100 Points! Algebra question. Write a quadratic equation in standard form with -1/2, 3/4 as its roots. Photo attached. Thank you!

0.685 is a decimal that equals 68.5%.

Answer: Using the formula i = prt, we have:

i = (104292.12 - 80040) = 24252.12

p = 80040

t = 3

Substituting these values, we get:

24252.12 = 80040 * r * 3

Solving for r, we get:

r = 0.101 or 10.1%

Therefore, the interest rate is 10.1%.

Step-by-step explanation:

The range of the function in the graph is  6≤e≤12. So correct option is A.

In mathematics, range is a term used to refer to the set of all possible output values of a function. It is the set of values that the function can take as its input varies over its entire domain. In other words, the range of a function is the set of all output values that can be obtained by evaluating the function for all possible input values.

For example, consider the function f(x) = x². The domain of this function is all real numbers, but the range is only non-negative real numbers, since x² is always non-negative for any real number x.

The range of a function can be determined by analyzing its graph, which is a visual representation of the function. The range corresponds to the set of all y-values that appear on the graph. For instance, the range of the function f(x) = sin(x) is the closed interval [-1, 1], since the sine function oscillates between -1 and 1 as its input varies over all real numbers.

Sometimes, it is useful to restrict the domain of a function in order to obtain a specific range. This process is called domain restriction or range selection. For example, the inverse function of f(x) = x² can be obtained by restricting the domain of f to non-negative real numbers, which ensures that the inverse function is also a function. The resulting function is f^-1(x) = √x, whose domain is non-negative real numbers and range is the same as the domain of f.

The range of the function in the graph is  6≤e≤12. So correct option is A.

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The statement that (sin 2x) / (1 + cos 2x) = tan x can be proven.

To prove that (sin 2x) / (1 + cos 2x) = tan x, we will use trigonometric identities.

(sin 2x) / (1 + cos 2x)

(2sin x × cos x) / (1 + (cos²x - sin²x))

(2sin x × cos x) / (cos²x + 2sin x × cos x + sin²x)

We can rewrite the denominator using the Pythagorean identity sin²x + cos²x = 1:

(2sin x × cos x) / (1 + 2sin x × cos x)

(2sin x × cos x) × (1 - 2sin x × cos x) / (1 - (2sin x × cos x)²)

((2sin x × cos x) - (4sin²x × cos²x)) / (1 - 4sin²x × cos²x)

(2sin x - 4sin²x) / (1/cos²x - 4sin²x)

Since tan x = sin x / cos x, we can rewrite the expression:

(2tan x - 4tan²x) / (sec²x - 4tan²x)

(2tan x - 4tan²x) / (1 + tan²x - 4tan²x)

(2tan x - 4tan²x) / (1 - 3tan²x)

2tan x × (1 - 2tan²x) / (1 - 3tan²x)

tan x

So, we have proved that (sin 2x) / (1 + cos 2x) = tan x.

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if we change the value of 523 to 424 in the list of numbers, then the mean decreases by approximately 3.22 and the median stays the same.

How to calculate the mean?

To calculate the mean, we add up all the numbers in the list and divide by the total number of values. Before the change, the sum of the numbers is:

164 + 225 + 227 + 250 + 261 + 268 + 277 + 379 + 523 = 2494

And there are 9 numbers in the list. So the mean is:

2494 / 9 ≈ 277.11

If we change the value of 523 to 424, then the sum becomes:

164 + 225 + 227 + 250 + 261 + 268 + 277 + 379 + 424 = 2465

And there are still 9 numbers in the list. So the new mean is:

2465 / 9 ≈ 273.89

So the mean decreases by approximately 3.22.

To calculate the median, we find the middle value of the list. If the list has an odd number of values, then the median is the middle value. If the list has an even number of values, then the median is the average of the two middle values. In this case, the list has an odd number of values, so the median is:

261

If we change the value of 523 to 424, then the list becomes:

164, 225, 227, 250, 261, 268, 277, 379, 424

And the median is still:

261

So the median stays the same.

In summary, if we change the value of 523 to 424 in the list of numbers, then the mean decreases by approximately 3.22 and the median stays the same.

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The axis of symmetry is x = -1 and the vertex is (-1, -2).

A parabola is a U-shaped curve that can be formed by graphing a quadratic function. It is a type of conic section, which is a curve formed by intersecting a plane with a cone at a certain angle.

To graph the function F(x) = x² + 2x - 1, we can first find the vertex and axis of symmetry using the formula:

x = -b/2a

where a and b are the coefficients of the x² and x terms, respectively. In this case, a = 1 and b = 2, so:

x = -2/(2*1) = -1

So the axis of symmetry is x = -1. To find the vertex, we can substitute this value of x into the function to get:

F(-1) = (-1)² + 2(-1) - 1 = -2

So the vertex is (-1, -2).

To graph the function, we can plot the vertex and then use symmetry to plot points on either side of the axis. We can also note that since a = 1 is positive, the parabola opens upward.

A rough sketch of the graph is shown.

The x-axis represents the axis of symmetry, and the y-axis represents the values of the function F(x). The vertex (-1, -2) is marked with an "x", and other points on the graph are marked with "x" to show the general shape of the parabola.

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When X = 5, Y is approximately equal to 27.34.

What is proportion?

The size, number, or amount of one thing or group as compared to the size, number, or amount of another. The proportion of boys to girls in our class is three to one.

We are given that "Y varies directly as the cube of X", which can be written as:

Y = kX³

where k is a constant of proportionality. We need to find the value of k to solve for Y when X = 5.

Using the values given in the problem, we can write:

7 = k(4³)

Simplifying this equation, we get:

7 = 64k

Dividing both sides by 64, we get:

k = 7/64

Now that we know the value of k, we can solve for Y when X = 5:

Y = (7/64)(5³) = 27.34 (rounded to two decimal places)

Therefore, when X = 5, Y is approximately equal to 27.34.

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Answer: 119

Step-by-step explanation:

We know that we must find the least common multiple of 15 and 27 in order to solve the problem because when we are finding remainders that are the same, there must be some relationship between the integer and the two dividing numbers.

Thus, we have the least common multiple 135 and its multiples which will all have a 0 remainder when divided by 15 and by 27.

We can take each of the numbers (7) and the 15 consecutive numbers after each of them, because of modulo becoming the same after 15. If we take the total of all these numbers, which will have the same remainder after division by 15 and 27, we are left with:

15 * 8 - 1 = 119

Therefore, the wire Ryan wants to walk is approximately 107.9 ft long.

Trigonometric functions are mathematical functions that relate the angles of a right triangle to the lengths of its sides. The most commonly used trigonometric functions are sine, cosine, and tangent, often abbreviated as sin, cos, and tan respectively.

The sine function (sinθ) gives the ratio of the length of the side opposite an angle θ in a right triangle to the length of the hypotenuse (the longest side). The cosine function (cosθ) gives the ratio of the length of the adjacent side to θ to the length of the hypotenuse, while the tangent function (tanθ) gives the ratio of the length of the opposite side to θ to the length of the adjacent side.

Other trigonometric functions include cosecant (cscθ), secant (secθ), and cotangent (cotθ), which are the reciprocals of the sine, cosine, and tangent functions, respectively.

A represents the top of the tower, B represents the point where the wire is attached 30 ft away from the tower, and the wire itself is represented by the line segment AB. The angle between the wire and the ground is θ, and we want to find the length of the wire, which is represented by y.

We can use the trigonometric function tangent to relate the angle θ to the sides of the triangle:

tan(θ) = y/x

Solving for y, we get:

y = x * tan(θ)

We know that x = 30 ft, and θ = 74º, so we can plug those values in and calculate y:

y = 30 ft * tan(74º) = 107.9 ft (rounded to the nearest tenth)

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The values are;

1. x = 6 ,y = 6√2

2. x = 9√2, y = 18

3. x = y = 9

4. x = 12, y = 12√2

5. x = y = 4√2

6. x = y =( 3√2)/2

Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle.

There are some special angles , in which 45 is part of them.

sin 45 = 1/√2

cos 45 = 1/√2

tan 45 = 1

1. x = 6 ( isosceles triangle)

y = 6 × √2 = 6√2

2. x = 9√2 ( isosceles triangle)

y = 9√2 × √2 = 9×2 = 18

3. x = 9√2/√2 = 9

x = y = 9 ( isosceles triangle)

4. x = 12 ( isosceles triangle)

y = 12×√2 = 12√2

5. x = 8/√2 = 8√2/2 = 4√2

x = y = 4√2( isosceles triangle)

6. x = 3/√2 = (3√2)/2

x = y =( 3√2)/2 ( isosceles triangle)

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===================================================

Explanation:

Focus on triangle BDH

The three inside angles of any triangle always add to 180 degrees.

B+D+H = 180

47+31+H = 180

78+H = 180

H = 180-78

H = 102

Angle BHD is 102 degrees.

It adds to angle x, aka angle BHC, to get 180 degrees. These two adjacent angles combine to a straight line.

(angle BHD) + (angle BHC) = 180

102 + x = 180

x = 180-102

--------------

Shortcut:

Focus on triangle BDH.

Use the remote interior angle theorem to add the given interior angles.

B+D = 47+31 = 78

This is equal to the exterior angle that is not adjacent to either interior angle mentioned. This refers to angle BHC, aka angle x.

The surface area of the triangular prism with rectangular base is 736 cm².

Surface area refers to the total area that the surface of an object covers. It is a measure of how much exposed area there is on the outside of an object. The surface area of an object is usually expressed in square units, such as square meters, square centimeters, or square feet.

Here,

The formula for the surface area of a triangular prism with rectangular base is:

SA = 2lw + lh + wh

where l is the length of the rectangular base, w is the width of the rectangular base, h is the height of the triangular prism.

Given:

Side of triangle = 16 cm and 12 cm

Side of rectangle = 20 cm and 10 cm

Since we are not given the height of the triangular prism, we cannot calculate the exact surface area. However, we can give a general formula for the surface area in terms of the height h:

SA = 2(20)(10) + (16)(h) + (12)(h)

SA = 400 + 28h

So the surface area of the triangular prism with rectangular base is 400 + 28h, where h is the height of the triangular prism in cm.

h=12 cm

400+28h=400+28*12

=736 cm²

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Answer: the speed of the boat is approximately 9.43 miles per hour.

Step-by-step explanation: To find boat speed, add vectors. Boat goes east at 8 mph, with 5 mph north wind. Boat velocity is a vector eastward and current velocity is a vector northward. To combine velocities, use vector addition. Use Pythagorean theorem to find resultant velocity: resultant speed = √89 ≈ 9.43 mph.

This sentence relating to the quotient can be expressed mathematically as:

(30 / x) - 2

This sentence can be expressed mathematically as:

(30 / x) - 2

where x represents the unknown number.

The word "quotient" indicates that we are dividing 30 by the unknown number x. The phrase "is decreased by 2" means that we need to subtract 2 from the quotient.

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Therefore, we can predict that approximately 5,845,200 people left on cruises from this state in the year 2010.

A linear equation is an algebraic equation of the form y=mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included.

This linear equation depicts the number of passengers embarking on cruises from a certain state between 2005 and 2009.

It goes like this:

y = 117,000x + 4,919,200

For example, if x = 0, which represents the year 2005, then y = 4,919,200. If x = 1, which represents the year 2006, then y = 4,919,200 + 117,000 = 5,036,200. Similarly, if x = 2, which represents the year 2007, then y = 4,919,200 + 2(117,000) = 5,153,200.

y = 117,000(5) + 4,919,200

y = 5,845,200

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Question:

The number of people, y, leaving on cruises from a certain state from 2005 to 2009 can be approximated by y=117,000x+4,919,200, where x is the number of years after 2005.

A graph of the points is shown in the image below.

If ABCD is rotated 90° clockwise (U) about the origin to form A'B'C'D', the coordinates of the vertices of A'B'C'D'

In Mathematics, a rotation is a type of transformation which moves every point of the object through a number of degrees around a given point, which can either be clockwise or counterclockwise (anticlockwise) direction.

Additionally, the rotation of a point 90° about the center (origin) in a clockwise direction would produce a point that has these coordinates (y, -x).

Next, we would apply a rotation of 90° clockwise to the coordinate of quadrilateral ABCD as follows;

(x, y)                               →            (y, -x)

Coordinate A = (-3, 4) → Coordinate A' = (4, -(-3)) = (4, 3)

Coordinate B = (1, 6) → Coordinate B' = (6, -(1)) = (6, -1)

Coordinate C = (5, -2) → Coordinate C' = (-2, -(5)) = (-2, -5)

Coordinate D = (1, -4) → Coordinate D' = (-4, -(1)) = (-4, -1)

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Answer:

x = √17

y = 10.1

Step-by-step explanation:

x² + 8² =  9²

x² = 81 - 64 = 17

x = √17

sin∅ = √17/9

∅ = 27.27°

9/y = cos(27.27)

y = 9/cos(27.27) = 10.13

y = 10.1

Answer: Option A:

To calculate the total amount Owen will spend on Option A, we need to calculate the monthly payment and then multiply it by the number of months:

First, we need to calculate the total amount of the loan. Owen is making a down payment of $3200, so he will be borrowing $28,820 (the price of the car minus the down payment).

Next, we can use the formula for calculating the monthly payment for a loan:

P = (r * A) / (1 - (1 + r)^(-n))

where P is the monthly payment, r is the monthly interest rate, A is the total amount of the loan, and n is the number of months.

For Option A, the monthly interest rate is 1.3% / 12 = 0.01083, the total amount of the loan is $28,820, and the number of months is 36. Plugging these values into the formula, we get:

P = (0.01083 * 28,820) / (1 - (1 + 0.01083)^(-36)) = $860.45

Therefore, the total amount Owen will spend on Option A is:

36 * $860.45 = $30,975.98

Option B:

For Option B, we need to take into account the $1500 cash back that Owen will receive as part of the down payment. This means that the total amount of the loan will be $32,020 - $3200 - $1500 = $27,320.

To calculate the monthly payment, we can use the same formula as before:

P = (r * A) / (1 - (1 + r)^(-n))

For Option B, the monthly interest rate is 5.2% / 12 = 0.04333, the total amount of the loan is $27,320, and the number of months is 36. Plugging these values into the formula, we get:

P = (0.04333 * 27,320) / (1 - (1 + 0.04333)^(-36)) = $825.53

Therefore, the total amount Owen will spend on Option B is:

36 * $825.53 + $1500 = $30,316.08

Therefore, Option A will cost Owen a total of $30,975.98, and Option B will cost him a total of $30,316.08. Therefore, Option B is the cheaper option for Owen.

Step-by-step explanation:

The ratio of the radii is equal to the ratio of the distances between corresponding points on the two circles, we can conclude that the two circles are similar.

A circle is a two-dimensional geometric shape that consists of all points in a plane that are equidistant from a fixed point called the center. The distance between the center and any point on the circle is called the radius. The line passing through the center of the circle and connecting two points on the circumference is called the diameter, which is twice the length of the radius.

The circle is a fundamental concept in geometry and has many important properties. One of the most well-known properties of a circle is that the circumference, which is the distance around the circle, is proportional to the diameter, and this proportionality constant is pi (π).

To prove that the two circles are similar, we need to show that their radii are proportional.

Let's first find the radius of circle A. The center of circle A is at (3, 4), and one point on the circle is (0, 8). The distance between these two points is the radius of the circle:

r_A = √((0 - 3)² + (8 - 4)²) = 5

Now let's find the radius of circle C. The center of circle C is at (0, -1), and one point on the circle is (0, 1). The distance between these two points is the radius of the circle:

r_C = √((0 - 0)² + (1 - (-1))²) = 2

So the ratio of the radii of the two circles is:

r_A / r_C = 5 / 2

Next, we need to show that the distance between corresponding points on the two circles is also proportional to the ratio of the radii. Let's take the point (0, 0) on circle A and the corresponding point (2, 0) on circle C. The distance between these two points is:

d = √((0 - 2)² + (0 - (-1))²) = √(5)

So the ratio of the distances is:

d_A / d_C = √(5) / 2

Since the ratio of the radii is equal to the ratio of the distances between corresponding points on the two circles, we can conclude that the two circles are similar.

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The solution of the quadratic equation is option C {-3, 2}.

Numerous techniques, such as factoring, completing the square, and the quadratic formula, can be used to solve quadratic equations. The values of x that make a quadratic equation true are represented by the solutions to the equation, which are known as its roots or zeros.

In several disciplines, including physics, engineering, economics, and finance, quadratic equations are frequently employed to simulate real-world occurrences. They can be used to depict a variety of phenomena, including a projectile's motion, an electrical circuit's behaviour, a satellite's trajectory, the cost of production, and many more.

The solutions of the quadratic equation are the x-intercepts of the graph. That is the point at which the graph, crosses the x-axis.

For the given graph the x-intercepts are {-3, 2}.

Hence, the solution of the quadratic equation is option C {-3, 2}.

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Answer: To calculate the probability that the shipment is rejected, we need to find the probability that at least one of the three components selected is defective.

First, we can find the probability that none of the three components selected is defective, which would mean the shipment is accepted. To do this, we use the hypergeometric distribution formula:

P(X = k) = (C(N-k,n-K) * C(K,k)) / C(N,n)

where:

N = total number of components in the shipment (160)

K = number of defective components in the shipment (4)

n = number of components selected for testing (3)

k = number of non-defective components among the selected components (i.e., all 3 components are non-defective)

Plugging in the values, we get:

P(all 3 components are non-defective) = (C(160-4-3,3-0) * C(4,0)) / C(160,3)

= (C(153,3) * C(4,0)) / C(160,3)

= (22,446 * 1) / 6,384,320

= 0.00352

Therefore, the probability that the shipment is accepted is 0.00352.

To find the probability that the shipment is rejected, we can subtract this probability from 1, since the probabilities of all possible outcomes must add up to 1:

P(at least one component is defective) = 1 - P(all 3 components are non-defective)

= 1 - 0.00352

= 0.99648

Therefore, the probability that the shipment is rejected is 0.99648, or approximately 99.65%.

Step-by-step explanation:

A: The Water Balloon height increase  from 0 to 2 seconds B: Water Balloon's height stay same from 2 to 4 seconds. C: Height decreasing fastest at 4 to 6 seconds. D: Almost near to ground as resistance.

A graph is a visual representation of data or mathematical relationships between variables. Graphs are used to help people better understand the information they are working with by displaying it in a more clear and organized way.

There are many different types of graphs, including bar graphs, line graphs, scatter plots, pie charts, and histograms. Each type of graph has its own strengths and weaknesses, and is used for different purposes.

In general, graphs consist of two axes: a horizontal x-axis and a vertical y-axis. The x-axis represents the independent variable, while the y-axis represents the dependent variable. Data points are plotted on the graph at the intersection of the x and y coordinates, and lines or curves are often drawn to connect the points and show trends or patterns in the data.

Part A

Ae seen from graph increases from 0 to 2 seconds.

Part B

Water Balloon's height stay same from 2 to 4 seconds

Part C

Height decreasing fastest at 4 to 6 seconds because the slope is steepest downward from 4 to 6 second as compared to 6 to 10 sec.

Part D

Height of balloon will we almost near to ground as resistance will play its role. but it will almost touch the ground.

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The answer of the given question based on the  twelve-tone technique is This is equivalent to 479,001,600 possible tone rows.

The twelve-tone technique is a method of musical composition developed by Arnold Schoenberg and his disciples in the Second Viennese School in the early 20th century. It is also known as dodecaphony, which means "twelve sounds" in Greek. The technique involves arranging the twelve notes of the chromatic scale in a specific order called a tone row or series, and then using that row as the basis for the composition.

Using the twelve-tone technique, we can create a tone row of 12 pitches from the chromatic scale in any order, but with no pitch repeated in the row. Since there are 12 pitches to choose from for the first note, 11 pitches for the second note (since we can't repeat the first pitch), 10 pitches for the third note (since we can't repeat either the first or second pitch), and so on, the total number of possible tone rows is:

12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1

which simplifies to:

12!

This is equivalent to 479,001,600 possible tone rows.

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The correct polynomial function of lowest degree with only real coefficients and having the zeros √7, -√7, and 5 is:

OB. f(x) = x³ - 5x² - 7x + 35

What is polynomial?

A polynomial is a mathematical expression that consists of variables and coefficients, which are combined using arithmetic operations such as addition, subtraction, multiplication, and non-negative integer exponents.

The polynomial function of lowest degree with only real coefficients and having the given zeros can be obtained by multiplying the factors (x - √7), (x + √7), and (x - 5) since the zeros are √7, -√7, and 5.

Expanding the product, we get:

(x - √7)(x + √7)(x - 5) = (x² - 7)(x - 5) = x³ - 5x² - 7x + 35

Therefore, the correct polynomial function of lowest degree with only real coefficients and having the zeros √7, -√7, and 5 is:

OB. f(x) = x³ - 5x² - 7x + 35

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Question A) A 99% confidence interval estimate of his population mean finishing time of 100 meters race is (12.8081,15.5919)

Question B) Option II) narrower is the correct Option.

What is  linear equation?

An algebraic equation with simply a constant and a first- order( direct) element, similar as y = mx b, where m is the pitch and b is the y- intercept, is known as a linear equation.

                         The below is sometimes appertained to as a" direct equation of two variables," where y and x are the variables. Equations whose variables have a power of one are called direct equations. One illustration with only one variable is where layoff b = 0, where a and b are real values and x is the variable.

Let X be the time required to finish 100 meters race competition with Tom ( in seconds.)

A) A 99% confidence interval estimate of his population mean finishing time of 100 meters race is (12.8081,15.5919)

x = 14.2

s^2= 0.9867

s= 0.9933

α/2, n-1 = 3.7074

Margin of error= 1.3919

lower bound= 12.8081

upper bound= 15.5919

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Answer: ∡1 = 153°

∡2 = 27°

Step-by-step explanation:

Answer:

angle 2=27°

angle 1= 153°

Step-by-step explanation:

6x-9+x=180

7x-9=180

7x=189

x=27

angle 2=27°

angle 1= 27×6-9=153°

angle 1= 153°

Answer:

39 divided by 8 is equal to 4 with a remainder of 7.

The quotient is the number of times the divisor goes into the dividend. In this case, 8 goes into 39 4 times with a remainder of 7.

The remainder is the number that is left over after the divisor has been divided into the dividend. In this case, 7 is left over after 8 has been divided into 39.

Here is the long division of 39 by 8:

```

39 / 8

4

32

7

```

Step-by-step explanation:

The quotient of 39 divided by 8 is 4, and the remainder is 7.

We have,

When performing long division, we divide the dividend (39) by the divisor (8) to find the quotient and remainder.

  4

--------

8 | 39

  - 32

    ---

     7

Here's how the long division process works for 39 divided by 8:

-We start by dividing the first digit of the dividend (3) by the divisor (8). Since 3 is less than 8, we can't divide it evenly, so we move to the next digit (9).

- We now have 39 as the remaining portion of the dividend. We divide 39 by 8. The largest multiple of 8 that fits into 39 is 4. We place the quotient, which is 4, above the line.

- We multiply the quotient (4) by the divisor (8), which gives us 32. We subtract 32 from 39, which leaves us with a remainder of 7.

- Since there are no more digits to bring down from the dividend, and the remainder (7) is less than the divisor (8), we stop the division process.

Therefore,

The quotient of 39 divided by 8 is 4, and the remainder is 7.

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Answer:

(x - 5)² + (y + 6)² = 16

Step-by-step explanation:

assuming you require the equation of the circle

the equation of a circle in standard form is

(x - h)² + (y - k)² = r²

where (h, k ) are the coordinates of the centre and r is the radius

here (h, k ) = (5, - 6 ) and r = 4 , then

(x - 5)² + (y - (- 6) )² = 4² , that is

(x - 5)² + (y + 6)² = 16

The x⁶y³ term in the expansion will be:  35 x⁶y³.

The x⁸y² term in the expansion will be: 21x⁸y².

The binomial expansion of (x + y)ⁿ is given by the binomial theorem, which states:

(x + y)ⁿ = C(n, 0) * xⁿ * y⁰ + C(n, 1) * xⁿ⁻¹ * y¹ + C(n, 2) * xⁿ⁻² * y² + ... + C(n, k) * xⁿ⁻ᵏ * yᵏ + ... + C(n, n) * x⁰ * yⁿ

where;

Given that the term 210x⁴y⁶ appears in the expansion, we can infer that it corresponds to C(n, k) * x⁴ * y⁶, where k is the number of times y appears in the term, and (n - k) is the number of times x appears in the term.

Comparing this with the given term, we can deduce the values of n, k, and x in the following way:

C(n, k) = 210

x⁴ = x⁴

y⁶ = y³ * y³

Comparing the exponents on x and y, we can set up the following equations:

n - k = 4 (1)

k = 3 (2)

Solving equation (2) for k, we get:

k = 3

Substituting this value of k into equation (1), we can solve for n:

n - 3 = 4

n = 7

So, the value of n is 7.

Now, we can use the binomial coefficient formula to calculate C(n, k):

C(n, k) = C(7, 3) = 7! / (3! * (7 - 3)!) = 35

Finally, substituting the values of n, k, and C(n, k) into the general term of the expansion, we can find the specific terms:

The x⁶y³ term in the expansion will be:

C(7, 3) * x⁶ * y³ = 35 * x⁶ * y³

The x⁸y² term in the expansion will be:

C(7, 2) * x⁸ * y² = 21 * x⁸ * y²

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Sin 7π/4 is the value of sine trigonometric function for an angle equal to 7π/4 radians. The value of sin 7π/4 is -(1/√2) or -0.7071 (approx).

315° is comparable to 7 / 4 radians. In general, we multiply the angle measurement in radians by 180/ to translate an angle measurement given in radians to degrees. Therefore, we multiply 7 / 4 by 180 / to convert to radians. We discover that 7/4 radians equals 315 degrees.

We can first convert the angle to degrees as follows:

7π/4 radians = (7/4) × 180 degrees/π ≈ 315 degrees

The trigonometric ratios for 315 degrees (or 7/4 radians) can therefore be calculated using the reference angle of 45 degrees (which is /4 radians), as shown below.

sin(7π/4) = -sin(π/4) = -1/√2

cos(7π/4) = -cos(π/4) = -1/√2

tan(7π/4) = tan(π/4) = 1

csc(7π/4) = csc(-π/4) = -√2/2

sec(7π/4) = sec(-π/4) = -√2/2

cot(7π/4) = cot(-π/4) = 1

Therefore, the trigonometric ratios for the angle 7π/4 radians are:

sin(7π/4) = -1/√2

cos(7π/4) = -1/√2

tan(7π/4) = 1

csc(7π/4) = -√2/2

sec(7π/4) = -√2/2

cot(7π/4) = 1

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