In how many ways can $9$ friends sit at a circular table if two of them, Anna and Bob, insist on having exactly two people seated between them? (As usual, two seatings are considered the same if one is a rotation of the other.)

Step-by-step explanation:

look at the attachment above

Answer:

[tex]\textsf{1)} \quad -\dfrac{1}{16}e^{-4x}\left(4x+1\right)+\text{C}[/tex]

[tex]\textsf{2)} \quad - \cos x+\dfrac{2}{3} \cos^3 x - \dfrac{1}{5} \cos^5 x +\text{C}[/tex]

Step-by-step explanation:

Question 1

[tex]\boxed{\begin{minipage}{5 cm}\underline{Integration by parts} \\\\$\displaystyle \int u \dfrac{\text{d}v}{\text{d}x}\:\text{d}x=uv-\int v\: \dfrac{\text{d}u}{\text{d}x}\:\text{d}x$ \\ \end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{5 cm}\underline{Integration of $e^{ax}$} \\\\$\displaystyle \int e^{ax}\:\text{d}x=\dfrac{1}{a}e^{ax}+\text{C}$\\\\for $a\neq 0$\\\end{minipage}}[/tex]

Given integral:

  [tex]\displaystyle \int xe^{-4x}\:\text{d}x[/tex]

Using Integration by parts:

[tex]\textsf{Let }\:u=x \implies \dfrac{\text{d}u}{\text{d}x}=1[/tex]

[tex]\textsf{Let }\:\dfrac{\text{d}v}{\text{d}x}=e^{-4x} \implies v=-\dfrac{1}{4}e^{-4x}[/tex]

Therefore:

[tex]\begin{aligned}\displaystyle \int u \dfrac{\text{d}v}{\text{d}x}\:\text{d}x & =uv-\int v\: \dfrac{\text{d}u}{\text{d}x}\:\text{d}x\\\\\implies \displaystyle \int xe^{-4x}\:\text{d}x & =-\dfrac{1}{4}xe^{-4x}-\int -\dfrac{1}{4}e^{-4x}\: \text{d}x\\\\& =-\dfrac{1}{4}xe^{-4x}+\int \dfrac{1}{4}e^{-4x}\: \text{d}x\\\\& =-\dfrac{1}{4}xe^{-4x}-\dfrac{1}{16}e^{-4x}+\text{C}\\\\& =-\dfrac{1}{16}e^{-4x}\left(4x+1\right)+\text{C}\end{aligned}[/tex]

Question 2

[tex]\boxed{\begin{minipage}{4 cm}\underline{Integrating $x^n$}\\\\$\displaystyle \int x^n\:\text{d}x=\dfrac{x^{n+1}}{n+1}+\text{C}$\\ \end{minipage}}[/tex]    

[tex]\boxed{\begin{minipage}{5 cm}\underline{Integrating a constant}\\\\$\displaystyle \int n\:\text{d}x=nx+\text{C}$\\(where $n$ is any constant value)\end{minipage}}[/tex]

Rewrite the given integral:

[tex]\begin{aligned}\displaystyle \int \sin^5 x \: \text{d}x & =\int (\sin x)^4 \cdot \sin x \: \text{d}x\\& =\int (\sin^2 x)^2 \cdot \sin x \: \text{d}x\end{aligned}[/tex]

Use the trig identity  [tex]\sin^2x+\cos^2x \equiv 1[/tex]  to rewrite  [tex]\sin^2x[/tex] :

[tex]\implies \displaystyle \int \sin^5 x \: \text{d}x = \int (1-\cos^2 x)^2 \cdot \sin x \: \text{d}x[/tex]

Integration by substitution

[tex]\textsf{Let }\:u=\cos x \implies \dfrac{\text{d}u}{\text{d}x}=-\sin x \implies \text{d}x=-\dfrac{1}{\sin x}\: \text{d}u[/tex]

Therefore:

[tex]\begin{aligned}\implies \displaystyle \int \sin^5 x \: \text{d}x & = \int (1-u^2)^2 \cdot \sin x \cdot -\dfrac{1}{\sin x}\: \text{d}u\\& = \int -(1-u^2)^2 \: \text{d}u\\ & =\int -1+2u^2-u^4 \: \text{d}u\\& =-u+\dfrac{2}{3}u^3-\dfrac{1}{5}u^5+\text{C}\end{aligned}[/tex]

Finally, substitute  [tex]u = \cos x[/tex]  back in:

[tex]\implies \displaystyle \int \sin^5 x \: \text{d}x=- \cos x+\dfrac{2}{3} \cos^3 x - \dfrac{1}{5} \cos^5 x +\text{C}[/tex]

Answer:

eyyyyyyyyyy wassupppppppp

Answer:

240%

Step-by-step explanation:

Volume of a prism is Bh (area of the base * height)

Let's start with the small prism.

B=125

h=12

Volume = 125*12 = 1500 cubic cm

Bigger prism base are is 160% of 125. 1.6 * 125 = 200

Height is 6 cm more = 12+6 = 18

V = 200 * 18 = 3600

To find the percentage, divide 3600 by 1500. You get 2.4, which is 240%

Answer:

There is no line plot

Step-by-step explanation:

our system of inequalities is:

x + y ≤ 13.

x*$15 + y*$10 ≥ $180.

The graph can be seen at the end.

Here we have the variables:

We know that she can only work 13 hours per week, then we have:

x + y ≤ 13.

And she wants to earn at least $180 per week, then:

x*$15 + y*$10 ≥ $180.

So our system of inequalities is:

x + y ≤ 13.

x*$15 + y*$10 ≥ $180.

The graph of the system can be seen below. The solution set is defined by the zone where the two shaded regions intersect (where we also should add the restrictions x >0, y > 0).

If you want to learn more about inequalities:

https://brainly.com/question/9774970

#SPJ1

Answer: 2 3/4 > 2.68 is correct (only)

Step-by-step explanation: 2 3/4 = 2.75, and 2.75 is greater than 2.68

The rest are incorrect. 7.895 * 10^2 (100) = 789.5, and 5.43 * 10^3 (1000) = 5430. So 789.5 > 5430 is false.

Sqrt(63) = 7.93, and 7.93 > 8.11 is false.

Absolute values ( | | ) mean that no matter if it is negative or positive, it will become positive (unless the negative sign is outside the absolute value). 1.5 > 3.2 is false, and even without the absolute value, -1.5 is still less than 3.2

Answer:

all of the above

Step-by-step explanation:

please mark me as brainliest

The point on the number line represents the product (- 4), (- 2), and (- 1) will be the negative 8. Then the correct option is A.

It is also known as the product. If the object n is given to m times, then we just simply multiply them.

The point on the number line represents the product (- 4), (- 2), and (- 1) will be

Then the product will be

-4 x -2 x -1 = -8

Then the correct option is A.

More about the multiplication link is given below.

https://brainly.com/question/19943359

#SPJ1

Answer:

1 HR ---> 42 km

? .----> 119 km

119 * 1/42

2.833

Answer:

[tex]\boxed {2 \frac{5}{6}}[/tex]

Step-by-step explanation:

Formula used :

[tex]\boxed {Time = \frac{Distance}{Speed}}[/tex]

Solving :

Time = 119/42

Time = 17/6

Time = 2 5/6 hours

Answer:

17

Step-by-step explanation:

→ Substitute 7 f(x) into the function

7 + 10

→ Solve

17

Answer:

f(7) = 17

Step-by-step explanation:

To find f(7), substitute x = 7 in the function. Thus, you will receive:

[tex]\displaystyle{f(7)=7+10}\\\\\displaystyle{f(7)=17}[/tex]

Henceforth, the value of f(7) is 17.

Learn More:

First lets simplify to slope intercept (y=mx+b)

y=[tex](2x-7)/-3[/tex]

y=-2/3x+7/3

Point 1: (0,7/3) as when x is 0, y=7/3.
Point 2: lets plug x in as 3 to cross cancel.

[tex]\frac{-2}{3} *\frac{3}{1} = \frac{-2}{1}[/tex]

So (3,-2).

Lets do point 3: plug in -3.

[tex]\frac{-2}{3} *\frac{-3}{1} = \frac{2}{1} = 2[/tex]

So (-3,2).

Your three points are (0,2.33 repeat) or (0,7/3), (3,-2) and (-3,2)

Hopefully this helps you out, heart if it was helpful and please do not hesitate to ask any questions. I have also attached an image of the line

#learnwithbrainly

Answer:

5,208,469

Step-by-step explanation:

5,000,000 + 200,000+ 8,000 + 400+60+9

5,000,000

  200,000

    00,000

       8,000

          400

            60

              9

5,208,469

The length of the box made by Mr. Baker is 10-inch

Mr. Baker has strip of 48- inch long  oak, by this strip he is making the rectangle box of width 14-inch.

The rectangle is 4 sided geometric shapes whose opposites are equal in lengths and all angles are about 90°.

As Mr. Baker has a strip that is 48-inch long and he made a rectangle box with it.

The width of the  box = 14 inches
now the length of the box = (total length of the strip - 2* width of the box)/2
⇒ length of the box = (48 - 2*14)/2
⇒ length of the box = 10 inches

Thus the Mr. Baker made the box which is 10 inches long.

learn more about rectangles here:
https://brainly.com/question/15019502

#SPJ1

Answer:

m=0 , b=41

Step-by-step explanation:

Answer:

[tex]4\sqrt{26}[/tex]

Step-by-step explanation:

The distance formula is [tex]\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex].

Thus we can plug the numbers in:

[tex]\sqrt{(-2-(-6))^2+(10-(-10))^2}[/tex]

[tex]\sqrt{(4)^2+(20)^2}[/tex]

[tex]\sqrt{16+400}[/tex]

[tex]\sqrt{416}[/tex]

[tex]4\sqrt{26}[/tex]

By applying the theory of separable ordinary differential equations we conclude that the solution of the differential equation [tex]\frac{dy}{dx} \cdot (\ln y)^{2} + y\cdot \sqrt{x} = 0[/tex] with y(0) = e is [tex]y = e^{\sqrt [3]{-2\cdot x^{\frac{3}{2} }+1}}[/tex].

In this question we must separate each variable on each side of the equivalence, integrate each side of the expression and find an explicit expression (y = f(x)) if possible.

[tex]\frac{dy}{dx} \cdot (\ln y)^{2} + y\cdot \sqrt{x} = 0[/tex]

[tex](\ln y)^{2}\,dy = -y \cdot \sqrt{x}\, dx[/tex]

[tex]-\frac{(\ln y)^{2}}{y}\, dy = \sqrt{x} \,dx[/tex]

[tex]-\int {\frac{(\ln y)^{2}}{y} } \, dy = \int {\sqrt{x}} \, dx[/tex]

If u = ㏑ y and du = dy/y, then:

[tex]-\int {u^{2}\,du } = \int {x^{\frac{1}{2} }} \, dx[/tex]

[tex]-\frac{1}{3}\cdot u^{3} = \frac{2\cdot x^{\frac{3}{2} }}{3} + C[/tex]

[tex]u^{3} = -2\cdot x^{\frac{3}{2} } + C[/tex]

[tex](\ln y)^{3} = - 2\cdot x^{\frac{3}{2} } + C[/tex]

[tex]C = (\ln e)^{3}[/tex]

[tex]C = 1[/tex]

And finally we get the explicit expression:

[tex]\ln y = \sqrt [3]{-2\cdot x^{\frac{3}{2} }+ 1}[/tex]

[tex]y = e^{\sqrt [3]{-2\cdot x^{\frac{3}{2} }+1}}[/tex]

By applying the theory of separable ordinary differential equations we conclude that the solution of the differential equation [tex]\frac{dy}{dx} \cdot (\ln y)^{2} + y\cdot \sqrt{x} = 0[/tex] with y(0) = e is [tex]y = e^{\sqrt [3]{-2\cdot x^{\frac{3}{2} }+1}}[/tex].

To learn more on ordinary differential equations: https://brainly.com/question/14620493

#SPJ1

The foci of the hyperbola represented by (0, –6) and (0, 6). Then the correct option is B.

The hyperbola is the locus of a point such that the difference of distance from point P to two non-movable points. These two points are called foci of hyperbola.

The equation of the parabola is given below.

5y² – 4x² – 80 = 0

Then the equation can be written as

y² / 4 – x² / 5 – 4 = 0

  y² / 16 – x² / 20 = 1

Then the value of eccentricity will be

a² = 16 and b² = 20

Then we have

e = √[1 + (b² / a²)]

e = √[1 + (20 / 16)]

e = 1.5

Then the focus of the parabola will be

⇒ (0, ±ae)

⇒ (0, ± 4 x 1.5)

⇒ (0, ±6)

⇒ (0, –6) and (0, 6)

Then the correct option is B.

More about the hyperbola link is given below.

https://brainly.com/question/12919612

#SPJ1

Answer:

y=55x

Step-by-step explanation:

(y1-y2)/(x1-x2)=(165-55)/(3-1)=55

55 is the slope

To find y-intercept:

y=55x+b

substitute 1 for x and 55 for y

55=55(1)+b

b=0

So y= 55x

the equation of the line passing through the points (3, 165) and (1, 55) is y = 55x. To write the equation of the line that passes through the points (3, 165) and (1, 55), you can use the point-slope form of a linear equation, which is: y - y₁ = m(x - x₁)

where (x₁, y₁) is one of the points on the line, m is the slope of the line.

First, calculate the slope (m) using the given points (3, 165) and (1, 55):

m = (y₂ - y₁) / (x₂ - x₁) = (55 - 165) / (1 - 3) = (-110) / (-2) = 55

Now that you have the slope (m = 55) and one of the points (let's use (3, 165)), you can plug them into the point-slope form:

y - 165 = 55(x - 3)

Now, you can simplify and rewrite this equation in slope-intercept form (y = mx + b):

y - 165 = 55x - 165

To isolate y, add 165 to both sides:

y = 55x

So, the equation of the line passing through the points (3, 165) and (1, 55) is y = 55x.

to know more about equation:

https://brainly.com/question/32543980

#SPJ3

Answer:

$16,400

Step-by-step Explanation:

Step 1: We multiply $10,000 by 16%, getting $1,600.

Step 2: We then multiply $1,600 by 4 (which is given). That gives us $6,400.

Step 3: Add that to $10,000. We get $16,400 as the answer!

Answer:

Simple interest = P  x  r  x  t

                         = 10,000  x  16/100  x   4

                         =  $6400

Final amount = P (1 +rt)

                       = 10,000 (1 + 16/100  x  4)

                       = $16400

(or you can just add the simple interest to the principle amount to get the final amount)

Answer:

No solution

Step-by-step explanation:

The attached graph represents the piecewise function.

The complete question is in the image

The function is given as:

[tex]f(x) = \left[\begin{array}{cc}3&x \le -3\\2x + 1&-3 < x < 4 \\-2&x \ge 4\end{array}\right[/tex]

Next, we plot the functions by their domain

See attachment for the piecewise function.

Read more about piecewise function at:

https://brainly.com/question/27262465

#SPJ1

Answer:

okay so I can't see the last option on ur sheet but when u graph this equation the point that lies on the curve is (2,-3) so that is the vertex but if I move up this curved line it passes through the Y axis at (0,-1), the Last answer that I can't see it wouldn't happen to be (-1,6) would it

Answer:  A B C

Step-by-step explanation:

A whole number is like 1,500 and a nonwhole number is like 33.2 or like 55/100.  

And have a good day.

In the expressions that we are given, we can note that there is a negative in all of them. This means that we can factor out the negative.

With the negative factored out, it will look like this: [tex]-(\frac{8}{11} + \frac{3}{4} + \frac{1}{4} )[/tex]
The reasoning for this is because you distribute the negative to each term. For example 8/11 is a term, and so is 3/4, and 1/4. So our first answer is D.

Another way we can do this is by combining like terms. We see that in the denominator for -3/4 and -1/4 they both share 4. We can combine it so it becomes -4/4 or -1. It would be E.

The third way we can do this is by rewriting it. Answer A rewrites it so that it is + -, effectively just -. This allows us to treat the equation as if it were the original one.

Hopefully this helps you out, brainliest would help me out.
#GetVerifiedAnswers #LearnWithBrainly

Your answer is correct. The two "bases" of the trapezoid are the line segments (0, 0)-to-(y, 0) and (0, x)-to(z, x), with lengths y and z, respectively. Then the average of the bases is [tex]\frac{y+z}2[/tex]. (We know they're the bases because the line segments are parallel.)

We multiply this by the height, which is given by the length of the line segment (0, 0)-to-(0,x), or x.

Hence the area is

[tex]\dfrac{x(y+z)}2[/tex]

(which is identical to what you have).

A set is a mathematical model for a collection of items. The set of W' is {5, 20, 30}.

A set is a mathematical model for a collection of items; it comprises elements or members, which may be any mathematical object: numbers, symbols, points in space, lines, other geometrical structures, variables, or even other sets.

The complete question is given below.

Q13.

T ∪ V = Data points that are in T or V or both.

T ∪ V = {5, 10, 20, 30}

Q14.

T ∩ W = Data points that are in T and W both.

T ∩ W = {10}

Q15.

W' = Data points that are not in W.

W' = {5, 20, 30}

Hence, the set of W' is {5, 20, 30}.

Learn more about Set:

https://brainly.com/question/8053622

#SPJ1

Using proportions, it is found that the coordinates of the treasure as given as follows: (7.6, 8.8).

A proportion is a fraction of a total amount, and the measures are related using a rule of three.

Researching the problem on the internet, the coordinates are as follows:

The treasure is buried between a rock and a tree in a 5:9 ratio, hence the expression is:

[tex]Ts - R = \frac{5}{14}(Tr - R)[/tex]

Hence, for the x-coordinate:

[tex]x - 3 = \frac{5}{14}(16 - 3)[/tex]

x - 3 = 5 x 13/14

x = 7.6.

For the y-coordinate:

[tex]y - 2 = \frac{5}{14}(21 - 2)[/tex]

y - 2 = 5 x 19/14

y = 8.8.

More can be learned about proportions at https://brainly.com/question/24372153

#SPJ1

Answer: the answer is b

Step-by-step explanation:

Answer:

Step-by-step explanation:

volume of a cuboid = Length × Width × Height Cubic units

6 × 11 × 6 =

66 * 6 =

396 Km³

to the nearest hundred

400 Km³