how many ways are there for $8$ people to sit around a circular table if none of alice, bob, and eve (three of the $8$ people) want to sit next to each other? two seatings are considered the same if one is a rotation of the other.

The polynomial x⁵ - 4x³ + x² - 4 will have a remainder of -8 when divided by x² - x + 1 and the result can be written in the form (x³ + x² - 4x + 4) - 8/(x² - x + 1).

Using the long division method will require us to; divide, multiply, subtract, bring down the next number and repeat the process to end at zero or arrive at a remainder

We shall divide the polynomial x⁵ - 4x³ + x² - 4 by x² - x + 1 as follows;

x⁵ divided by x² equals x³

x² - x + 1 multiplied by x³ equals x⁵ - x⁴ + x³

subtract x⁵ - x⁴ + x³ from x⁵ - 4x³ + x² - 4 will result to x⁴ - 5x³ + x² - 4

x⁴ divided by x² equals x²

x² - x + 1 multiplied by x² equals x⁴ - x³ + x²

subtract x⁴ - x³ + x² from x⁴ - 5x³ + x² - 4 will result to -4x³ - 4

-4x³ divided by x² equals -4x

x² - x + 1 multiplied by -4x equals -4x³ + 4x² - 4x

subtract 4x³ + 4x² - 4x from -4x³ - 4 will result to 4x² - 4x - 4

4x² divided by x² equals 4

x² - x + 1 multiplied by 4 equals 4x² - 4x - 4

subtract 4x² - 4x - 4 from 4x² - 4x - 4 will result to a remainder of -8.

Therefore, the polynomial x⁵ - 4x³ + x² - 4 divided by x² - x + 1 gives a quotient x³ + x² - 4x + 4 and a remainder of -8 and can be written as (x³ + x² - 4x + 4) - 8/(x² - x + 1).

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Answer: number 1: 0.1111 and number 2: 1.0909

Step-by-step explanation: I hope I did this right

12 13 14 15 16 17 18 19 20 at the fisher farm, the weights of zucchini squash are normally distributed,  then Weight represents the top 10% of the zucchinis is 5.9

The formula for calculating a z-score is z = (x-μ)/σ,

where x is the raw score

μ is the population mean

σ is the population standard deviation.

We're looking for:

Weight is 10% of zucchinis at the top.

Top 10% = 90% of the population

90th percentile Z score is 1.282.

Hence

1.282 = x - 5 / 0.7

1.282 × 0.7 = x - 5

0.8974 = x - 5

x = 5 + 0.8974

x = 5.8974

5.9 ounces, roughly.

The top 10% of the zucchinis have a weight of 5.9 ounces.

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The area of the shaded portion of the figure is 272 units squared.

The figure above is a triangle inside a trapezium. The area of the shaded portion can be found when we subtract the area of the triangle from the area of the trapezium.

Therefore,

area of the shaded portion = area of the trapezium - area of the triangle

area of the triangle = 1 / 2 bh

where

Therefore,

area of the triangle = 1 / 2 × 18 × 16

area of the triangle = 288 / 2

area of the triangle = 144 units²

area of the trapezium = 1 / 2 (a + b)h

where

Therefore,

area of the trapezium = 1 / 2 (18 + 34)16

area of the trapezium = 1 / 2 (52)16

area of the trapezium =  832 / 2

area of the trapezium = 416 unit²

Therefore,

area of the shaded portion = 416 - 144

area of the shaded portion = 272 units²

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6 points were scored in the 11th game.

Mean is the arithmetic average of a set of given numbers. Therefore the mean in math is often referred to as simply the average.

The mean is the total number of points scored divided by the number of games.

So if the mean is 17 points per game for the first 10 games, the total number of points scored in the first 10 games is 17 * 10 = 170.

Similarly,

if the mean is 16 points per game after the 11th game, the total number of points scored in the first 11 games is 16 * 11 = 176.

To find the number of points scored in the 11th game

we can subtract the total number of points scored in the first 10 games from the total number of points scored in the first 11 games:

176 - 170 = 6

Therefore, 6 points were scored in the 11th game.

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Answer:

y = -4x

Step-by-step explanation:

When equations are in the form

y = mx  the slope m (is y/x)

We can create a total number of 10 prime numbers from the set {1, 3, 6, 7}.

Given that,

The number is either 1-digit or 2-digit.

In the case of 1-digit, the only 1-digit primes are 3 and 7, for a total of 2 primes.

In the case of 2-digit, We have the following combinations of numbers: 13, 16, 17, 36, 37, 67, 76, 73, 63, 71, 61, 31. Out of these 12 numbers, it is easier to count the composites: 16, 36, 76, and 63 for a total of 4 composites, which we subtract from the original 12 numbers to yield 12-4=8 prime numbers.

Thus, We can create a total number of 10 prime numbers from the set {1, 3, 6, 7}.

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The expected value of the number of marbles Zuri removes from the bag is the sum of the number of red marbles plus the number of blue marbles.

The expected value of the number of marbles Zuri removes from the bag is the sum of the number of red and blue marbles. This is because the expected value of an event is the sum of the probability of each outcome multiplied by the associated outcome. Since Zuri is randomly removing marbles from the bag, one at a time and without replacement, the probability of each outcome (red or blue) is equal. Therefore, the expected value of the number of marbles Zuri removes from the bag is the sum of the number of red marbles plus the number of blue marbles. This is because each marble has an equal chance of being removed and, therefore, each marble has the same expected value.

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For given function f(x) the y in the graph are f(2) = 4,f(0) = 3 ,f(-6) = -6 and f(4) = 5

A function is described as a relationship between a set of inputs, each of which has a single output. The simplest definition of a function is an association between inputs where each input is connected to a single, unique output. There is a codomain or range for each function. A function is frequently referred to as f(x), where x is the input.

Here,

In the given graph

when x = 2

y = 4

and x = 0

y = 3

Put it in slope formula

=> m =y2 - y1 / x2 -x1

=>  m =1/2

m = 1/2

Now we can just put them into slope formula

y - y₁ = m(x - x₁)

y - 4 = 1/2(x - 2)

y = 1/2(x) -1 + 4

y = 1/2(x) + 3

Now we can just put them into slope formula

y - y₁ = m(x - x₁)

y - 3 = 1/2(x - 0)

y = 1/2(x)  + 3

Thus, the function for given f(x) is solved

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Answer:

Step-by-step explanation:

true true false

Answer:

We can use the following steps to determine the different amounts of money that can be made from a collection of coins that consists of two pennies, one nickel, and one dime:

Step 1: List out all the different combinations of coins that can be used to make the different amounts of money.

Two pennies

One penny and one nickel

One penny and one dime

Two pennies and one nickel

Two pennies and one dime

One nickel and one dime

Two pennies, one nickel, and one dime

Step 2: Calculate the value of each combination of coins.

Two pennies = $0.02

One penny and one nickel = $0.06

One penny and one dime = $0.11

Two pennies and one nickel = $0.07

Two pennies and one dime = $0.12

One nickel and one dime = $0.15

Two pennies, one nickel, and one dime = $0.17

Step 3: Write down the total amount of money that can be made from each combination.

Final Answer: There are 7 different amounts of money that can be made from a collection of coins that consists of two pennies, one nickel, and one dime, they are $0.02, $0.06, $0.11, $0.07, $0.12, $0.15, and $0.17

The supplied statement states that the function f(maverage )'s rate of change is 0.78 cm, representing the fish's original length.

A function in mathematics is an expression, rule, or law that determines the link between two variables, one of which is an independent variable (the dependent variable). Functions may be present everywhere in mathematics, and they are essential for building physical connections in the sciences.

The fish's original length, which is represented by the graph's y-intercept, is 5 cm.

The function f(maverage )'s rate of change at m = 3 to m = 7

m = 3 => f(3) = 5(1.08) = 5.4 cm

m = 7 => f(7) = 5(1.08)⁷ = 8.55 cm

Change in 9 - 1 = 8 months = 8.55 - 5.4 = 3.15 cm

The function's average variation at m = 3 through m = 7 is given.

= (3.15/4)

= 0.78 cm per month

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Answer:

[tex]y=-\frac{2}{3}x+4[/tex]

Step-by-step explanation:

[tex]y=\frac{4-0}{0-6}=-\frac{2}{3} \\ \\ b=4 \implies y=-\frac{2}{3}x+4[/tex]

Answer:

A.  293 dolls each

B.  710 beads

C.  (n+n)/3 >2xy

Step-by-step explanation:

1.  Let A be the number of dolls collected by Gelyn and B the number collected by Geraldine.  

At the start:

 A = 1,178, and

 B = 588

Girlie gets the same number of dolls from both, which we'll say is x.  The situation is then:

Gelyn:  A - x

Geraldine = B - x

Girlie = 2x

After the transfer, we find that A-x = 3(B-x)  [Gelyn had 3 times as many dolls left as Geraldine]

Lets rearrange this last expression:

A-x = 3(B-x)

A-x = 3B-3x

2x=3B-A

Use the values of A and B to find x:

2x=3(588)-(1,178)

2x = 586

x =  293

Gelyn  and Geraldine each gave Girlie 293 dolls.

========

2.  Let A, B, and C stand for the number of beads in Boxes A, B, and C, respectively.  We are told that:

 A + B + C = 4342 [Box A, Box B, and Box C contain 4,342 beads altogether]

and learn that:  

A+18 = B, and  [There are 18 more beads in Box B than Box A]

C = 3B              [There are 3 times as many beads in Box C as in Box B]

How many beads in Box A?

We have three equations.  Lets find a way to substitute so that we can eliminate the unknowns B and C.

Look at:   A + B + C = 4342

If we can rearrange the other equations to express the variables B and C in terms of A, we could solve the problem.

A+18 = B becomes B=A+18  [We can now calculate B from A]

The second, C = 3B, does not have A, but we can use the above expression for B, which will allow C to be expressed as a function of A:

C = 3B

C = 3(A+18)

C = 3A + 64   [We can now calculate C from A]

Use these definitions of B and C in the first equation (substitution):

A + B + C = 4342

A + (A+18) + (3A + 64) = 4342

6A + 82 = 4342

6A = 4260

A = 710   Box A has 710 beads

3.  The sum of x and n divided by 3 more than twice the product of x times y

          (n+n)/3        [The sum of x and n divided by 3]

         >2xy            [more than twice the product of x times y]

Put these together:

    (n+n)/3 >2xy

The sequence of numbers for the expression -n² - 2n + 3 is

0, -5, -12, -21, and -32.

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

-n² - 2n + 3

The sequence of numbers for n = 1, 2, 3, 4, 5 are:

n = 1,

-1 - 2 + 3

= -3 + 3

= 0

n = 2,

-4 - 4 + 3

= -8 + 3

= -5

n = 3,

-9 - 6 + 3

= -15 + 3

= -12

n = 4,

-4² - 2 x 4 + 3

= -16 - 8 + 3

= -21

n = 5,

-25 - 10 + 3

= -35 + 3

= -32

Thus,

The sequence of numbers is 0, -5, -12, -21, and -32.

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The inequality that could represent the possible values for the number of tickets sold will be 6.5s + 11.5a ≥ 1300.

Inequality is defined as an equation that does not contain an equal sign. Inequality is a term that describes a statement's relative size and can be used to compare these two claims.

The dramatization club is offering passes to their play to fund-raise for the show's costs. Every understudy ticket sells for $6.50 and every grown-up ticket sells for $11.50. The dramatization club should make at least $1300 from ticket deals to take care of the show's expenses.

Let 's' and 'a' be the number of tickets sold to students and adults, respectively. Then the inequality is given as,

6.5s + 11.5a ≥ 1300

The inequality that could represent the possible values for the number of tickets sold will be 6.5s + 11.5a ≥ 1300.

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Answer:

Step-by-step explanation:

The power of 10 is the exponent of places we move the decimal point to get the first factor. For any number greater than 1, the power of 10 in scientific notation will be positive.

First Non-zero Power of 10

Example: Let’s look at the mass of Earth in kilograms.

What power? 9

Numbers can also be converted from scientific notation to standard notation.

Look at the number 6.51 × 108 . The power of 10 denotes the number of places the decimal point has to move. The power of 10 in this number is positive, so we move the decimal point to the right 8 places.

The number 6.51 × 108 written in standard notation is 651000000.

Answer:

57 grey plates

Step-by-step explanation:

304 = 3 : 8 : 5

304 = 16x

[tex] \frac{ \\ 304}{16} = \frac{16x}{16} [/tex]

x = 19

by solving systematically for the grey plate

3 × 19 = 57 grey plates.

Additional solvings.

by solving systematically for the white plates

8 × 19 = 152 white plates

by solving systematically for the black plates

5 × 19 = 95 black plates

Total plates

57 + 152 + 95 = 304

I hope this helped

First of all, we need to know what the units are in this problem.

We can see that there is yards, inches, and feet.

(Keep in mind that yd = yard, ft = feet, and in = inches)

1yd = 3ft

1ft = 12in

Let's figure out how many inches are in a yard.

If we know that 3 feet are in every yard, and 12 inches are in every foot, we can make an equation that looks like this:

(1yd × 3ft) × 12 = 36

This means that there are 36 inches in every foot.

Now that we know how many inches are in a yard, lets figure out how many inches are in 10 yards.

36 × 10 = 360

There are 360 inches in 10 yards.

Now, let's figure out how much 360 inches exceeds 192 inches.

360 - 192 = 168

To convert this to feet, we need to use an equation that looks like this:

168 ÷ 12 = 14

Therefore, the amount of feet that the length of the room exceeds the width is 14 feet.

The above statement suggests that perhaps the curve's approximate length is 2.

Curve, An abstract phrase used only to describe the trajectory of a linearly stretching point in mathematics (see continuity). Usually, an equation will create such a route. The term may also be used to describe a linear model or a collection of connected line segments. The curve is referred to as a route, or a plot line (or just arc). If a curve is the continuous injective picture of either an interval or indeed a circle, it is considered simple.

Considering function is,

[tex]y=\sqrt{x-x^2}+\sin ^{-1} \sqrt{x}[/tex]

To determine the curve's actual length

First ,  Parametrise  the Curve  by x = sin²u

[tex]\begin{aligned}& y=\sqrt{\sin ^2 u-\sin ^4 u}+\sin ^{-1} \sqrt{\sin ^2 u} \\& y=\sin u \sqrt{1-\sin ^2 u}+\sin ^{-1} \sin u\end{aligned}[/tex]

y = sinucosu + u

[tex]\begin{aligned}& y=\frac{1}{2} \times 2 \sin u \cos u+u \\& y=\frac{\sin 2 u}{2}+u \\& \frac{\mathrm{d} y}{\mathrm{~d} u}=\frac{\mathrm{d}}{\mathrm{d} u}\left[\frac{\sin 2 u}{2}+u\right] \\& \frac{\mathrm{d} y}{\mathrm{~d} u}=\left[\frac{2 \cos 2 u}{2}+1\right]\end{aligned}[/tex]

[tex]\frac{\mathrm{d} y}{\mathrm{~d} u}=[\cos 2 u+1][/tex]

and, x = sin²u

[tex]\begin{aligned}& \frac{\mathrm{d} x}{\mathrm{~d} u}=\frac{\mathrm{d}}{\mathrm{d} u}\left[\sin ^2 u\right] \\& \frac{\mathrm{d} x}{\mathrm{~d} u}=[2 \sin u \cos u] \\& \frac{\mathrm{d} x}{\mathrm{~d} u}=[\sin 2 u] \\& \left(\frac{\mathrm{d} s}{\mathrm{~d} u}\right)^2=\left(\frac{\mathrm{d} x}{\mathrm{~d} u}\right)^2+\left(\frac{\mathrm{d} y}{\mathrm{~d} u}\right)^2 \\& \left(\frac{\mathrm{d} s}{\mathrm{~d} u}\right)^2=(\sin 2 u)^2+(1+\cos 2 u)^2\end{aligned}[/tex]

[tex]\begin{aligned}\left(\frac{\mathrm{d} s}{\mathrm{~d} u}\right)^2 & =\left[\sin ^2 2 u+1+\cos ^2 2 u+2 \cos 2 u\right] \\\left(\frac{\mathrm{d} s}{\mathrm{~d} u}\right)^2 & =[1+1+2 \cos 2 u] \\\left(\frac{\mathrm{d} s}{\mathrm{~d} u}\right)^2 & =[2+2 \cos 2 u] \\\left(\frac{\mathrm{d} s}{\mathrm{~d} u}\right)^2 & =4 \cos ^2 u \\\left(\frac{\mathrm{d} s}{\mathrm{~d} u}\right)^2 & =2 \cos u\end{aligned}[/tex]

Currently, the curve's exact length is,

[tex]\begin{aligned}& =\int_0^{\Pi / 2} 2 \cos u d u \\& =2 \int_0^{\Pi / 2} \cos u d u \\& =2[\sin u]_0^{\Pi / 2} \\& =2\left[\sin \frac{\Pi}{2}-\sin 0\right]\end{aligned}[/tex]

= 2

As a result, the curve's exact length is 2

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To find the domain and range of a system of inequalities, you need to solve the system of inequalities, plot the solution, determine inequality sign for solution, shade area of solution, then the x-values of the area represent domain and y values represent range.

Solve the system of inequalities for x and y in terms of each other.

Plot the solutions on a coordinate plane. Each solution will be a boundary line for the solution set.

Determine the inequality signs for each solution. Solutions with < or > signs will be represented as a dashed line, while solutions with ≤ or ≥ signs will be represented as a solid line.

Shade the portion of the coordinate plane that satisfies all of the inequalities.

The shaded area represents the solution set for the system of inequalities.

The domain is the set of all x-values that are included in the solution set and the range is the set of all y-values that are included in the solution set.

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Answer:

[tex]\boxed{\displaystyle f^{-1}(x) =\left(\dfrac{x-2}{4}\right)^5 + 1\\}[/tex]

This is the last choice on the screenshot you provided

Step-by-step explanation:

Use y to represent f(x)

[tex]y=4\sqrt[5]{x-1}+2[/tex]

Replace x with y and y with x:

[tex]x=4\sqrt[5]{y-1}+2[/tex]

Solve for y in terms of x and that will be the inverse of f(x)

Answer: Last choice on the screen, namely

[tex]\boxed{\displaystyle f^{-1}(x) =\left(\dfrac{x-2}{4}\right)^5 + 1\\}[/tex]

Answer:

1. Write the number as the numerator of the fraction.

2. Write 1 as the denominator of the fraction.

Step-by-step explanation:

Example-

To turn the number 3 into a fraction, write 3 as the numerator and 1 as the denominator. The resulting fraction is 3/1.

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  The ant's entire distance travelled along its semicircular journey is its total distance travelled overall. The ant is travelling around a circle with radius R, thus the distance it has travelled is equal to half of the circle's circumference, πR.

The change in the ant's position is represented by a vector quantity called the displacement of the ant. The ant in this scenario begins at one end of the semi-circular path and moves 2R away from the beginning point to reach the other end. The direction of the displacement vector is from the starting point to the ending point, and the displacement's magnitude is equal to the 2R distance between the two points.

    So, the option c is most suitable option.

Therefore, πR and 2R are the ant's displacement and the distance travelled, respectively.

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Check the picture below.

If the random variable X is distributed normally with a mean of 0 and a standard deviation of 1, the probability P(x = 0) =.50 is not correct.

Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has included probability to forecast the likelihood of certain events. The degree to which something is likely to happen is basically what probability means.

The probability of all the events in a sample space adds up to 1.

Thus, If the random variable X is distributed normally with a mean of 0 and a standard deviation of 1, the probability P(x = 0) =.50 is not correct.

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Listing of the angles and the sides in the order of smallest to largest is given by :

Side opposite to obtuse angle > Side opposite to acute angle and

side opposite to right angle > side opposite to acute angle.

As given in the question,

In the given triangle,

Ordering of the angles and sides from smallest to largest.

As per the property of the triangle:

Side opposite to the greatest angle is the largest side of the given triangle.

If the given triangle is obtuse angle triangle,

Then other two angles are acute.

Compare all the angles and sides arrange in ascending order.

Side opposite to obtuse angle  > Side opposite to acute angle

In the right angled triangle,

Other two angles are acute.

Same compare and write in ascending order.

side opposite to right angle > side opposite to acute angle

Therefore, the sides and angles can be written in smallest to largest order by comparison and is given by:

Side opposite to obtuse angle  > Side opposite to acute angle and

side opposite to right angle > side opposite to acute angle.

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0.5x + 6 = 4x - 1

3.5x = 7

x = 2

Only one solution exists for the given equation.

Answer:

one

Step-by-step explanation

For a polynomial function, the number of solutions, or zeros, is generally equivalent to the highest degree. For example, a cubic function (e.g. [tex]x^3+2x^2[/tex]) has 3 real zeros (x=-2, x=0, and x=0) and a quadratic (e.g. [tex]x^2-1[/tex]) has 2 real zeros (x=-1 and x=1).

For your problem, [tex]\frac{1}{2} (x+12)=4x-1[/tex], the highest degree is 1 since this is a linear function. To prove there is only one solution, let's bring all the terms to one side, simplify, and solve for x.

There only one solution to the equation as shown above and as can be seen in step 4, the x is only to the first degree ([tex]x=x^1[/tex]) which indicates that there is only one solution, 2.

Answer:

  -4y²

Step-by-step explanation:

You want to simplify the expression -√(16y⁴), given that y < 0.

The square root can be simplified by removing squares from under the radical.

  [tex]-\sqrt{16y^4}=-\sqrt{(4y^2)^2}=\boxed{-4y^2}[/tex]

The sign of y is irrelevant.

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Yes, it is possible for a data set to have no mode.

A mode is the most common value in a data set, so if there are no observations that occur more than once, the data set would have no mode. For example, a data set of the numbers {1, 2, 3, 4, 5} would have no mode because none of the numbers appear more than once.

Additionally, a dataset can also have multiple modes, or a bimodal distribution, which means that there are two or more observations that occur with the same highest frequency. A dataset can have a multimodal distribution as well if it has more than two modes.

In summary, a dataset can have no mode if no observations occur more than once, or have multiple/multimodal modes if more than one observations occur with the same highest frequency.

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