Tüm alıştırma soruları

Soru 2141Soru →

A theater crew is building a wooden stage prop in the shape of a right triangle. The vertical height of the prop is $12$ feet and the horizontal base is $5$ feet. What is the length of the slanted top edge of the prop?

169

feet

17

feet

13

feet

119

feet

Cevabı ve açıklamayı göster

Cevap: $13$ feet

Cevap

The length of the slanted top edge is

13

feet.

The slanted edge represents the hypotenuse of a right triangle. According to the Pythagorean theorem, the square of the hypotenuse equals the sum of the squares of the two perpendicular legs (

a^2 + b^2 = c^2

). Substituting the given legs (

5

and

12

), we calculate

5^2 + 12^2 = 25 + 144 = 169

. Taking the square root of

169

yields

13

feet.

Adım Adım Çözüm

1

Identify the given values as the legs of a right triangle.

The legs are

a = 5

and

b = 12

. The unknown slanted edge is the hypotenuse,

c

.

The Pythagorean theorem relates the two perpendicular legs to the longest side.

2

Substitute the leg values into the Pythagorean theorem formula,

a^2 + b^2 = c^2

.

5^2 + 12^2 = c^2

This sets up the equation to solve for the missing hypotenuse.

3

Square the numbers and add them together.

25 + 144 = c^2

, which simplifies to

169 = c^2

.

This isolates

c^2

on one side of the equation.

4

Take the square root of both sides to find the value of

c

.

c = 13

Taking the square root cancels the exponent and gives the final length of the edge.

Anahtar Kavram

Using the Pythagorean theorem to calculate the hypotenuse of a right triangle.

Soru 2142Soru →

A military supply depot receives a shipment of $5,000$ equipment crates. Upon initial sorting, $20\%$ of the crates are assigned to the aviation unit. Of the remaining crates, $35\%$ are assigned to the infantry unit. How many crates are assigned to the infantry unit?

140

1,400

1,750

2,600

Cevabı ve açıklamayı göster

Cevap: 1,400

Cevap

1,400

The problem requires a two-step percentage calculation. First, find the remaining crates after

20\%

are taken from the initial

5,000

crates, leaving

4,000

crates. Second, calculate

35\%

of that remaining

4,000

crates, which equals

1,400

.

Adım Adım Çözüm

1

Calculate the number of crates assigned to the aviation unit.

5,000 \times 0.20 = 1,000

To find how many crates are removed from the initial shipment.

2

Subtract the aviation crates from the total to find the remaining crates.

5,000 - 1,000 = 4,000

The infantry receives a percentage of the remaining crates, not the original total.

3

Calculate the number of crates assigned to the infantry unit from the remaining amount.

4,000 \times 0.35 = 1,400

This answers the specific question asked.

Anahtar Kavram

Calculating percentages of a remaining quantity in a multi-step scenario.

Alternatif Yöntem

Instead of calculating the

20\%

and subtracting, you can find the remaining percentage directly:

100\% - 20\% = 80\%

. Then,

5,000 \times 0.80 = 4,000

. Finally,

4,000 \times 0.35 = 1,400

.

Tahmini Süre:1m 0s

Soru 2143Soru →

A circular hatch on a maritime vessel has a diameter of $14$ inches. What is the area of the hatch in square inches?

14\pi

28\pi

49\pi

196\pi

Cevabı ve açıklamayı göster

Cevap: $49\pi$

Cevap

The correct area is

49\pi

square inches.

The correct answer is found by first determining the radius, which is half of the given diameter (

14 \div 2 = 7

inches). Applying the area formula

A = \pi r^2

yields

\pi \times 7^2 = 49\pi

square inches.

Adım Adım Çözüm

1

Determine the radius of the circular hatch.

Radius =

14 \div 2 = 7

inches.

The formula for the area of a circle requires the radius, which is exactly half of the diameter.

2

Apply the area formula for a circle.

Area =

\pi \times 7^2 = 49\pi

square inches.

The area of a circle is found using the formula

A = \pi r^2

.

Anahtar Kavram

Calculating the area of a circle by first deriving the radius from the diameter.

Soru 2144Soru →

A radar operator tracks the number of aircraft detected in their sector during $5$ consecutive hours. The hourly counts are $24$ , $18$ , $35$ , $21$ , and $12$ . What is the median number of aircraft detected per hour?

21

22

35

23

Cevabı ve açıklamayı göster

Cevap: $21$

Cevap

The median number of aircraft detected per hour is

21

.

The median is found by arranging the values in ascending order (

12

,

18

,

21

,

24

,

35

) and identifying the middle value. The middle number is

21

.

Adım Adım Çözüm

1

Order the hourly aircraft counts from least to greatest.

12

,

18

,

21

,

24

,

35

The median is the middle value of an ordered dataset.

2

Identify the middle number in the sorted sequence.

The third number is

21

.

Since there are

5

data points (an odd number), the median is exactly in the middle.

Anahtar Kavram

The median is the middle value of a sorted dataset. If a dataset has an odd number of values, the median is the single middle number.

Soru 2145Soru →

A high-speed passenger train is scheduled to travel a $300$ -mile route between two cities with an overall average speed of $60$ miles per hour. During the first leg of the trip, the train travels $150$ miles at an average speed of $50$ miles per hour. At what average speed must the train travel for the remaining $150$ miles to meet its scheduled overall average speed?

75 mph

70 mph

100 mph

30 mph

Cevabı ve açıklamayı göster

Cevap: 75 mph

Cevap

The train must travel at 75 mph for the remaining distance.

To solve for the required speed, you must work with time. The entire 300-mile trip at an average of 60 mph allows a total of 5 hours (

300 \div 60 = 5

). The first 150 miles at 50 mph consumed 3 hours (

150 \div 50 = 3

). This leaves exactly 2 hours to cover the remaining 150 miles. Dividing the remaining distance by the remaining time (

150 \div 2

) gives the necessary speed of 75 mph.

Adım Adım Çözüm

1

Calculate the total time allowed for the entire trip.

300 \text{ miles} \div 60 \text{ mph} = 5 \text{ hours}

Average speed is defined as total distance divided by total time. We need the total time limit to figure out how much time is left.

2

Calculate the time spent on the first leg of the trip.

150 \text{ miles} \div 50 \text{ mph} = 3 \text{ hours}

Using the distance and speed given for the first segment, we determine how much of our total time budget has been used.

3

Determine the time remaining for the second leg.

5 \text{ hours} - 3 \text{ hours} = 2 \text{ hours}

Subtracting the time already used from the total allowed time gives the time remaining to complete the journey.

4

Calculate the required average speed for the remaining distance.

150 \text{ miles} \div 2 \text{ hours} = 75 \text{ mph}

Speed is distance divided by time. Applying this to the remaining segment gives the required speed.

Anahtar Kavram

To find an unknown speed in a multi-segment trip, you must base calculations on Total Distance and Total Time, rather than averaging the individual speeds.

Soru 2146Soru →

Match each circular geometry scenario with its corresponding simplified algebraic expression.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct pairings connect each geometric scenario to its simplified algebraic derivation by applying circle formulas, expanding binomials, and rationalizing denominators.

Each expression is correctly derived by applying the fundamental formulas for circle area and circumference, followed by rigorous algebraic simplification such as factoring, binomial expansion, or radical rationalization.

Adım Adım Çözüm

1

Analyze the annular pathway area scenario.

\pi x(2r + x)

Requires subtracting the inner pool area from the total outer area and expanding the squared binomial

(r+x)^2

.

2

Determine the semicircular perimeter expression.

r(\pi + 2)

Requires adding the semicircular arc length to the straight diameter base and factoring out the radius.

3

Calculate the algebraic expression for tire revolutions.

\frac{D}{\pi d}

Requires dividing the total linear distance by the circumference of the tire, utilizing the diameter.

4

Solve for the half-area radius.

\frac{R\sqrt{2}}{2}

Requires setting up an area ratio equation, solving for the new radius, and rationalizing the denominator of the resulting square root.

Anahtar Kavram

Algebraic representation and derivation of advanced circular measurements

Soru 2147Soru →

A rectangular garden gate needs a diagonal wooden brace for support. The gate is $4$ feet tall and $3$ feet wide. What is the length, in feet, of the diagonal brace?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

The diagonal brace forms the hypotenuse of a right triangle where the width (

3

feet) and height (

4

feet) are the legs. According to the Pythagorean theorem (

a^2 + b^2 = c^2

), we substitute the given lengths:

3^2 + 4^2 = c^2

. This simplifies to

9 + 16 = 25

. Taking the square root of

25

gives a diagonal length of

5

feet. This forms a common

3-4-5

Pythagorean triple.

Adım Adım Çözüm

1

Identify the right triangle formed by the gate

The legs of the right triangle are

3

and

4

, and the hypotenuse is the unknown diagonal brace.

The corners of a rectangular gate form right angles, so drawing a diagonal creates a right triangle.

2

Apply the Pythagorean theorem

3^2 + 4^2 = c^2

The Pythagorean theorem (

a^2 + b^2 = c^2

) is used to find the missing side length of a right triangle.

3

Calculate the squares of the legs

9 + 16 = c^2

Squaring the given lengths prepares the equation for addition.

4

Solve for the hypotenuse

25 = c^2

, which means

c = 5

Taking the square root of the sum yields the length of the diagonal brace.

Anahtar Kavram

Using the Pythagorean theorem to find the hypotenuse of a right triangle.

Soru 2148Soru →

A piece of wire is bent to form a square with an area of $144$ square centimeters. If the same wire is straightened and then bent to form a rectangle whose length is $4$ centimeters longer than its width, what is the area of this new rectangle?

48

square centimeters

140

square centimeters

144

square centimeters

572

square centimeters

Cevabı ve açıklamayı göster

Cevap: $140$ square centimeters

Cevap

140

square centimeters

First, find the side length of the square. Since the area is

144

, the side length is

\sqrt{144} = 12

. The total length of the wire is the perimeter of the square, which is

12 \times 4 = 48

. This wire is then used to form a rectangle, so the rectangle's perimeter is also

48

. Let the width of the rectangle be

W

. The length is

W + 4

. The perimeter formula is

2(L + W) = P

. Substituting the values gives

2(W + W + 4) = 48

, which simplifies to

4W + 8 = 48

. Solving for

W

yields

4W = 40

, so

W = 10

. The length is

10 + 4 = 14

. The area of the new rectangle is

10 \times 14 = 140

square centimeters.

Adım Adım Çözüm

1

Calculate the total length of the wire using the square's area.

The side of the square is

\sqrt{144} = 12

cm. The perimeter is

12 \times 4 = 48

cm.

The total length of the wire remains constant and equals the perimeter of both shapes.

2

Set up an equation for the rectangle's perimeter.

Let width be

W

. Length is

W + 4

. The perimeter is

2(W + (W + 4)) = 48

.

The rectangle is made from the same wire, so its perimeter is

48

cm, and its length is

4

cm longer than its width.

3

Solve for the width (

W

) and length (

L

) of the rectangle.

2(2W + 4) = 48 \Rightarrow 4W + 8 = 48 \Rightarrow 4W = 40 \Rightarrow W = 10

cm. The length is

10 + 4 = 14

cm.

Solving the linear equation provides the dimensions needed to find the area.

4

Calculate the area of the rectangle.

Area =

10 \times 14 = 140

square centimeters.

The area of a rectangle is its length multiplied by its width.

Anahtar Kavram

Perimeter and Area of 2D Figures

Soru 2149Soru →

A rectangular shipping crate has a length of $6$ feet, a width of $4$ feet, and a height of $3$ feet. What is the volume of the crate in cubic feet?

Cevabı ve açıklamayı göster

Cevap: 72

Cevap

72

The volume of a rectangular prism is calculated by multiplying its length, width, and height. Multiplying

6

feet by

4

feet by

3

feet results in exactly

72

cubic feet.

Adım Adım Çözüm

1

Recall the volume formula for a rectangular prism.

V = l \times w \times h

To find the volume, we must multiply the length, width, and height of the 3D figure.

2

Substitute the given dimensions into the formula.

V = 6 \times 4 \times 3

The problem states the length is

6

feet, the width is

4

feet, and the height is

3

feet.

3

Multiply the values to find the total volume.

V = 24 \times 3 = 72

Completing the arithmetic gives the final volume in cubic feet.

Anahtar Kavram

Volume of a Rectangular Prism

Soru 2150Soru →

A base defense unit receives a shipment of $7,500$ sandbags to reinforce security perimeters. If $18\%$ of the sandbags are deployed to the north gate and $34\%$ are deployed to the east gate, how many sandbags remain in the reserve depot?

Cevabı ve açıklamayı göster

Cevap: 3600

Cevap

There are 3,600 sandbags remaining in the reserve depot.

The correct answer is found by first determining the total percentage of sandbags allocated to the gates, which is

18\% + 34\% = 52\%

. Subtracting this from the total

100\%

leaves

48\%

for the reserve depot. Multiplying

0.48

by the initial

7,500

sandbags gives the final remaining amount of

3,600

sandbags.

Adım Adım Çözüm

1

Calculate the total percentage of deployed sandbags.

18\% + 34\% = 52\%

This consolidates the parts of the whole that have already been allocated.

2

Determine the percentage of sandbags that remain.

100\% - 52\% = 48\%

Subtracting the allocated portion from the total whole (100%) gives the remaining fraction.

3

Calculate 48% of the initial 7,500 sandbags.

0.48 \times 7500 = 3600

This translates the remaining percentage back into a specific physical quantity.

Anahtar Kavram

Calculating the remaining percentage of a quantity by subtracting the used percentages from

100\%

, then multiplying by the original total.

Soru 2151Soru →

A kite designer is creating a kite in the shape of a rhombus. The two crossing wooden dowels that form the internal diagonals of the kite measure $16$ inches and $30$ inches. What is the total perimeter of the kite's outer edge, in inches?

Cevabı ve açıklamayı göster

Cevap: 68

Cevap

68

The correct answer is found by recognizing that a rhombus's diagonals bisect each other perpendicularly. This allows the use of the Pythagorean theorem with half-diagonal lengths (

8

and

15

) to find a side length of

17

. Multiplying this single side length by the four equal sides of the rhombus yields a total perimeter of

68

inches.

Adım Adım Çözüm

1

Determine the properties of the diagonals in a rhombus.

The diagonals bisect each other at right angles, forming four identical right triangles.

This breaks down the rhombus into simpler geometric shapes (right triangles), allowing for side length calculation.

2

Calculate the lengths of the legs for one of the right triangles.

The legs are

8

inches (

16 \div 2

) and

15

inches (

30 \div 2

).

The diagonals cut each other exactly in half at the intersection.

3

Use the Pythagorean theorem to find the length of the rhombus's outer edge.

The side length is

\sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17

inches.

The outer edge forms the hypotenuse of the right triangle created by the half-diagonals.

4

Calculate the total perimeter.

17 \times 4 = 68

inches.

A rhombus has four equal sides, so the total perimeter is four times the length of one side.

Anahtar Kavram

Perimeter of a Rhombus and Pythagorean Theorem

Soru 2152Soru →

A military supply convoy is tasked with delivering equipment to an outpost $210$ miles away. The convoy commander is ordered to maintain an overall average speed of $35$ miles per hour for the entire journey. If the convoy completes the first $90$ miles in $2$ hours, what average speed, in miles per hour, must they maintain for the remainder of the trip to exactly meet their overall time target?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

The convoy must maintain a speed of 30 mph for the remainder of the trip.

The overall target requires the

210

-mile trip to be completed in

6

hours (

210 \div 35

). Because the first

90

miles took

2

hours, there are

120

miles left to travel and

4

hours remaining to do so. Therefore, the required speed is

120

miles

\div

4

hours =

30

mph.

Adım Adım Çözüm

1

Calculate the total allowable time for the convoy's trip.

Total time =

210 \div 35 = 6

hours.

Overall average speed is the total distance divided by total time. Knowing the total distance and target average speed allows us to find the total time allowed.

2

Calculate the remaining distance for the second part of the trip.

Remaining distance =

210 - 90 = 120

miles.

We need to find out how much further the convoy has to travel.

3

Calculate the remaining time available to complete the journey.

Remaining time =

6 - 2 = 4

hours.

Subtracting the time already spent from the total time gives the time left to cover the remaining distance.

4

Calculate the required average speed for the final leg.

Required speed =

120 \div 4 = 30

mph.

Average speed for any leg is the distance of that leg divided by the time spent on that leg.

Anahtar Kavram

Average speed is calculated by dividing the total distance by the total time, and the properties of individual legs (distance and time) sum to the total distance and total time.

Soru 2153Soru →

A recruiting station commander records the number of new applicants processed each day for $5$ consecutive days. The daily counts are $3$ , $3$ , $5$ , $8$ , and $11$ . Match each measure of central tendency to its correct calculated value for this data set.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct matches are: Mode pairs with

3

, Median pairs with

5

, and Mean pairs with

6

.

Based on the data set of

3, 3, 5, 8, 11

, the mode is

3

because it appears most frequently. The median is

5

because it is the central value of the ordered list. The mean is

6

because the sum (

30

) divided by the count (

5

) equals

6

.

Adım Adım Çözüm

1

Determine the mode of the data set.

Mode =

3

The mode is the number that appears most frequently.

3

is the only number that appears more than once.

2

Determine the median of the data set.

Median =

5

The data is already arranged in ascending order:

3, 3, 5, 8, 11

. The middle number (third position out of five) is

5

.

3

Calculate the mean of the data set.

Mean =

6

Calculate the sum of the numbers (

3 + 3 + 5 + 8 + 11 = 30

) and divide by the number of days (

5

):

30 \div 5 = 6

.

Anahtar Kavram

Calculating the primary measures of central tendency (mean, median, and mode) for a data set.

Tahmini Süre:45s

Soru 2154Soru →

A game at a school carnival involves drawing two wooden blocks from a bag without looking. The bag initially contains 9 red blocks, 7 yellow blocks, and 5 blue blocks. If a player draws two blocks one after the other without replacing the first block, what is the probability that the first block drawn is blue and the second block drawn is yellow?

\frac{1}{12}

\frac{5}{63}

\frac{4}{7}

\frac{12}{41}

Cevabı ve açıklamayı göster

Cevap: $\frac{1}{12}$

Cevap

\frac{1}{12}

The correct answer is found by multiplying the probability of the first event by the probability of the second event, accounting for the reduced total pool. The probability of drawing a blue block first is

\frac{5}{21}

. Because this block is not replaced, only 20 blocks remain in the bag. The probability of then drawing a yellow block is

\frac{7}{20}

. Multiplying these probabilities gives

\frac{5}{21} \times \frac{7}{20} = \frac{35}{420}

, which simplifies to

\frac{1}{12}

.

Adım Adım Çözüm

1

Calculate the total number of blocks initially in the bag.

9 \text{ (red)} + 7 \text{ (yellow)} + 5 \text{ (blue)} = 21 \text{ total blocks}

The denominator for the first probability requires the total count of all possible outcomes.

2

Determine the probability of the first event (drawing a blue block).

\frac{5}{21}

There are 5 blue blocks out of the 21 total blocks.

3

Determine the probability of the second event (drawing a yellow block) given that a blue block was already removed.

\frac{7}{20}

Because the first block was not replaced, there are now

21 - 1 = 20

total blocks. There are still 7 yellow blocks in the bag.

4

Multiply the probabilities of the two dependent events and simplify.

\frac{5}{21} \times \frac{7}{20} = \frac{35}{420} = \frac{1}{12}

To find the compound probability of both events occurring in sequence, you must multiply their individual probabilities.

Anahtar Kavram

Dependent Compound Probability

Soru 2155Soru →

A cylindrical canister has a radius of $3$ inches and a height of $5$ inches. Which of the following represents the total volume of the canister in cubic inches?

45\pi

30\pi

75\pi

180\pi

Cevabı ve açıklamayı göster

Cevap: $45\pi$

Cevap

45\pi

The correct answer is

45\pi

. The volume of a cylinder is found using the formula

V = \pi r^2 h

. Squaring the given radius of

3

results in

9

. Multiplying

9

by the given height of

5

yields

45\pi

cubic inches.

Adım Adım Çözüm

1

Identify the correct formula for the volume of a cylinder.

V = \pi r^2 h

The question asks for the volume of a cylindrical shape.

2

Substitute the given values into the formula.

V = \pi (3)^2 (5)

The radius (r) is 3 inches and the height (h) is 5 inches.

3

Square the radius according to the order of operations.

3^2 = 9

Exponents must be evaluated before multiplication.

4

Multiply the squared radius by the height and pi.

9 \times 5 \times \pi = 45\pi

This completes the volume calculation.

Anahtar Kavram

Volume of a Cylinder

Soru 2156Soru →

In a right triangle, the two shorter sides (legs) measure $10$ inches and $24$ inches. What is the length of the hypotenuse in inches?

34

60

26

676

Cevabı ve açıklamayı göster

Cevap: 26

Cevap

The length of the hypotenuse is 26 inches.

The hypotenuse is found by applying the Pythagorean theorem (

a^2 + b^2 = c^2

). When you square the given legs (

10

and

24

), you get

100

and

576

. Adding these gives

676

. Finally, taking the square root of

676

results in

26

. This is also recognized as a scaled version of the fundamental 5-12-13 Pythagorean triple (multiplied by 2).

Adım Adım Çözüm

1

Recall the Pythagorean theorem formula.

a^2 + b^2 = c^2

This formula connects the lengths of the legs (

a

and

b

) to the hypotenuse (

c

) in a right triangle.

2

Substitute the given leg lengths into the formula.

10^2 + 24^2 = c^2

The lengths of the two shorter sides are given as 10 and 24.

3

Square the lengths of the legs.

100 + 576 = c^2

Following the order of operations, evaluate the exponents before adding.

4

Add the squared values together.

676 = c^2

Combine the values to find the square of the hypotenuse.

5

Take the square root of both sides to solve for c.

c = \sqrt{676} = 26

Taking the square root provides the final length of the hypotenuse.

Anahtar Kavram

Pythagorean Theorem

Soru 2157Soru →

A sonar technician records the number of acoustic anomalies detected during $5$ consecutive watch shifts. The counts are $7, 2, 8, 2$ , and $6$ . What is the median number of anomalies detected per shift?

2

5

6

8

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

6

To find the median, the values must first be ordered from least to greatest:

2, 2, 6, 7, 8

. Because there is an odd number of values (

5

), the median is the single middle number. The third number in this sorted list is

6

.

Adım Adım Çözüm

1

Order the given numbers from least to greatest.

2, 2, 6, 7, 8

The median must always be found from a sorted list of values.

2

Identify the middle value in the sorted list.

Since there are

5

values, the middle value is the

3

rd number, which is

6

.

For an odd number of data points, the median is exactly the single middle number.

Anahtar Kavram

The median is the middle value of a data set when the numbers are arranged in numerical order.

Soru 2158Soru →

Calculate the solution $(x, y)$ for the system of linear equations shown below.

3x - 2y = 7

2x + 4y = 10

What is the value of the expression $x - y$ ?

3

6

2

8

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The correct value of the expression is 2.

The system can be efficiently solved using either substitution or elimination. By dividing the second equation by 2, we get

x + 2y = 5

, which means

x = 5 - 2y

. Substituting this into the first equation gives

3(5 - 2y) - 2y = 7

. Distributing the 3 results in

15 - 6y - 2y = 7

, which simplifies to

15 - 8y = 7

. Subtracting 15 from both sides gives

-8y = -8

, so

y = 1

. Substituting

y = 1

back into

x = 5 - 2y

yields

x = 5 - 2(1) = 3

. Finally, the requested expression is

x - y

, which evaluates to

3 - 1 = 2

.

Adım Adım Çözüm

1

Simplify the second equation by dividing all terms by 2.

x + 2y = 5

Dividing the entire equation by a common factor makes the coefficients smaller and allows us to easily isolate

x

for the substitution method.

2

Isolate

x

in the simplified equation.

x = 5 - 2y

Expressing

x

in terms of

y

prepares us to substitute this expression into the first equation.

3

Substitute the expression for

x

into the first equation and solve for

y

.

3(5 - 2y) - 2y = 7 \Rightarrow 15 - 6y - 2y = 7 \Rightarrow 15 - 8y = 7 \Rightarrow -8y = -8 \Rightarrow y = 1

This step eliminates

x

entirely, creating a single-variable linear equation that can be solved to find the exact value of

y

.

4

Substitute the value of

y

back into the isolated

x

equation to find

x

.

x = 5 - 2(1) \Rightarrow x = 5 - 2 \Rightarrow x = 3

Now that we have the numerical value for

y

, we can plug it back in to find the corresponding value for

x

.

5

Evaluate the final expression

x - y

using the calculated values.

3 - 1 = 2

The question asks specifically for the value of

x - y

, not just the ordered pair.

Anahtar Kavram

To solve a system of linear equations, you can use substitution by isolating a variable in one equation and substituting it into the other, or use elimination by combining the equations to cancel out a variable.

Soru 2159Soru →

An aviation squadron is allotted a total of $12,000$ flight hours for the training year. During the first quarter, the squadron uses $28\%$ of its total allotted hours. During the second quarter, it uses an additional $35\%$ of the total allotted hours. How many flight hours remain for the rest of the year?

4,440

5,616

7,560

444

Cevabı ve açıklamayı göster

Cevap: $4,440$

Cevap

The squadron has

4,440

flight hours remaining.

The total percentage of flight hours used in the first two quarters is the sum of

28\%

and

35\%

, which equals

63\%

. To find the remaining percentage, subtract this from

100\%

, leaving

37\%

. Finally, to find the exact number of hours left, calculate

37\%

of the total

12,000

hours by multiplying

12,000

by

0.37

, which results in

4,440

hours.

Adım Adım Çözüm

1

Add the percentages of flight hours used in the first and second quarters.

28\% + 35\% = 63\%

This determines the total percentage of the allotted hours that have already been consumed.

2

Subtract the total used percentage from

100\%

to find the remaining percentage.

100\% - 63\% = 37\%

This represents the percentage of the total flight hours that are still available for the rest of the year.

3

Calculate

37\%

of the total

12,000

flight hours.

0.37 \times 12,000 = 4,440

Converting the remaining percentage to a decimal and multiplying it by the total allocation yields the actual number of remaining hours.

Anahtar Kavram

Calculating Percentages of Quantities

Soru 2160Soru →

A navigation briefcase contains $15$ identical-looking USB drives. $9$ of the drives contain map data, and $6$ contain communication protocols. If a pilot randomly selects two drives from the briefcase one after the other without replacing the first, what is the probability that both selected drives contain map data? Express your answer as a simplified fraction.

Cevabı ve açıklamayı göster

Cevap: 12/35; 12 / 35

Cevap

12/35

The correct answer is derived by multiplying the initial probability of drawing a map drive (

\frac{9}{15}

) by the updated probability of drawing a second map drive from the remaining inventory (

\frac{8}{14}

). Simplifying the fractions to

\frac{3}{5}

and

\frac{4}{7}

before multiplying gives the final result of

\frac{12}{35}

.

Adım Adım Çözüm

1

Determine the probability of selecting a map data drive on the first draw.

P(First Map) =

9

map drives /

15

total drives =

\frac{9}{15} = \frac{3}{5}

.

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

2

Determine the probability of selecting a map data drive on the second draw, given that one map drive has already been removed.

P(Second Map) =

8

map drives remaining /

14

total drives remaining =

\frac{8}{14} = \frac{4}{7}

.

Since the first drive is not replaced, both the number of map drives and the total number of drives decrease by one. This is a dependent event.

3

Calculate the compound probability of both dependent events occurring.

\frac{3}{5} \times \frac{4}{7} = \frac{12}{35}

.

To find the probability of two dependent events happening in sequence, you multiply the probability of the first event by the conditional probability of the second event.

Anahtar Kavram

Compound probability of dependent events (without replacement)

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