find the centre and radius of the following Cycles 9 x square + 9 y square +27 x + 12 y + 19 equals 0​

The probability of drawing a purple marble from the bag is 44%.

The probability of drawing a purple marble can be determined by subtracting the sum of the probabilities of drawing an orange and a white marble from 1, since these three events are mutually exclusive and exhaustive.

Given that there is a 36% chance of drawing an orange marble and a 20% chance of drawing a white marble, the sum of these probabilities is 36% + 20% = 56%.

To find the probability of drawing a purple marble, we subtract this sum from 100% (or 1):

1 - 56% = 44%.

Therefore, the probability of drawing a purple marble from the bag is 44%.

In summary, when randomly choosing a marble from the bag, there is a 36% chance of selecting an orange marble, a 20% chance of selecting a white marble, and a 44% chance of selecting a purple marble. These probabilities add up to 100%, ensuring that one of the three outcomes will occur.

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The difference in the addition of both fractions is that they have different lowest common multiples.

We want to add the fraction expression given as:

¹/₂ + ¹/₄

Taking the Lowest common multiple of 8, we have:

(4 + 2)/8 = 6/8

However, for the second fraction expression, we have:

¹/₂ + ¹/₃

Taking the lowest common multiple which is 6, we have:

(3 + 2)/6 = 5/6

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(1) (x + 1)/(x - 2) - (x + 1)/(x^2 + 1)

(2) -2/(2x - 1) + (3/2)/(2x + 1)

To decompose the rational expressions into partial fractions, we first need to factorize the denominators. Let's start with the first expression:

Factorizing the denominator:

x^3 - 2x^2 + x - 2 = (x - 2)(x^2 + 1)

Decomposing the fraction:

We have a linear factor and a quadratic factor, so the partial fraction decomposition will be of the form:

A/(x - 2) + (Bx + C)/(x^2 + 1)

Finding the values of A, B, and C:

Multiplying both sides of the equation by the common denominator (x - 2)(x^2 + 1) gives:

x^5 - 2x^4 + x^3 + x + 5 = A(x^2 + 1) + (Bx + C)(x - 2)

By equating coefficients of corresponding powers of x, we get:

A = 1

-2A + B = -2

A - 2B + C = 1

Solving this system of equations, we find A = 1, B = -1, and C = 0.

Therefore, the partial fraction decomposition is:

(x + 1)/(x - 2) - (x + 1)/(x^2 + 1)

Now let's move on to the second expression:

Factorizing the denominator:

4x^2 - 1 = (2x - 1)(2x + 1)

Decomposing the fraction:

Since we have two linear factors, the partial fraction decomposition will be of the form:

A/(2x - 1) + B/(2x + 1)

Finding the values of A and B:

Multiplying both sides of the equation by the common denominator (2x - 1)(2x + 1) gives:

4x^2 - 14x + 2 = A(2x + 1) + B(2x - 1)

By equating coefficients of corresponding powers of x, we get:

4A + 4B = 4 (coefficients of x^2)

A - B = -7 (coefficients of x)

A - B = 1 (constant term)

Solving this system of equations, we find A = -2 and B = 3/2.

Therefore, the partial fraction decomposition is:

-2/(2x - 1) + (3/2)/(2x + 1)

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The Area of the shaded portion is: 28 in²

The formula for the area of a circle is:

A = πr²

Where:

A is Area

R is radius

Thus:

Area of circle = π * 10² = 314.2 in²

Area of quadrant = 314.2 in²/4 = 78.5 in²

Area of triangle is given by the formula:

A = ¹/₂ * base * height

Thus:

A = ¹/₂ * 10 * 10

A = 50 in²

Area of shaded portion = 78.5 in² - 50 in²

Area = 28 in²

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Answer:

Here are the steps for conducting a survey in your school to find out which mode of transportation students prefer the most, arranged in order from start to finish:

1) Define the objective: Clearly define the objective of the survey, which is to determine the preferred mode of transportation among students in your school.

2) Determine the sample size: Decide on the number of students you want to include in your survey. This will depend on factors such as the size of your school and the resources available for conducting the survey.

3) Design the survey questionnaire: Create a well-designed survey questionnaire that includes relevant questions about different modes of transportation. Ensure that the questions are clear, unbiased, and cover all the necessary aspects.

4) Seek necessary permissions: If required, seek permission from school authorities or relevant individuals to conduct the survey on school premises and to involve students.

5) Pre-test the survey: Before distributing the survey to the entire student population, pre-test the questionnaire on a small group of students to identify any potential issues or areas for improvement.

6) Distribute the survey: Once the questionnaire has been finalized and pre-tested, distribute it to the selected sample of students. Ensure that the survey is distributed in a fair and unbiased manner, taking into account the diversity of the student population.

7) Collect the survey responses: Set a specific timeframe for students to complete the survey and collect the responses. Consider using a combination of online platforms and physical collection methods to maximize participation.

8) Analyze the data: Once you have collected all the survey responses, analyze the data to determine the preferred mode of transportation among students. Use appropriate statistical tools and techniques to interpret the results accurately.

9) Present the findings: Prepare a report or presentation summarizing the survey findings. Include key insights, trends, and any significant findings related to the preferred mode of transportation among students.

10) Share the results: Share the survey results with the school community, including students, teachers, and administrators. This can be done through presentations, newsletters, or other suitable communication channels.

By following these steps, you will be able to conduct a comprehensive survey to determine the preferred mode of transportation among students in your school.

Hope this helps!

I can not do it for you but I can give you what you should do

Explanation: take a survey

The measure of angle 1 is given as 130 degrees.the measure of angle 1 is 130 degrees provides specific information about the amount of rotation between the two rays or lines

An angle is a geometric figure formed by two rays or lines that share a common endpoint called the vertex. The measure of an angle is determined by the amount of rotation between the two rays or lines.

In this case, angle 1 has a measure of 130 degrees. This means that if we were to rotate one of the rays or lines forming the angle by 130 degrees, it would coincide with the other ray or line.

The degree is a unit of measurement for angles, and it is based on dividing a full circle into 360 equal parts. Each part, or degree, corresponds to a specific amount of rotation. In this case, angle 1 is measured to be 130 degrees, which is less than half of a full circle.

When interpreting the measure of angle 1, it's important to consider the context in which it is being used. Angles can be found in various settings, such as geometry, trigonometry, or real-world applications. Depending on the context, the measure of an angle can have different interpretations and implications.

In geometry, angles are used to describe the relationships between lines, shapes, and spatial configurations. They are often classified based on their measures, such as acute (less than 90 degrees), right (exactly 90 degrees), obtuse (greater than 90 degrees but less than 180 degrees), or straight (exactly 180 degrees).

In trigonometry, angles are used to define the ratios of sides in right triangles and to study periodic functions such as sine and cosine.

In real-world applications, angles can be used to measure directions, inclinations, or orientations of objects or phenomena.

Therefore, knowing that the measure of angle 1 is 130 degrees provides specific information about the amount of rotation between the two rays or lines

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Step-by-step explanation:

To eliminate x in the system of equations:

1. Multiply Equation 1 by 9 and multiply Equation 2 by -4, this gives:

Equation 1: 36x -54y = 90

Equation 2: -36x - 8y = -28

2. Add the two equations together to eliminate x:

(36x - 54y) + (-36x - 8y) = 90 - 28

Simplifying, we get:

-62y = 62

3. Solve for y:

y = -1

4. Substitute y = -1 into one of the original equations, say Equation 1:

4x - 6(-1) = 10

Simplifying, we get:

4x + 6 = 10

5. Solve for x:

4x = 4

x = 1

Therefore, the solution to the system of equations is x = 1 and y = -1. We can check that these values are correct by substituting them back into the original equations and verifying that they satisfy both equations.

1. The area of the Earth's surface covered in water is approximately 0.71 * 201,061,929 = 142,550,781 square miles.

2. Approximately 1.76% of the Earth's surface lies in the United States.

3. The volume of the entire Earth is approximately 268,082,573,106 cubic miles.

To answer Alonzo's questions, let's calculate the required values using the given information:

1. The area of the Earth's surface that is covered in water:

The Earth's surface area can be calculated using the formula for the surface area of a sphere: 4πr^2. Given that the radius of the Earth is approximately 4,000 miles, we have:

Surface area of the Earth = 4π(4,000)^2 = 4π(16,000,000) ≈ 201,061,929 square miles.

Alonzo can research and find that about 71% of the Earth's surface is covered in water. Thus, the area of the Earth's surface covered in water is approximately 0.71 * 201,061,929 = 142,550,781 square miles.

2. The percent of the Earth's surface that lies in the United States:

The land area of the United States is approximately 3,537,438 square miles. To calculate the percentage, we divide the land area of the United States by the total surface area of the Earth and multiply by 100:

Percentage = (3,537,438 / 201,061,929) * 100 ≈ 1.76%.

Therefore, approximately 1.76% of the Earth's surface lies in the United States.

3. The volume of the entire Earth:

The volume of a sphere can be calculated using the formula: (4/3)πr^3. Substituting the radius of the Earth, we have:

Volume of the Earth = (4/3)π(4,000)^3 = (4/3)π(64,000,000,000) ≈ 268,082,573,106 cubic miles.

Thus, the volume of the entire Earth is approximately 268,082,573,106 cubic miles.

These calculations provide Alonzo with the approximate values he needs regarding the Earth's surface area covered in water, the percentage of the Earth's surface within the United States, and the volume of the entire Earth.

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T satisfies both the additive and scalar multiplication properties, it can be concluded that T is a linear transformation.

To prove that T is a linear transformation, we need to show that it satisfies two properties: additive property and scalar multiplication property.

Additive property:

Let z = (x1, x2) be an arbitrary vector in R2. We want to confirm that T(z1 + z2) = T(z1) + T(z2).

Let z1 = (x1, x2) and z2 = (y1, y2) be arbitrary vectors in R2.

T(z1 + z2) = T((x1 + y1, x2 + y2)) = (x1 + y1, x2 + y2)

T(z1) + T(z2) = T(x1, x2) + T(y1, y2) = (x1, x2) + (y1, y2) = (x1 + y1, x2 + y2)

We can see that T(z1 + z2) = T(z1) + T(z2), thus satisfying the additive property.

Scalar multiplication property:

Let z = (x1, x2) be an arbitrary vector in R2 and k be an arbitrary scalar. We want to confirm that T(kz) = kT(z).

T(kz) = T(kx1, kx2) = (kx1, kx2)

kT(z) = kT(x1, x2) = k(x1, x2) = (kx1, kx2)

We can see that T(kz) = kT(z), thus satisfying the scalar multiplication property.

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Carey's graph will be a straight line with a slope of -1 and a y-intercept of 3.

Carey correctly graphs a linear function with a slope of -1 and a y-intercept of 3. A linear function is represented by the equation y = mx + b, where m is the slope and b is the y-intercept.

In this case, the slope is -1, which means that for every unit increase in the x-coordinate, the y-coordinate decreases by 1 unit. The y-intercept is 3, indicating that the graph intersects the y-axis at the point (0, 3).

To plot the graph, Carey starts by marking the point (0, 3) on the graph. Then, for every unit increase in the x-coordinate, Carey moves one unit downward. Similarly, for every unit decrease in the x-coordinate, Carey moves one unit upward. These steps ensure the correct slope of -1.

After connecting the points, Carey will obtain a line that starts at the y-intercept (0, 3) and slants downward, with a slope of -1. The resulting graph will be a straight line extending to both sides of the coordinate plane.

In conclusion, it is represented by the equation y = -x + 3.

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The central angle measure, x, for the Baseball slice in the circle graph is 50.4 degrees.

Given that, a survey was conducted in which 2400 people were asked to name their favorite sport to watch and the table below shows their answers: Sport percentage of people are:

Football 29%

Soccer 9%

Baseball 14%

Basketball 8%

Hockey 8%

Other 32%

We need to find the central angle measure, x, for the Baseball slice in the circle graph.

For this, we need to use the formula that gives the central angle measure in degrees for a sector of a circle:

Central angle measure = (Percentage/100) × 360°

Now, using the above formula, we can calculate the central angle measure for each sport as shown below:

Football: (29/100) × 360° = 104.4°

Soccer: (9/100) × 360° = 32.4°

Baseball: (14/100) × 360° = 50.4°

Basketball: (8/100) × 360° = 28.8°

Hockey: (8/100) × 360° = 28.8°

Other: (32/100) × 360° = 115.2°

The sum of all central angle measures should be 360°, which is the measure of a full circle.

So we can check if the calculations are correct: 104.4° + 32.4° + 50.4° + 28.8° + 28.8° + 115.2° = 360°

We see that the sum is indeed 360°, so the calculations are correct. The central angle measure for the Baseball slice is 50.4°.

Therefore, the central angle measure, x, for the Baseball slice in the circle graph is 50.4 degrees.

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The Slope of the line passing through the points (-2, -3) and (-4, -9) is 3.

The slope of a line, we can use the formula:

slope = (change in y) / (change in x)

Given the coordinates of two points on the line: (-2, -3) and (-4, -9), we can calculate the slope.

Let's denote the coordinates of the first point as (x₁, y₁) = (-2, -3) and the coordinates of the second point as (x₂, y₂) = (-4, -9).

The change in y is equal to y₂ - y₁, and the change in x is equal to x₂ - x₁.

change in y = -9 - (-3) = -9 + 3 = -6

change in x = -4 - (-2) = -4 + 2 = -2

Now we can substitute these values into the slope formula:

slope = (-6) / (-2)

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 2:

slope = 6 / 2

Simplifying further:

slope = 3

Therefore, the slope of the line passing through the points (-2, -3) and (-4, -9) is 3.

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The perimeter of the figure, rounded to the nearest tenths place, is 41.1 units.

To find the perimeter of the figure composed of a square and four semicircles, we need to determine the lengths of the square's sides and the semicircles' arcs.

Given that the side length of the square is 4 units, the perimeter of the square is simply the sum of all four sides, which is 4 + 4 + 4 + 4 = 16 units.

Now, let's focus on the semicircles. Each semicircle's diameter is equal to the side length of the square, which is 4 units. Therefore, the radius of each semicircle is half of the diameter, or 2 units.

The formula to find the arc length of a semicircle is given by θ/360 * 2πr, where θ is the angle of the arc and r is the radius. In this case, the angle of the arc is 180 degrees since we are dealing with semicircles.

Using the formula, the arc length of each semicircle is 180/360 * 2π * 2 = π * 2 = 2π units.

Since there are four semicircles in the figure, the total length of the arcs is 4 * 2π = 8π units.

Finally, we can calculate the perimeter by adding the length of the square's sides and the length of the semicircles' arcs:

Perimeter = Length of square's sides + Length of semicircles' arcs

= 16 units + 8π units

To round the perimeter to the nearest tenths place, we need to determine the approximate value of π. Taking π as approximately 3.14, we can calculate the approximate perimeter as:

Perimeter ≈ 16 + 8 * 3.14 ≈ 16 + 25.12 ≈ 41.12 units.

Therefore, the perimeter of the figure, rounded to the nearest tenths place, is approximately 41.1 units.

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Answer:

0.2369

Step-by-step explanation:

To find the probability of exactly 3 out of 12 randomly selected college students working full time, we can use the binomial probability formula.

The formula for the probability of exactly k successes in n trials, where the probability of success is p, is:

P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)

In this case, n = 12 (number of trials), k = 3 (number of successes), and p = 0.20 (probability of success, i.e., percentage of college students working full time).

Plugging in the values:

P(X = 3) = (12 choose 3) * 0.20^3 * (1 - 0.20)^(12 - 3)

Calculating the expression:

P(X = 3) = (12! / (3! * (12 - 3)!)) * 0.20^3 * (0.80^9)

= (12! / (3! * 9!)) * 0.008 * 0.134217728

≈ 0.2369 (rounded to 4 decimal places)

Therefore, the probability that exactly 3 out of the 12 randomly selected college students work full time is approximately 0.2369.

Hope this helps!

Jordan bikes at a Faster speed than Hayden because he covers a greater distance in the same amount of time.

To determine who bikes at a faster speed, we can compare the rates at which Hayden and Jordan cover distance over a given time period.

Hayden bikes 1.8 miles in 6 minutes, which can be expressed as a rate of 1.8 miles / 6 minutes = 0.3 miles per minute.

Jordan bikes 3.2 miles in 8 minutes, which can be expressed as a rate of 3.2 miles / 8 minutes = 0.4 miles per minute.

Comparing the two rates, we can see that Jordan bikes at a faster speed. Jordan covers a greater distance (3.2 miles) in the same amount of time (8 minutes) compared to Hayden, who only covers 1.8 miles in 6 minutes. Therefore, Jordan's rate of 0.4 miles per minute is greater than Hayden's rate of 0.3 miles per minute, indicating that Jordan bikes at a faster speed.

In summary, Jordan bikes at a faster speed than Hayden because he covers a greater distance in the same amount of time.

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To find the expected number of times you would wait between 19 and 23 minutes, we need to calculate the z-scores for these values and use them to find the area under the normal distribution curve between those values.

First, we calculate the z-score for 19 minutes:

z = (19 - 22) / 3 = -1

Next, we calculate the z-score for 23 minutes:

z = (23 - 22) / 3 = 0.33

Using a standard normal distribution table or calculator, we can find the area under the normal distribution curve between these z-scores:

P(-1 < z < 0.33) = 0.4082 - 0.3413 = 0.0669

This means that there is a probability of 0.0669 of waiting between 19 and 23 minutes for a single visit to the restaurant.

To find the expected number of times you would wait between 19 and 23 minutes over 37 visits, we multiply the probability for a single visit by the number of visits:

Expected number of times = 0.0669 x 37 ≈ 2.47

Rounding to the nearest whole number, we would expect to wait between 19 and 23 minutes about 2 times over 37 visits to the restaurant.

The pulley ratio is approximately 0.6667, calculated by dividing the Diameter of pulley A (5.74 cm) by the diameter of pulley B (8.61 cm).

The pulley ratio, we need to compare the diameters of the two pulleys. The pulley ratio is the ratio of the diameters of pulley A to pulley B.

Given that the diameter of pulley A is 5.74 cm and the diameter of pulley B is 8.61 cm, we can calculate the pulley ratio using the following steps:

Step 1: Write down the diameters of pulley A and pulley B.

  Diameter of pulley A = 5.74 cm

  Diameter of pulley B = 8.61 cm

Step 2: Calculate the pulley ratio.

  Pulley ratio = Diameter of pulley A / Diameter of pulley B

Substituting the given values, we have:

  Pulley ratio = 5.74 cm / 8.61 cm

Step 3: Simplify the ratio if possible.

  In this case, the ratio cannot be simplified further since the diameters do not have any common factors other than 1.

Step 4: Calculate the final result.

  Pulley ratio = 5.74 cm / 8.61 cm ≈ 0.6667 (rounded to four decimal places)

Therefore, the pulley ratio is approximately 0.6667.

When discussing technical concepts and calculations, it is important to maintain academic integrity and avoid plagiarism. Plagiarism involves using someone else's work or ideas without proper attribution. To ensure originality, it is essential to express the information in your own words and provide accurate calculations based on the given data.

In conclusion, the pulley ratio is approximately 0.6667, calculated by dividing the diameter of pulley A (5.74 cm) by the diameter of pulley B (8.61 cm).

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The angle measure of an arc π meters is 45°.

Given that, the diameter of a circle is 8 meters.

Radius of a circle = 8/2 = 4 meters

The formula to find the arc length of a circle is θ/360° ×2πr.

Here, θ/360° ×2πr.

π=θ/360° ×2π×4

1=θ/360° ×8

360°=8θ

θ=360°/8

θ=45°

Therefore, the angle measure of an arc π meters is 45°.

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Answer:kkkkkkkkkkkkkkkkkkkkkkkkkkk mean ok 26 times

The two pairs of polar coordinates for the point (3, -3) when x² - y² = 4 in polar coordinates are (3√2, -45°) or (3√2, 315°) and (-3√2, 135°).

Given the Cartesian coordinate (3, -3), and the equation x² - y² = 4.To convert Cartesian coordinates to polar coordinates, we use the formula:r = sqrt(x² + y²)θ = tan⁻¹(y/x)

The first step is to substitute the given Cartesian coordinates into the formula:

r = sqrt(3² + (-3)²)r = sqrt(18)r = 3√2

To determine the angle θ, we first look at the sign of both the x and y coordinates. Since x is positive and y is negative, the angle θ is in the fourth quadrant.

To determine the angle θ, we use the formula:θ = tan⁻¹(y/x)θ = tan⁻¹(-3/3)θ = -45°Alternatively, we can add 360° to get the angle in the fourth quadrant:θ = 315°

Therefore, one pair of polar coordinates is (3√2, -45°) or (3√2, 315°).To determine the second pair of polar coordinates, we can add 180° to the angle and negate the radius,

since this will give us the same point but in the opposite direction.θ = -45° + 180° = 135°r = -3√2

Therefore, the second pair of polar coordinates is (-3√2, 135°).

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The value of x is calculated as 30.

The value of y is calculated as 28.

The measure of x and y is calculated by applying the following circle theorem as follows;

If line XZ is the diameter of the circle, then angle XYZ will be equal to 90 degrees.

The value of x is calculated as;

3x = 90

x = 90 / 3

x = 30

The value of y is calculated as follows;

2y + 34 = 90 (complementary angles sum up to 90 degrees)

2y = 56

y = 56/2

y = 28

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The calculation of this can be done by first determining the future value of the monthly payments of $327.50

The future value of an annuity can be determined using a financial calculator, mathematical formula, or spreadsheet software. The future value of an annuity is calculated by multiplying the periodic payment amount by the future value factor,

which is based on the number of payments and the interest rate.For example, suppose we want to know the future value of a $500 end-of-month deposit into an annuity that pays 6% interest compounded monthly for five years.

The future value factor for 60 periods at 0.5 percent per month is 80.9747, which can be multiplied by the monthly deposit amount to find the future value of the annuity.500 × 80.9747 = 40,487.35

This means that a $500 end-of-month deposit into an annuity paying 6% interest compounded monthly for five years will have a future value of $40,487.35.

Therefore, to accumulate a $20,000 down payment for a home in five years, you would need to deposit $327.50 per month into the annuity.

 for 60 months using the formula and then solving for the monthly payment amount where FV = $20,000 and n = 60, r = 0.5%.FV = PMT [(1 + r)n – 1] / r$20,000 = PMT [(1 + 0.005)60 – 1] / 0.005PMT = $327.50 (rounded to the nearest cent).

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The percent of the front page taken up by the prom story is 20%

From the question, we have the following parameters that can be used in our computation:

The front page

From the front page, we have

Area front page = 5 * 4

Area front page = 20

Also, we have

Prom = 2 * 2

Prom = 4

So, we have

Percentage = 4/20 * 100%

Evaluate

Percentage = 20%

Hence, the percent of the front page taken up by the prom story is 20%

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Total mangoes: 15

Leftover mangoes after monkey took two fifth:

2/5 * 15

= 6

So, two-fifths of the mangoes are 6.

Now calculating leftover:

15 - 6

= 9 mangoes(ANSWER)

Step-by-step explanation:

First    2x * 2x =   4x^2

Outer   2x * -3 =  -6x

Inner     3 * 2x = 6x

Last    3 * - 3 = -9

4x^2 -6x + 6x - 9   =  4x^2 - 9

Answer

4x²-9

Step-by-step explanation:

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The measure of the arc angle WX is 100 degrees.

The arc angle WX can be found as follows:

The measure of an inscribed angle is half of the measure of the intercepted arc and half the measure of the central angle intersecting the same arc.

Therefore,

arc WY = 2(75)

arc WY =  150 degrees

Therefore, let's find the arc  angle  WX as follows:

arc  angle  WX  = 360 - 150 - 110

arc  angle  WX  = 210 - 110

arc  angle  WX  =  100 degrees

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Find the common ratio, r, of the sequence

12 000 * r = 9600

r = 9600/12000 =  0.8

12000 ( 0.8)^6   = $ 3145.73  in year 7  ( I used '6' instead of '7' because 12 000 is listed as year '1'   and not year '0' )

The exact value of sin(pi/3) is √3. By definition, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. Therefore, sin(pi/3) = √3/1 = √3.

The exact value of sin(pi/3) can be determined using trigonometric properties and identities.

First, we know that pi/3 is equivalent to 60 degrees. In a unit circle, the point corresponding to 60 degrees forms an equilateral triangle with the origin and the x-axis. This triangle has side lengths of 1, 1, and √3.

To find the sine of pi/3, we consider the side opposite the angle (pi/3) in the triangle. In this case, the opposite side has a length of √3. The hypotenuse of the triangle is 1, as it is the radius of the unit circle.

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We can use the logarithmic identity log_a(b) + log_a(c) = log_a(bc) to simplify the left side of the equation:

log_2(x) + log_4(x+1) = log_2(x) + log_2((x+1)^(1/2))

Using the rule log_a(b^c) = c*log_a(b), we can simplify further:

log_2(x) + log_2((x+1)^(1/2)) = log_2(x(x+1)^(1/2))

Now we can rewrite the equation as:

log_2(x(x+1)^(1/2)) = 3

Using the rule log_a(b^c) = c*log_a(b), we can rewrite this as:

x(x+1)^(1/2) = 2^3

Squaring both sides, we get:

x^2 + x - 8 = 0

This is a quadratic equation that can be solved using the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 1, b = 1, and c = -8. Plugging in these values, we get:

x = (-1 ± sqrt(1^2 - 4(1)(-8))) / 2(1)

x = (-1 ± sqrt(33)) / 2

x ≈ -2.54 or x ≈ 3.54

However, we must check our solutions to make sure they are valid. Plugging in x = -2.54 to the original equation results in an invalid logarithm, so this solution is extraneous. Plugging in x = 3.54 yields:

log_2(3.54) + log_4(4.54) = 3

0.847 + 0.847 = 3

So x = 3.54 is the valid solution to the equation.

The Function  f(g(x)) = g(f(x)) = x + 8.

The functions are: f(x) = x + 4 and g(x) = x + 4. We can find f(g(x)) by substituting g(x) in place of x in f(x).

f(g(x)) = f(x + 4) = (x + 4) + 4 = x + 8

Similarly, we can find g(f(x)) by substituting f(x) in place of x in g(x).g(f(x)) = g(x + 4) = (x + 4) + 4 = x + 8

Thus, we can see that f(g(x)) and g(f(x)) are equal to each other,

which is x + 8.

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