To find out how much raw material will be removed from a product your company manufactures, find the common denominator needed to subtract the following fractions. The common denominator is _____ five-sixths minus two-sevenths.

We can follow Arithmetic series, then the number of cubic feet concrete is required for the first 12 steps = 140.4 cubic feet.

Given that,

The first step requires 1. 8 cubic feet of concrete,

The first 5 steps require a total of 27 cubic feet.

An arithmetic series is the sum of the terms in an arithmetic sequence with a definite number of terms. Following is a simple formula for finding the sum: Formula 1: If [tex]S_{n}[/tex] represents the sum of an arithmetic sequence with terms , then. This formula requires the values of the first and last terms and the number of terms.

For an arithmetic series

[tex]a , a+d , a+2d , a+3d , .......[/tex]  

where a is the first term and d is the common difference, the sum of n terms equals,

[tex]S_{n}[/tex] = [tex]a_{n} +(\frac{n(n-1)}{2} ) d[/tex]         (Equation-1)

Since the first step requires 1.8 cubic feet of concrete, a = 1.8.

Now, the first 5 steps require 27 cubic feet of concrete, that is,

[tex]S_{5}[/tex] = 27

a = 1.8

n = 5

We can substitute values in equation-1,

[tex]S_{n}[/tex] = [tex]a_{n} +(\frac{n(n-1)}{2} ) d[/tex]  

[tex]S_{5}[/tex] = (5) (1.8) + [tex](\frac{5(5-1)}{2} )d[/tex]

[tex]S_{5}[/tex] = (5) (1.8) + [tex](\frac{5*4}{2} )d[/tex]

[tex]S_{5}[/tex] = (5) (1.8) + [tex]\frac{20}{2} d[/tex]

[tex]S_{5}[/tex] = (5) (1.8) + 10 d

[tex]S_{5}[/tex] = 9 + 10 d

Then, [tex]S_{5}[/tex] = 27

27 = 9 + 10d

27 - 9 = 10d

18 = 10d

d = [tex]\frac{18}{10}[/tex]

d = 1.8

Find the concrete required for 12 steps.

Then, we can substitute n = 12

So,

We can write,

[tex]S_{n}[/tex] = [tex]a_{n} +(\frac{n(n-1)}{2} ) d[/tex]  

[tex]S_{12}[/tex] = (12) (1.8) + [tex](\frac{12(12-1)}{2} )d[/tex]

[tex]S_{12}[/tex] = (12) (1.8) + [tex]\frac{12*11}{2} d[/tex]

[tex]S_{12}[/tex] = (12) (1.8) + [tex]\frac{132}{2} d[/tex]

[tex]S_{12}[/tex] = (12) (1.8) + 66 d

[tex]S_{12}[/tex] = 21.6 + 66 d

We can substitute d = 1.8

[tex]S_{12}[/tex] = 21.6 + 66 (1.8)

[tex]S_{12}[/tex] = 21.6 + 118.8

[tex]S_{12}[/tex] = 140.4

Therefore,

We can follow Arithmetic series, then the number of cubic feet concrete is required for the first 12 steps = 140.4 cubic feet.

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Answer: 3.64

Step-by-step explanation:

3.64

What is the Nearest Hundredth?

Nearest hundredth is the second digit after the decimal point.

How to Calculate Rounding to the Nearest 100th?

If the digit after hundredth is greater than or equal to 5, add 1 to hundredth. Else remove the digit. Example

124.586

The third digit of right of decimal point is 6

The second digit after decimal point is 8 which is greater than 5

So add 1 to 8

Result = 124.59

its 3.6400 I believe since you are rounding to the nearest hundredth which would be in the 2nd place value to the right of the decimal and 32 would round down to 0.

Answer:

[tex]36.77150[/tex], or 37 (rounded)

Step-by-step explanation:

1. [tex]\sqrt{97} = 9.84885[/tex]

2. [tex]9.84885[/tex] + [tex]8.169178744[/tex] = [tex]18.01803[/tex]

3. [tex]\frac{18.01803}{0.49}[/tex]

4. = [tex]36.77150[/tex]

You can simplify 36.77150 to 37

... Hope this helped! Have a wonderful New Year!

The value of the equation is A = 36.77

What is an Equation?

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the equation be represented as A

Now , the value of A is

A = ( √97 + 2.014³ ) / 0.49   be equation (1)

On simplifying the equation , we get

√97 = 9.8488578

2.014³ = 8.169178744

Substituting the values in the equation , we get

A = ( 9.8488578 + 8.169178744 ) / 0.49

A = 18.018036544 / 0.49

A = 36.77150

Therefore , the value of A is 36.77

Hence , the equation is A = 36.77

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Answer:

Vertex = (0,0)

Focus = (0, -1/8)

Axis of symmetry; x = 0

directrix, y = 1/8

graph: (attached picture)

Step-by-step explanation:

Answer:

Vertex = (0,0)

Focus = (0, -1/8)

Axis of symmetry; x = 0

directrix, y = 1/8

Step-by-step explanation:

Answer: Square Root Function

The inverse of a quadratic function is a square root function because

The amount of earned income the Salazars can report for EITC purposes in 2021 is $21,000.

The maximum amount of earned income the Salazars can report for EITC purposes in 2021 is $21,000. This is calculated by adding Braden's nontaxable combat pay ($14,000) and his wife's earned income ($7,000).

Formula:

Maximum Earned Income for EITC = Braden's Nontaxable Combat Pay + Wife's Earned Income

Maximum Earned Income for EITC = $14,000 + $7,000

Maximum Earned Income for EITC = $21,000

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A series of transformations that would create a congruent shape would include any combination of rotations, reflections, and translations that preserve the size and shape of the original figure.

For example, rotating a square by 90 degrees and then reflecting it across a line of symmetry would create a congruent square. Similarly, translating a triangle to a new location on the plane while preserving its orientation would also create a congruent triangle.

The structures, cultures, and business models of an institution must be realigned in order to create a learning environment that produces substantial and equitable increases in outcomes and educational value. Transformation describes this procedure.

Any activity or collection of activities that takes one or more inputs, changes and adds value to them, and produces outputs for consumers or clients is referred to as a transformation process.

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Three types of numbers fall under the heading of "real numbers," which we will discuss in a moment. Real numbers include all types of numbers, including whole, rational, and irrational numbers.

A real number is a quantity that may be represented mathematically by an endless decimal expansion. In contrast to the counting-derived natural numbers 1, 2, 3,..., real numbers are used to quantify constantly altering quantities such as size and time. Rational and irrational numbers are both included in real numbers. Real numbers include both irrational and rational numbers such as 3, 22/7, and integers such as (-2, 0, 1), fractions such as (1/2), and decimal numbers such as (1/2).

whole = 0,1,2,3,4,5,6,7,8,9...….

rational = 1,2,3,3/4,5/6,...…

irrational =  [tex]\sqrt{8}[/tex], etc

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Answer:

50000

Step-by-step explanation:

We know it increase 27% since it was last measured

100% + 27% = 127%

We have to find 1%

We take 63500 divided by 127 = 500

500 times 27 = 13500

Now we have to take 63500 and subtract by 13500 to find the previous population.

63500 - 13500 = 50000

So, the previous population is 50000

10 more than 20 will be Addition of 10 and 20 which is supposed to be 10 plus 20 = 30.

Addition is the process of combining items and counting them as a single large group. The process of adding two or more numbers together is known as addition in mathematics. The terms addends and total both refer to the numbers that are added and the result of the operation.

The SUM is the result or conclusion that results from adding two or more integers. Addends are the numbers that are added.

We simply changed "more than" to "plus" and found the answer to the straightforward math issue. It is equivalent to 10 + 20 to add 10 to 20. Therefore, the answer is 30 once more.

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The coordinates' weighted average is 4

The given parameters are:

Coordinate 2 has a weight of 2

Coordinate 3 has a weight of 2

Coordinate 10 has a weight of 1

The weight average is then calculated as:

Weight average = Sum of (Weight * Coordinate)/Sum of Weights

So, we have:

Weight average = (2 × 2 + 3 ×2 + 10 × 1)/(2 + 2 + 1)

Evaluate the quotient

Weight average = 4

Hence, the weight average of the coordinates is 4.

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Answer:

All the options except for option B.

Answer:

A,C and D are not.

where as B is solution

Angle can be copied by following 6 steps.

These are the steps to copy an angle:

1.Make point B the center of the working line (l).

2.Create an arc (A, r) that intersects the two sides of angle A at locations S and T by setting your compass to any radius r.

3.Create an arc (B, r) with a point V where it intersects the line l.

4.Build an arc (S, ST).

5.Create an arc at point W where (V, ST) intersects (B, r).

6.You're done after you draw line BW.

Angle can be copied by following 6 steps.

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Step-by-step explanation:

This law is called the closure property of real numbers. It states that if a and b are real numbers, then their sum, a + b, is also a real number. This is because the set of real numbers is closed under addition, which means that any operation performed on real numbers will result in another real number.

For example, if a = 3 and b = 4, then a + b = 7, which is a real number. Similarly, if a = -2 and b = 3, then a + b = 1, which is also a real number.

The closure property of real numbers holds for many other mathematical operations as well. For example, the product of two real numbers is also a real number, as is the quotient of two real numbers (assuming that the divisor is not zero). The set of real numbers is also closed under subtraction and negation.

Correct Question:-

Find the equations of lines normal to the curve [tex]\rm{y=3x-x^3}[/tex] at points where it cuts the x axis.

[tex]\quad[/tex]

Solution:-

The given curve,

[tex]\longrightarrow\rm{y=3x-x^3}[/tex]

Differentiating wrt x we get,

[tex]\longrightarrow\rm{\dfrac{dy}{dx}=3-3x^2}[/tex]

This term is slope of tangent on the curve at a point (x, y).

Taking reciprocal,

[tex]\longrightarrow\rm{\dfrac{dx}{dy}=\dfrac{1}{3-3x^2}}[/tex]

and multiplying by -1,

[tex]\longrightarrow\rm{-\dfrac{dx}{dy}=\dfrac{1}{3x^2-3}}[/tex]

Now this term is slope of normal on the curve at a point (x, y).

[tex]\longrightarrow\rm{n(x)=\dfrac{1}{3(x^2-1)}}[/tex]

[tex]\quad[/tex]

The points where the curve [tex]\rm{y=3x-x^3}[/tex] cuts x axis is obtained by equating y = 0.

[tex]\longrightarrow\rm{0=y}[/tex]

[tex]\longrightarrow\rm{0=3x-x^3}[/tex]

[tex]\longrightarrow\rm{0=x(3-x^2)}[/tex]

[tex]\longrightarrow\rm{0=x(\sqrt3-x)(\sqrt3+x)}[/tex]

[tex]\Longrightarrow\rm{x\in\left\{-\sqrt3,\ 0,\ \sqrt3\right\}}[/tex]

So the points where the curve cuts x axis are (-√3, 0), (0, 0) and (√3, 0).

[tex]\quad[/tex]

Consider the point (-√3, 0).

The slope of normal at this point is,

[tex]\longrightarrow\rm{n(-\sqrt3)=\dfrac{1}{3((-\sqrt3)^2-1)}}[/tex]

[tex]\longrightarrow\rm{n(-\sqrt3)=\dfrac{1}{6}}[/tex]

Then equation of normal at this point is,

[tex]\longrightarrow\rm{y-0=\dfrac{1}{6}(x+\sqrt3)}[/tex]

[tex]\longrightarrow\bf{x-6y+\sqrt3=0}[/tex]

[tex]\quad[/tex]

Consider the point (0, 0).

The slope of normal at this point is,

[tex]\longrightarrow\rm{n(0)=\dfrac{1}{3((0)^2-1)}}[/tex]

[tex]\longrightarrow\rm{n(0)=-\dfrac{1}{3}}[/tex]

Then equation of normal at this point is,

[tex]\longrightarrow\rm{y-0=-\dfrac{1}{3}(x-0)}[/tex]

[tex]\longrightarrow\bf{x+3y=0}[/tex]

[tex]\quad[/tex]

Consider the point (√3, 0).

The slope of normal at this point is,

[tex]\longrightarrow\rm{n(\sqrt3)=\dfrac{1}{3((\sqrt3)^2-1)}}[/tex]

[tex]\longrightarrow\rm{n(\sqrt3)=\dfrac{1}{6}}[/tex]

Then equation of normal at this point is,

[tex]\longrightarrow\rm{y-0=\dfrac{1}{6}(x-\sqrt3)}[/tex]

[tex]\longrightarrow\bf{x-6y-\sqrt3=0}[/tex]

[tex]\quad[/tex]

Hence the equations of lines normal to the curve [tex]\rm{y=3x-x^3}[/tex] at points where it cuts x axis are,

[tex]\longrightarrow\left\{\begin{array}{ll}\bf{x-6y-\sqrt3}&\bf{=0}\\\bf{x+3y}&\bf{=0}\\\bf{x-6y+\sqrt3}&\bf{=0}\end{array}\right.[/tex]

Answer:

To complete each welcome packet, it will take a total of 5 minutes to print + 2 minutes to organize + 5 minutes to proofread = 12 minutes.

If it takes 3 minutes to change the ink in the printer every 5 packets, you will need to change the ink in the printer every 5 packets * 12 minutes per packet = 60 minutes.

In total, it will take you 60 minutes to change the ink in the printer + 12 minutes per packet to complete each packet, for a total of 60 minutes + (12 minutes/packet * number of packets) = 72 minutes per packet.

In the time allotted, you have 3 hours * 60 minutes/hour = 180 minutes.

Therefore, you will be able to fully complete 180 minutes / 72 minutes per packet = 2.5 packets.

Rounded down to the nearest whole number, you will be able to fully complete 2 welcome packets in the time allotted.

Step-by-step explanation:

Answer:

J

Step-by-step explanation:

[tex]x^3-x^2+4x-4=0 \\ \\ x^2(x-1)+4(x-1)=0 \\ \\ (x^2+4)(x-1)=0 \\ \\ (x-2i)(x+2i)(x-1)=0 \\ \\ x=1, 2i, -2i[/tex]

The probability with which bob plays l in the mixed strategy equilibrium = 0.9090

What is Probability?

The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true. An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty.

Here suppose Alice plays U with probability P and D * with probability ( 2 - P)

Therefore, when Bob plays L, Bob gets payoff 8 with probability P and (2) with probability ( 2 - P). Hence

= 8 * P + 2 * ( 2 - P)

= 8P +4 - 4P

= 4 + 8P

Also, when Bob plays R, Bob gets pay off 4 with probability P and 7 probability (2-P). Hence expected payoff is

= 4*P+7(2-P)

= 4P+14-7P

=14-3P

At mixed strategy nash equilibrium the expected payoffs are equal. hence,

4 +8P = 14-3P

3P+8P = 14-4

11P = 10

P = 10/11

P = 0.9090

In the mixed strategy equilibrium, Bob's chances of playing l are equal to 0.9090.

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Here suppose Alice plays U with probability P and D * with probability ( 2 - P)

Therefore, when Bob plays L, Bob gets payoff 8 with probability P and (2) with probability ( 2 - P). Hence

= 8 * P + 2 * ( 2 - P)

= 8P +4 - 4P

= 4 + 8P

Also, when Bob plays R, Bob gets pay off 4 with probability P and 7 probability (2-P). Hence expected payoff is

= 4*P+7(2-P)

= 4P+14-7P

=14-3P

At mixed strategy nash equilibrium the expected payoffs are equal. hence,

4 +8P = 14-3P

3P+8P = 14-4

11P = 10

P = 10/11

P = 0.9090

In the mixed strategy equilibrium, Bob's chances of playing l are equal to 0.9090.

The factors of the expression 2[tex]x^{2}[/tex] -5x -12 are (x - 4) and (2x+3)

given that

2[tex]x^{2}[/tex] -5x -12 =0

in the above equation

a = 2[tex]x^{2}[/tex]

b =-5x

c = -12

now we need to multiply a and c

a * c = 2[tex]x^{2}[/tex] * -12 = -24[tex]x^{2}[/tex]

now split b into two terms such that sum is equal to -5x  and their product is -24

-5x = -8x +3x

2[tex]x^{2}[/tex] -5x -12 =0

2[tex]x^{2}[/tex] -8x + 3x -12 =0

2x(x - 4)+3(x-4) =0

(2x+3)(x-4) =0

The factors of the expression 2[tex]x^{2}[/tex] -5x -12 are (x - 4) and (2x+3)

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When a point or object is moved but its size and shape don't change, it is said to have undergone rigid motion, sometimes referred to as a rigid transformation.

The separation between each pair of the geometric object's points is preserved during a rigorous change of the object. In other words, a rigid transformation maintains the object's size and shape. For instance, the rigid transformation of a triangle retains the angles and lengths of the triangle. Imagine a stiff shape distortion, similar to a cardboard shape. The form remains the same after being pushed left and right, rotated, and flipped.

When the mold was created using putty, the opposite was true. The thing can then be twisted and stretched. These conversions, nevertheless, are not exact. Stretching a form alters the distances between each pair of its points.

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A maximum of 1,048,576 rows are allowed in Microsoft Excel 2007. 15,384 columns. Columns and rows make up an Excel spreadsheet.

The arrangement of items next to or horizontally across one another is referred to as a row. As a vertical grouping of items according to category, a column can be said to exist. Left to right is the direction of arranging. From top to bottom, the set up is made. At far right is where the total is displayed.

here

Numbers 1, 2, 3, and 4 are used to designate the rows, which are horizontally oriented.

A, B, C, D, etc., are used to denote the letters that run horizontally along the columns.

Cells are what happen when a column and row intersect.

A maximum of 1,048,576 rows are allowed in Microsoft Excel 2007. 15,384 columns.

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a) 55 people expect to be between 165 and 175 cm.

b) 76 people to be taller than 166 cm.

What is Standard normal distribution?

The standard normal distribution is a special type of normal distribution where the mean is 0, and the standard deviation is 1.

a) To answer this question, we can use the standard normal distribution table to find the proportion of heights that fall within certain ranges of the mean.

The standard normal distribution table gives us the proportion of values within a certain number of standard deviations of the mean.

Since the standard deviation of the heights is 5, we can find the proportion of heights between 165 and 175 cm by looking at the range of values that are one standard deviation below and one standard deviation above the mean.

The mean of the heights is 170, so one standard deviation below the mean is 170 - 5 = 165. One standard deviation above the mean is 170 + 5 = 175. Therefore, the range of heights that are one standard deviation below and one standard deviation above the mean is 165 to 175.

If we look at the standard normal distribution table, we find that approximately 68% of the values fall within one standard deviation of the mean. Since 80 is approximately 68% of 118, we would expect approximately 55 people to have heights between 165 and 175 cm.

Hence, 55 people expect to be between 165 and 175 cm.

b) For the second part of the question, we can find the proportion of heights that are taller than 166 cm by looking at the range of values that are more than one standard deviation above the mean.

The mean of the heights is 170, so more than one standard deviation above the mean is anything greater than 175. If we look at the standard normal distribution table, we find that approximately 95% of the values fall within two standard deviations of the mean.

Since 80 is approximately 95% of 84, we would expect approximately 76 people to be taller than 166 cm.

Hence, 76 people to be taller than 166 cm.

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Go to the Home tab and choose Number from the drop-down menu. In addition, you have the following choices: When choosing Number, press CTRL + 1. Right-click the cell or cell range, choose Format Cells..., then choose Number.

To calculate and represent a quantity, a number, an arithmetic value, is needed. Numbers are represented by mathematical inscriptions, such the letter "3." Using logical symbols or digits to represent numbers is known as a number system.

here ,

The cells you want to format should be selected.

Place your cursor over the Dialog Box Launcher next to Number on the Home tab.

Click Fraction in the Category list.

Select the desired fraction format type from the Type list by clicking it.

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Answer:

bro what u doin this on a quiz u aint goin do this on time

Step-by-step explanation:

The graph of the equation given is mentioned below.

What is plot point in graph ?

Whenever we draw a point on our grid, we'll call it "plotting a point." The grid is really called a "graph" and the points we plot are called "coordinates." These are really locations on a plane -- we're just finding and labeling them.

Equation given,

y = 3/5x - 1    ----eq(i)

y = 2/5x        -----eq(ii)      

Now, for this two equation we infinite points for real numbers.

Now, take some points to plot the graph.

First for the eq(i)

Put x = 0, so y = -1

Put x = 5, so y = 2

Points for eq(i) is (0, -1) and (5, 2)

Now, for eq(ii)

Put x = 0, so y = 0

Put x = 5, so y = 2

Points for eq(ii) is (0, 0) and (5, 2)

Common point is (5, 2)

Now, plot this point on graph.

Hence, for more clarity graph of the equation given is mentioned below.

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Observing the graph and the lines drawn, The graph is inequality is known if it is showing the region for which the inequality exists. Usually it is shown with arrows at the end of the line.

An inequality in mathematics is a relation that compares two numbers or other mathematical expressions in an unequal way. The majority of the time, size comparisons between two numbers on the number line are made. Less than, larger than, less than or equal to, and more than or equal to are the four fundamental inequalities.

The graph of a function f in mathematics is the collection of ordered pairs where f(x)=y. These pairs are Cartesian coordinates of points in two-dimensional space and so form a subset of this plane in the typical situation when x and f(x) are real integers.

Using a given graph,

We know that the graph is inequality if it is showing the region where the inequality exists,

It is usually shown using arrows at the end of the line towards the direction where the inequality is satisfied.

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Consistency exists inside the linear equation system.

To determine whether a system of linear equations is consistent, you can try to solve the system to find the values of the variables that satisfy all of the equations. The system is consistent if it has at least one solution. If the system has no solutions, then it is inconsistent.

In this case, the system of linear equations is 2x + 3y - 9 = 0 and 4x + 6y - 18 = 0.

To solve this system, you can use algebraic methods such as substitution or elimination. Using substitution, you can solve one of the equations for one of the variables in terms of the other, and then substitute this expression into the other equation to solve for the other variable. For example, you could solve the first equation for y:

2x + 3y - 9 = 0

3y = 9 - 2x

y = (9 - 2x)/3

Then, you could substitute this expression for y into the second equation to solve for x:

4x + 6((9 - 2x)/3) - 18 = 0

4x + (54 - 12x)/3 - 18 = 0

(12x - 54)/3 + 4x - 18 = 0

(12x - 72)/3 = 0

12x - 72 = 0

12x = 72

x = 6

Substituting this value for x back into the first equation, you can solve for y:

2(6) + 3y - 9 = 0

12 + 3y - 9 = 0

3y = -3

y = -1

Therefore, the solution to this system of equations is x = 6 and y = -1. Since there is at least one solution, this system is consistent.

The complete question is:-

Is the system of linear equations 2x+3y-9=0 and 4x+6y-18=0 consistent? justify your answer.

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To find the mean deviation of a set of numbers, you need to first find the average (mean) of the set, and then for each number in the set, subtract the mean and take the absolute value of the result. The mean deviation is then the average of these absolute values.

For the first 10 even natural numbers (2, 4, 6, 8, 10, 12, 14, 16, 18, 20), the mean is (2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20) / 10 = 60/10 = 6.

To find the mean deviation, you would subtract the mean from each number, take the absolute value of the result, and then average those values. The mean deviation of the first 10 even natural numbers is therefore (|6 - 2| + |6 - 4| + |6 - 6| + |6 - 8| + |6 - 10| + |6 - 12| + |6 - 14| + |6 - 16| + |6 - 18| + |6 - 20|) / 10 = (4 + 2 + 0 + 2 + 4 + 6 + 8 + 10 + 12 + 14) / 10 = 5.5.

So the answer is B: 5.5.

Numbers in Nature When counting or organizing objects, the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 are utilized. Absolute numbers are integers and zeros. neither a fraction nor a decimal.

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The vertical asymptotes of a function can be found by setting the denominator equal to zero and solving for x. This will give the x-values that the function cannot approach as it approaches infinity.

The vertical asymptotes of a function can be found by setting the denominator equal to zero and solving for x. This will give the x-values that the function cannot approach as it approaches infinity.In other words, a vertical asymptote is a line that the graph of a function approaches but never touches. This line is the result of the denominator of a rational function becoming equal to zero. When this happens, the function is undefined at that value of x. To find the vertical asymptotes of a function, one must first factor the denominator and set each factor equal to zero. Then, solve each factor for x to get the x-values of the vertical asymptotes. It is important to note that a vertical asymptote can also appear if the numerator of a rational function is equal to zero. In this case, the vertical asymptote is the x-value at which the numerator is equal to zero. The vertical asymptotes of a function can help us to understand the behavior of the function and its graph.

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