essential for solving complex geometry problems.

Adım Adım Çözüm

1

Identify the definition of Complementary Angles.

Matches with 'Two angles whose measures add up to

90^\circ

'.

This is a fundamental geometric property for complementary pairs.

2

Identify the definition of Supplementary Angles.

Matches with 'Two angles whose measures add up to

180^\circ

'.

Supplementary pairs form a straight angle or line when combined adjacently.

3

Identify the definition of Vertical Angles.

Matches with 'Opposite angles formed by the intersection of two lines, which are always equal in measure'.

The intersection of two lines inherently creates these opposite, congruent angle pairs.

4

Identify the definition of Adjacent Angles.

Matches with 'Two angles that share a common vertex and side, but have no interior points in common'.

This physical layout defines adjacency in geometry without dictating a specific angle sum.

Anahtar Kavram

Basic Angle Relationships and Definitions

Soru 475Soru →

A military mess hall allocates $\frac{3}{40}$ of its weekly grocery budget to purchase fresh vegetables. What percentage of the weekly grocery budget is spent on fresh vegetables?

7.5%

75%

0.075%

13.3%

Cevabı ve açıklamayı göster

Cevap: 7.5%

Cevap

7.5%

To convert a fraction to a percentage, divide the numerator by the denominator to find the decimal equivalent, and then multiply by 100. Calculating

3 \div 40

gives

0.075

, and multiplying by 100 yields

7.5\%

.

Adım Adım Çözüm

1

Convert the fraction

\frac{3}{40}

into a decimal.

3 \div 40 = 0.075

To find a percentage, the fraction must first be expressed as a decimal value.

2

Convert the resulting decimal to a percentage.

0.075 \times 100 = 7.5\%

A decimal is converted to a percentage by multiplying by 100, which shifts the decimal point two places to the right.

Anahtar Kavram

Converting Fractions to Percentages

Tahmini Süre:1m 0s

Soru 476Soru →

Match each geometric description to the correct polygon or quadrilateral it uniquely describes.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The geometric descriptions correspond perfectly to the Dodecagon, Octagon, Rectangle, and Nonagon based on their algebraic angle properties and specific diagonal traits.

Each provided description translates to a specific algebraic or geometric property that uniquely identifies a single polygon. Setting up and solving the respective equations for

n

(number of sides) or applying logical diagnostic properties of quadrilaterals produces the correct matching pairs.

Adım Adım Çözüm

1

Analyze the polygon where the interior angle sum is five times the exterior angle sum.

180(n-2) = 5(360) \Rightarrow 180(n-2) = 1800 \Rightarrow n-2 = 10 \Rightarrow n=12

.

Applying the universal constant for exterior angle sums (

360^\circ

) allows solving algebraically for the

12

-sided dodecagon.

2

Solve for the polygon where the number of diagonals equals

2.5

times the number of vertices (

n

).

\frac{n(n-3)}{2} = 2.5n \Rightarrow n-3 = 5 \Rightarrow n=8

.

Equating the formula for the total number of diagonals to

2.5n

allows us to simplify and identify the

8

-sided octagon.

3

Deduce the quadrilateral with congruent, bisecting, non-perpendicular diagonals.

The shape is uniquely identified as a rectangle (excluding a square).

Bisecting diagonals prove it is a parallelogram, congruent diagonals prove it is a rectangle, and non-perpendicularity proves it is not a square.

4

Calculate the sides of the polygon where the interior angle is

100^\circ

larger than the exterior angle.

I + E = 180^\circ

and

I - E = 100^\circ

. Solving the system gives

E = 40^\circ

. Since

n = \frac{360}{E}

,

n = \frac{360}{40} = 9

.

Recognizing that interior and exterior angles form a linear pair lets us set up a system of equations. Finding the exterior angle easily leads to the

9

-sided nonagon.

Anahtar Kavram

Advanced Polygon Analysis and Quadrilateral Properties

Soru 477Soru →

A painter is marking a new parallelogram-shaped crosswalk on a street. The base of the parallelogram is $45$ feet long, and the perpendicular height across the street is $20$ feet. What is the total area of the crosswalk, in square feet?

Cevabı ve açıklamayı göster

Cevap: 900

Cevap

900

The area of a parallelogram is calculated by multiplying its base by its perpendicular height. Given a base of

45

feet and a height of

20

feet, the area is

45 \times 20 = 900

square feet.

Adım Adım Çözüm

1

Identify the relevant formula

A = b \times h

The area of a parallelogram is the product of its base and perpendicular height.

2

Substitute the given values into the formula

A = 45 \times 20

The base (

b

) is given as

45

feet and the height (

h

) is

20

feet.

3

Calculate the final area

900

Multiplying

45

by

20

gives

900

.

Anahtar Kavram

Area of a Parallelogram

Tahmini Süre:30s

Soru 478Soru →

In a geometric diagram, angle $\alpha$ and angle $\beta$ are complementary. Furthermore, angle $\beta$ and angle $\gamma$ are supplementary. If the measure of angle $\alpha$ is represented by the expression $(2x - 5)^\circ$ and the measure of angle $\gamma$ is represented by $(5x - 2)^\circ$ , what is the measure of angle $\beta$ ?

37^\circ

97^\circ

135^\circ

153^\circ

Cevabı ve açıklamayı göster

Cevap: $37^\circ$

Cevap

37^\circ

The correct answer is found by recognizing that angle

\beta

can be expressed in two ways:

90^\circ - \alpha

and

180^\circ - \gamma

. Setting

90 - (2x - 5)

equal to

180 - (5x - 2)

allows us to solve for

x = 29

. Substituting

x

back into the first expression yields

95 - 2(29) = 37^\circ

.

Adım Adım Çözüm

1

Express angle beta in terms of x using the complementary relationship.

\beta = 90 - (2x - 5) = 95 - 2x

Since alpha and beta are complementary, their sum is 90 degrees.

2

Express angle beta in terms of x using the supplementary relationship.

\beta = 180 - (5x - 2) = 182 - 5x

Since beta and gamma are supplementary, their sum is 180 degrees.

3

Set the two expressions for beta equal to each other and solve for x.

95 - 2x = 182 - 5x \Rightarrow 3x = 87 \Rightarrow x = 29

Both expressions represent the same angle, so they must be algebraically equivalent.

4

Substitute the value of x back into either expression to find the measure of angle beta.

\beta = 95 - 2(29) = 95 - 58 = 37^\circ

This provides the final numerical measure of the target angle.

Anahtar Kavram

Algebraic application of complementary and supplementary angle relationships

Tahmini Süre:2m 0s

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A cyclist is riding on a paved trail at a constant speed of $15 \text{ miles per hour}$ . At this rate, how many miles will the cyclist travel in $40 \text{ minutes}$ ?

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Cevap: 10

Cevap

10

The distance is found by matching the time units to the speed units. First, convert

40 \text{ minutes}

into

\frac{2}{3} \text{ hours}

. Then, multiply the speed (

15 \text{ mph}

) by the time (

\frac{2}{3} \text{ hours}

) to get exactly 10 miles.

Adım Adım Çözüm

1

Convert the travel time from minutes to hours.

40 \text{ minutes} = \frac{40}{60} \text{ hours} = \frac{2}{3} \text{ hours}

The speed is given in miles per hour, so the time must be converted to hours before using it in the distance formula.

2

Calculate the total distance traveled.

15 \times \frac{2}{3} = 10 \text{ miles}

Distance is equal to the rate (speed) multiplied by the time (

d = r \times t

).

Anahtar Kavram

Unit conversions for time in rate problems

Soru 480Soru →

A wholesale electronics representative earns a guaranteed weekly base salary of $\$800$ . In addition to the base salary, he earns a $5\%$ commission on any weekly sales that exceed a $\$2,000$ quota. If his total sales for the week were $\$6,500$ , what was his total gross pay for the week?

\$225

\$1,125

\$1,025

\$3,050

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Cevap: $\$1,025$

Cevap

\$1,025

The total gross pay is the sum of the base salary and the earned commission. First, subtract the quota from total sales to find the commissionable amount:

\$6,500 - \$2,000 = \$4,500

. Next, calculate the

5\%

commission on this amount:

\$4,500 \times 0.05 = \$225

. Finally, add the commission to the base salary:

\$800 + \$225 = \$1,025

.

Adım Adım Çözüm

1

Determine the amount of sales eligible for commission by subtracting the quota from total sales.

\$6,500 - \$2,000 = \$4,500

The

5\%

commission is only earned on sales that exceed the

\$2,000

quota.

2

Calculate the commission amount by converting

5\%

to a decimal and multiplying it by the eligible sales.

0.05 \times \$4,500 = \$225

This determines the additional earnings from sales performance.

3

Add the calculated commission to the guaranteed base salary to find the total gross pay.

\$800 + \$225 = \$1,025

Total gross pay is the sum of all fixed and variable earnings.

Anahtar Kavram

Calculating total gross pay involving a base salary and a threshold-based commission.

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