Multiply or divide. State any restrictions on the variables.

2x² + 5x + 2 / 4x² - 1 . 2x²+x-1 / x²+x-2

The slope-intercept equation of the line that passes through (-3, 1) and is perpendicular to x - 2y = 8 is y = -2x - 5.

To find the slope-intercept equation of the line that passes through (-3, 1) and is perpendicular to the line x - 2y = 8, we need to determine the slope of the given line and then find the negative reciprocal of that slope to obtain the slope of the perpendicular line.

First, let's rearrange the equation x - 2y = 8 to the slope-intercept form (y = mx + b):

-2y = -x + 8

y = (1/2)x - 4

The slope of the given line is 1/2. The negative reciprocal of 1/2 is -2, which is the slope of the perpendicular line.

Now, we can use the point-slope form of a linear equation, y - y1 = m(x - x1), with the point (-3, 1) and the slope -2:

y - 1 = -2(x - (-3))

y - 1 = -2(x + 3)

y - 1 = -2x - 6

y = -2x - 5

Therefore, the slope-intercept equation of the line that passes through (-3, 1) and is perpendicular to x - 2y = 8 is y = -2x - 5.

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The unit price for the 15.3-ounce box, given by the supermarket, can be determined by converting the unit price for the 24-ounce box from cost per pound to cost per ounce. The unit price for the 15.3-ounce box is approximately x cents per ounce (where x is the converted unit price per ounce).

To find the unit price for the 15.3-ounce box, we need to convert the unit price for the 24-ounce box from cost per pound to cost per ounce. Since there are 16 ounces in a pound, we can convert the cost per pound to cost per ounce by dividing it by 16.

Let's assume the unit price for the 24-ounce box, displayed by the supermarket, is y dollars per pound. To convert this to cost per ounce, we divide y by 16. The resulting value, y/16, represents the unit price in dollars per ounce.

Now, to express the unit price in cents per ounce (to the nearest cent), we multiply y/16 by 100 to convert it to cents. This gives us the converted unit price in cents per ounce, which is approximately x cents per ounce.

Therefore, the unit price for the 15.3-ounce box, given by the supermarket, is approximately x cents per ounce.

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The equation 2x + 7 - x = 3 + x + 4 is sometimes true.To determine whether the equation is always, sometimes, or never true, we can simplify and analyze both sides of the equation.

To determine whether the equation is always, sometimes, or never true, we can simplify and analyze both sides of the equation.

Simplifying the equation:

2x + 7 - x = 3 + x + 4

x + 7 = 7 + x

We notice that both sides of the equation are identical. This means that no matter what value of x we substitute, both sides of the equation will always be equal. Therefore, the equation is sometimes true for any value of x.

In summary, the equation 2x + 7 - x = 3 + x + 4 is sometimes true.

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To find two additional roots for the polynomial function P(x) = 0 with given roots -2i and √10, we consider the conjugate of -2i and the negative square root of 10.

The given roots are -2i and √10. Since the polynomial has rational coefficients, the additional roots must also be complex conjugates and negatives of the given roots, respectively.

The complex conjugate of -2i is 2i. Therefore, 2i is an additional root of P(x) = 0.

The negative square root of 10 is -√10. Hence, -√10 is the second additional root of P(x) = 0.

Therefore, the two additional roots for the polynomial function P(x) = 0, with given roots -2i and √10, are 2i and -√10.

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The polynomial -2x³ - 7x⁴ + x³ can be written in standard form as -7x⁴ - x³ - 2x³. It is a 4th-degree polynomial and has three terms.

To write the polynomial -2x³ - 7x⁴ + x³ in standard form, we rearrange the terms in descending order of the degree of the variable. Doing so, we get -7x⁴ - x³ - 2x³.

The highest degree of the variable, x, in the polynomial is 4, making it a 4th-degree polynomial.

The number of terms in the polynomial is determined by counting the separate algebraic expressions separated by addition or subtraction signs. In this case, we have three terms: -7x⁴, -x³, and -2x³.

Therefore, the polynomial -2x³ - 7x⁴ + x³ can be classified as a 4th-degree polynomial with three terms.

In summary, the given polynomial -2x³ - 7x⁴ + x³ is written in standard form as -7x⁴ - x³ - 2x³. It is a 4th-degree polynomial with three terms.

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The indicated type of proof for proving ∠A ⊕ ∠F is a Two-Column Proof. In a Two-Column Proof, we present the statements (or facts) on the left column and their corresponding justifications (or reasons) on the right column.

By systematically providing statements and their justifications, we demonstrate the logical progression of the proof, leading to the desired conclusion. To prove ∠A ⊕ ∠F, we would begin by listing the given information, such as "A B C H" and "D C G F are parallelograms," as the initial statements. Then, we would proceed with a series of logical deductions and theorems, referencing them as justifications. The goal is to establish the relationship or property that connects ∠A and ∠F, providing a step-by-step argument until we reach the desired conclusion.

Throughout the proof, we would use relevant geometric principles, definitions, postulates, and theorems to build a coherent and valid argument that supports the statement ∠A ⊕ ∠F. By following the structure of a Two-Column Proof, we can clearly present the logical progression of the proof and justify each step along the way, ultimately demonstrating the validity of the conclusion.

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These parametric equations represent the x and y coordinates of points on the line as the parameter t varies.

To write the vector equation and parametric equations of a line that contains the point (2,7), we need an additional piece of information: either another point on the line or the direction vector of the line.

Let's say we have another point on the line, such as (4,9). With this information, we can proceed to write the vector equation and parametric equations.

Vector Equation: The vector equation of a line passing through points P(2,7) and Q(4,9) can be written as: r = p + t⋅d

Here, r represents the position vector of any point on the line, p represents the position vector of point P(2,7), t is a parameter that varies along the line, and d represents the direction vector of the line, which can be obtained by subtracting the position vectors of points P and Q:

d = Q - P = (4,9) - (2,7) = (2,2)

Therefore, the vector equation becomes:

r = (2,7) + t⋅(2,2)

Parametric Equations: The parametric equations express the x, y, and z coordinates of a point on the line in terms of the parameter t. In this case, since we are dealing with a 2D line, there will be only x and y coordinates.

x = 2 + 2t

y = 7 + 2t

These parametric equations represent the x and y coordinates of points on the line as the parameter t varies.

Please note that if you have a different point or specific direction vector in mind, the equations would be modified accordingly.

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After solving the compound equation with the modulus function, we obtain the solution h = 5/2.

To solve this question, we use the modulus function's properties and learn how to solve equations in their presence.

The modulus function returns the absolute value of the operand, number, or expression, regardless of the sign they carry. Denoted by |x|, it works in the following ways.

|x| = x,  if x ≥ 0

|x| = -x,  if x < 0

This function is beneficial in the field of mathematics and sciences, as in various cases, it is necessary for only the magnitude to be used, and the sign to be avoided.

Ex: Distance, Speed, etc.

In the given problem, 7|8 - 3h| = 21h - 49

The left-hand side can go through two possible cases.

Case 1:

8 - 3h ≥ 0

Then |8 - 3h| = 8 - 3h

So,

7(8 - 3h) = 21h - 49

56 - 21h = 21h - 49

56 + 49 = 21h + 21h

105 = 42h

h = 105/42

h = 21*5/21*2

h = 5/2 OR h = 2.5

Case 2:

8 - 3h < 0

Then |8 - 3h| = -(8 - 3h) = 3h - 8

So,

7(3h - 8) = 21h - 49

21h - 56 = 21h - 49

(21h - 21h) -56 + 49 = 0

-7 = 0

This is an absurd result.

Thus, we conclude that such a case is not possible.

Finally, we can say that Case 1 is true, and h = 2.5 is a solution for the compound equation.

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The domain of the composition function f o g(x) can be expressed as (-∞, -4] ∪ (-4, 2) ∪ (2, +∞).

To find the domain of f o g(x), we need to consider two things: the domain of f(x) and the domain of g(x), and find their intersection.

The function f(x) = 1/(x-2) has a restricted domain because the denominator cannot be equal to zero. Thus, x-2 ≠ 0, which means x ≠ 2. So the domain of f(x) is all real numbers except x = 2.

The function g(x) = √(x+4) involves taking the square root of a real number. For the square root to be defined, the expression inside the radical (x+4) must be non-negative. Therefore, x+4 ≥ 0, which implies x ≥ -4. Hence, the domain of g(x) is all real numbers greater than or equal to -4.

To find the domain of f o g(x), we need to find the intersection of the domains of f(x) and g(x). Since f(x) cannot have x = 2 and g(x) must have x ≥ -4, the domain of f o g(x) is the set of real numbers greater than or equal to -4, excluding x = 2. In interval notation, the domain can be expressed as (-∞, -4] ∪ (-4, 2) ∪ (2, +∞).

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The scale factor and the perimeters are = 1/2 and 34 and 17 units.

Given that are two polygons are similar MNPQ ~ XYZW, we need to find the scale factor and the perimeter of each polygon.

Scale factor = ratio of the lengths of the sides of the similar polygons.

Scale factor = MQ / WX = 8/4 = 2

Now,

The perimeter of the polygon XYZW = 10 + 9 + 8 + 7 = 34 units.

We know that the ratio of the perimeters of the similar polygons is equal to the scale factor,

So,

The perimeter of the polygon MNPQ = 34 / 2 = 17 units.

Hence the scale factor and the perimeters are = 1/2 and 34 and 17 units.

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We need to add 25 to both sides of this equation to complete the square. Therefore, the correct option is option D.

The equation given to complete the square is;

[tex]x^2[/tex] - 10x = 3

We need to find out the quantity that we will add to both sides of the equation to complete the square.

[tex]x^2[/tex] - 10x - 3 = 0

([tex]x^2[/tex] - 10x +    ) - 3 = 0

In this equation, a = 1, b = -10, and c = -3.

b = -10

b/2a = -10/(2 * 1)  = -5

(b/2a[tex])^2[/tex] = [tex](-5)^2[/tex] = 25

([tex]x^2[/tex] - 10x + 25) - 3 - 25 = 0

(x - 5[tex])^2[/tex] - 28 = 0

(x - 5[tex])^2[/tex] = 28

Therefore, we need to add 25 to both sides of this equation to complete the square. The correct option is option D.

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The correct answer is d. y=|x| translated 5 units right and 3 units up.

The equation y=|x-3|+5 represents a translation of the absolute value function y=|x| to the right by 3 units and up by 5 units.

The expression "x-3" inside the absolute value represents the horizontal translation of 3 units to the right.

The "+5" term outside the absolute value represents the vertical translation of 5 units up. Therefore, option d, y=|x| translated 5 units right and 3 units up, accurately describes the translation of the given equation.

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The residual value at x = 2 will be 1.902 .

Given,

Data set : x 1 2 3 4 5 6 y 24 29 26 40 26 42

y^=2.714x + 21.67

From the equation the predicted value is when x = 2

Y = 2.714(2) + 21.67

Y = 27.098

From the data set given

[tex]Y_{2}[/tex] = 29.

So the residual value will be ,

[tex]Y_{2}[/tex] - Y

29 - 27.098

= 1.902

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To convert 18 feet to yards, we need to determine the equivalent length in yards.

To convert feet to yards, we use the conversion factor that 1 yard is equal to 3 feet. By dividing the given length of 18 feet by the conversion factor of 3, we can find the equivalent length in yards.

Dividing 18 feet by 3, we get 6 yards. Therefore, 18 feet is equal to 6 yards.

When converting units of length, it is important to understand the relationship between the two units. In this case, since there are 3 feet in 1 yard, dividing the length in feet by 3 gives us the length in yards. Thus, 18 feet is equivalent to 6 yards.

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The length of the rectangle is 15/16 cm, and the width is 9/16 cm, maintaining the ratio of 5:3 and resulting in a perimeter of 3 cm.

Let's denote the length of the rectangle as L and the width as W. According to the problem, the ratio between the length and width is given as 5:3.

We can express this relationship using the equation:

L/W = 5/3

To find the length and width, we also need to consider the perimeter of the rectangle. The formula for the perimeter of a rectangle is given by:

Perimeter = 2(L + W)

In this case, the perimeter is given as 3 cm. Substituting the values into the equation, we have:

2(L + W) = 3

Now, we have a system of two equations:

Equation 1: L/W = 5/3

Equation 2: 2(L + W) = 3

We can solve this system of equations to find the values of L and W.

From Equation 1, we get:

L = (5/3)W

Substituting this value of L into Equation 2, we have:

2((5/3)W + W) = 3

(10/3)W + 2W = 3

(16/3)W = 3

W = (3 * 3) / 16

W = 9/16

Now, substituting this value of W back into Equation 1, we can find L:

L = (5/3)(9/16)

L = (45/48)

L = 15/16

Therefore, the length of the rectangle is 15/16 cm, and the width is 9/16 cm, maintaining the ratio of 5:3 and resulting in a perimeter of 3 cm.

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The decimal value of the 8-bit binary number 10011110, when interpreted as an unsigned number, is 158.

To convert a binary number to decimal, you can assign each bit a weight based on its position and then calculate the sum. In an 8-bit number, the rightmost bit (bit 0) has a weight of 2^0 = 1, the next bit (bit 1) has a weight of 2^1 = 2, the next bit (bit 2) has a weight of 2^2 = 4, and so on.

In this case, we have:

1 * 2^7 + 0 * 2^6 + 0 * 2^5 + 1 * 2^4 + 1 * 2^3 + 1 * 2^2 + 1 * 2^1 + 0 * 2^0

= 128 + 0 + 0 + 16 + 8 + 4 + 2 + 0

= 158

So, the decimal value of the 8-bit binary number 10011110 is 158.

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The length of the rectangle is , the correct option is F.

We are given that;

The area =  [tex]6 x³-22x²+23 x-5 .[/tex]

Width=  3 x-5

Now,

The area of a rectangle is given by the formula `A = l*w`, where `l` is the length and `w` is the width.

We can substitute these values in the formula to get:

[tex]6x³-22x²+23x-5 = l * (3x-5)[/tex]

Simplifying this equation gives:

[tex]2x³ - 4x² + 3x + 1 = l[/tex]

Therefore, by the area answer will be [tex]2x²-4 x+1`.[/tex]

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The standard form of the equation of the ellipse with center at the origin, a vertex at (-9,0), and a co-vertex at (0,-2) is 9x²/81 + y²/4 = 1.


The standard form of the equation of an ellipse with center at the origin is x²/a² + y²/b² = 1, where “a” represents the length of the semi-major axis and “b” represents the length of the semi-minor axis. In this case, the vertex (-9,0) is located on the horizhorizontalontal axis, so the distance from the origin to the vertex is the length of the semi-major axis, “a”.

Therefore, a = 9. Similarly, the co-vertex (0,-2) is located on the vertical axis, so the distance from the origin to the co-vertex is the length of the semi-minor axis, “b”. Hence, b = 2. Plugging these values into the standard form equation gives us 9x²/81 + y²/4 = 1.

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The equation of the parabola with its vertex at the origin and the focus at (3, 0) is x^2 = 12y.

To write the equation of a parabola with its vertex at the origin (0, 0) and the focus at (3, 0), we can use the standard form of the parabola equation:

(x - h)^2 = 4p(y - k)

In this equation, (h, k) represents the vertex, and p represents the distance from the vertex to the focus or the directrix.

Since the vertex is at the origin (0, 0), we have h = 0 and k = 0. The focus is at (3, 0), which means p is the distance between the origin (vertex) and the focus. In this case, p = 3.

Substituting these values into the equation, we get:

(x - 0)^2 = 4(3)(y - 0)

Simplifying further:

x^2 = 12y

Therefore, the equation of the parabola with its vertex at the origin and the focus at (3, 0) is x^2 = 12y.

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The correct answer is 11-5i.  The expression [tex](8 - \sqrt{-1}) - (-3 + \sqrt{-16})[/tex]  can be -simplified as complex number 11 - 5i.

First, let's simplify the square roots:

  [tex]\sqrt{-1}[/tex]is equal to the imaginary unit "i," which is defined as the square root of -1.

[tex]\sqrt{-16}[/tex]   is equal to 4i because the square root of -16 is 4i.

Now let's substitute these values into the expression:

(8 - i) - (-3 + 4i)

To simplify, let's distribute the negative sign to both terms within the second set of parentheses:

(8 - i) + (3 - 4i)

Next, let's combine like terms:

8 + 3 = 11

-1i - 4i = -5i

Therefore, the simplified expression is 11 - 5i.

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Substituting a = -2 and b = -3 into the expression a(b-2a) results in -2. Therefore, the evaluated value of the expression is -2.


To evaluate the expression a(b-2a) for a = -2 and b = -3, we substitute the given values into the expression. Plugging in a = -2 and b = -3, we have -2((-3) – 2(-2)). To simplify the expression, we first simplify the inner brackets. The term -3 – 2(-2) can be rewritten as -3 + 4, which gives us 1.

Now, substituting this value back into the expression, we have -2(1). To find the result, we multiply -2 by 1. The product of -2 and 1 is -2. Therefore, when we evaluate the expression a(b-2a) for a = -2 and b = -3, we get -2 as the final answer.
Hence, by substituting the given values of a = -2 and b = -3 into the expression a(b-2a) and simplifying the resulting expression, we find that the evaluation yields -2 as the answer.

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The property that justifies the statement is the addition property of equality.

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

If 4x - 5 = x + 12, then 4x = x + 17.

From the above, we add 5 to both sides of the equation

So, we have

4x = x + 12 + 5

Evaluate

4x = x + 17

This means that the property is the addition property of equality.

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The volume of the square pyramid-shaped roof with a height of 5 feet and a side length of 6 feet is 60 cubic feet.

A square pyramid has a square base and four triangular sides that come together to form a single point. To calculate the volume of a square pyramid, you can use the formula: 1/3 x Base x Height, where the base is the area of the square base and the height is the height of the pyramid.

In the given scenario, the rooftops of the village are shaped like square pyramids. The height of the roof is 5 feet and the length of the sides is 6 feet. Let us calculate the volume of the roof using the formula mentioned above:

The base of the square pyramid = side * side= 6 * 6= 36 sq. ft, Height of the square pyramid = 5 ft. Volume of the square pyramid= 1/3 * Base * Height= 1/3 * 36 sq. ft * 5 ft= 60 cubic feet. Therefore, the volume of the roof is 60 cubic feet.

Summary: A square pyramid has a square base and four triangular sides that come together to form a single point. The formula to calculate the volume of a square pyramid is 1/3 x Base x Height. The rooftops of the village are shaped as square pyramids with a height of 5 feet and the length of the sides is 6 feet. To calculate the volume of the roof, we can use the formula and find the volume of the roof. The volume of the roof is 60 cubic feet.

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The value of x that satisfies the equation x - [6 1 -2 3] = [2 0 -3 1] is x = 9.

The equation x - [6 1 -2 3] = [2 0 -3 1] can be expanded as follows:

x - 6 - 1 - 2 - 3 = 2 0 - 3 + 1

Simplifying the left-hand side gives x - 12 = 5. Solving for x, we get x = 9.

To understand this equation intuitively, we can think of the vectors on either side of the equal sign. The vector on the left-hand side represents the difference between the vector x and the vector [6 1 -2 3]. The vector on the right-hand side represents the vector [2 0 -3 1]. Therefore, the equation is saying that the difference between x and [6 1 -2 3] is equal to [2 0 -3 1]. This is only possible if x is equal to 9.

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The data described, specifically the horsepower of motorcycles in a dealership, is quantitative data. Quantitative data is numerical in nature and represents a measurable quantity or attribute. In this case, the horsepower values assigned to motorcycles can be measured and compared using numeric values.

Quantitative data represents numerical values that can be measured or counted. It provides a quantitative measurement or description of a particular attribute or characteristic. In the case of the horsepower of motorcycles in a dealership, the data is quantitative because it involves numerical values that quantify the power output of the motorcycles.

Horsepower is a unit of measurement used to indicate the power or performance of an engine, including that of motorcycles. It is typically measured using standardized tests or calculations. The horsepower values assigned to motorcycles in the dealership can be expressed as numerical quantities, such as 50 horsepower, 100 horsepower, or 150 horsepower.

Quantitative data allows for meaningful mathematical operations and comparisons. For example, you can calculate the average horsepower of all the motorcycles in the dealership, compare the horsepower of different models, or analyze the distribution of horsepower across the available motorcycles.

In summary, the horsepower of motorcycles in a dealership represents quantitative data because it involves numerical values that can be measured, compared, and analyzed.

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In a 30°-60°-90° triangle, the shorter leg is √3 cm, the longer leg is 2√3 cm, and the hypotenuse is 2 cm.

A 30°-60°-90° triangle is a special right triangle with specific angle measures. In this triangle, the sides are related by certain ratios. The shorter leg is opposite the 30° angle, the longer leg is opposite the 60° angle, and the hypotenuse is opposite the 90° angle.

In a 30°-60°-90° triangle, the ratios of the side lengths are as follows:

The ratio of the shorter leg to the hypotenuse is 1:2, meaning the shorter leg is half the length of the hypotenuse.

The ratio of the longer leg to the hypotenuse is √3:2, meaning the longer leg is √3 times the length of the shorter leg.

The ratio of the longer leg to the shorter leg is √3:1, meaning the longer leg is √3 times the length of the shorter leg.

Given that the shorter leg is √3 cm, we can use these ratios to find the lengths of the other sides:

The longer leg is √3 * √3 = 3 cm.

The hypotenuse is 2 * √3 = 2√3 cm.

So, in this 30°-60°-90° triangle, the shorter leg is √3 cm, the longer leg is 3 cm, and the hypotenuse is 2√3 cm.

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The units-of-activity depreciation method calculates depreciation based on the usage or activity level of an asset. So, the units-of-activity depreciation for the year is $13,080.

To calculate the units-of-activity depreciation for the year, we start by subtracting the estimated residual value from the initial cost to determine the depreciable amount: $175,000 - $11,500 = $163,500.

Next, we calculate the depreciation rate per mile by dividing the depreciable amount by the estimated useful life in miles: $163,500 / 30,000 miles = $5.45 per mile.

Finally, we multiply the depreciation rate per mile by the actual miles driven during the year (2,400 miles) to determine the units-of-activity depreciation: $5.45 per mile * 2,400 miles = $13,080.

Therefore, the units-of-activity depreciation for the year is $13,080. This method is suitable for assets whose wear and tear or usage is directly related to the number of units or activities performed. In this case, the truck's depreciation expense is based on the actual miles driven during the year, reflecting its usage and decreasing value based on that usage.

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The remove interior angles of angle BCD are given as follows:

<A and <C.

Remote interior angles are defined as the angles of a triangle that do not share a vertex with a given exterior angle

The exterior angle for the triangle is angle C, hence angles <A and <B are the remote interior angles of the triangle relative to angel C.

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The mean is 1.354 seconds and sample standard deviation is approximately 0.0164 seconds

Let us find the mean which is sum of all the observations by total number of observations.

Mean = (1.36 + 1.04 + 1.22 + 1.29 + 1.48 + 1.55 + 0.97 + 1.18 + 1.13 + 1.32) / 10

= 13.54 / 10

= 1.354

The mean of the data set is 1.354 seconds.

We will use the formula for the sample standard deviation, which involves finding the differences between each value and the mean, squaring those differences, summing them up, dividing by (n-1), and taking the square root.

Deviation = [(1.36 - 1.354)² + (1.04 - 1.354)² + (1.22 - 1.354)² + (1.29 - 1.354)² + (1.48 - 1.354)² + (1.55 - 1.354)² + (0.97 - 1.354)² + (1.18 - 1.354)² + (1.13 - 1.354)² + (1.32 - 1.354)²] / 9

= [0.000036 + 0.000088 + 0.000157 + 0.000068 + 0.000167 + 0.000167 + 0.001409 + 0.000136 + 0.000074 + 0.000073] / 9

= 0.002415 / 9

= 0.0002683

Standard Deviation = √(0.0002683)

= 0.0164

Hence, the sample standard deviation of the data set is 0.0164 seconds.

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Answer:

a₁=15 ; aₙ=aₙ₋₁ + 4

Step-by-step explanation:

In this image, you can see that for each step made, 4 is added to the previous number. a₁ should be 15 because it is the first number seen in the mix. The formula should come out to aₙ=aₙ₋₁+4. This is because the common difference is only 4, and that's essentially all you needed to plug in.