Find the constant term in the expansion of (2x+1/x)8

The point where the boat should be landed is the point 3.4 miles from point P towards the town.

The point where the boat should be landed can be found by expressing

the distance traveled on the boat and walking as a function of time.

Given that x represents the distance from point P to the boat landing point.

Therefore, distance of rowing the boat = √((12 - x)² + 3²)

The total time, t, is, therefore;

[tex]t=\frac{12-x}{4} +\frac{\sqrt{x^2+3^2} }{3}-----------(1)[/tex]

differentiate the above equation we get:

[tex]\frac{dt}{dx} =\frac{d}{dx} (\frac{12-x}{4} +\frac{\sqrt{x^2+3^2} }{3} )\\\\\\=\frac{12.(-3+4*\frac{2x}{2*\sqrt{x^2+9} } )}{144} \\\\\\=\frac{12(-3+4*\frac{2x}{2*\sqrt{x^2+9} }) }{144}\\\\=-\frac{1}{4} +\frac{x}{3*\sqrt{x^2+9} } \\\\\frac{x}{3*\sqrt{x^2+9} }=\frac{1}{4} \\\\cross multiplying both terms we get:\\\\4x=3*\sqrt{x^2+9}[/tex]

squaring on both sides we get:

16x^2=9(X^2+9)

16x^2=9x^2+81

7x^2=81

[tex]x=\frac{9*\sqrt{x} }{7}[/tex]

Approximately,x=3.4 miles.

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Answer:

3

Step-by-step explanation:

Given functions:

[tex]\begin{cases}f(x)=2x+1\\g(x)=x^2-3\end{cases}[/tex]

Function composition is an operation that takes two functions and produces a third function.

The given composite function f o g(-2) means to substitute the function g(-2) in place of the x in function f(x):

[tex]\begin{aligned}(f \circ g)(-2)&=f[g(-2)]\\&=2[g(-2)]+1\\&=2(2^2-3)+1\\&=2(4-3)+1\\&=2(1)+1\\&=2+1\\&=3\end{aligned}[/tex]

The Green's function that solves the problem on a semicircular region is given by G(x, xo) = მ (x-xo)(cos(θ)sin(θ)).

What is Green's function?

Green's function is an important mathematical tool used to solve differential equations. It is defined as a solution to a homogeneous linear differential equation with a source, or delta, function as its input. Green's functions are also known as impulse response functions and can be used to describe the response of a system to an arbitrary input. Green's functions provide a way to solve differential equations without having to solve the entire equation all at once. They are particularly useful for solving equations with boundary conditions that are difficult to solve without the use of Green's functions. Green's functions can also be used to solve integral equations and provide a powerful tool for solving many physical problems.

The reflection method for finding a Green's function for a problem on a semicircular region can be used as follows. First, we solve the problem on the half-plane by finding a Green's function G(x, xo) that satisfies the boundary conditions given. Then, we reflect the solution across the boundary of the semicircle, namely T1 and T2, to obtain the Green's function on the entire semicircle.
The Green's function on the half-plane is given by G(x, xo) = მ (x-xo). Using this solution, we can reflect it across the boundary of the semicircle. For T1, the reflected solution is given by G(x, xo) = მ (x-xo)cos(θ), where θ is the angle between the normal vector to T1 and the vector from xo to x. For T2, the reflected solution is given by G(x, xo) = მ (x-xo)sin(θ), where θ is the angle between the normal vector to T2 and the vector from xo to x.
Therefore, the Green's function that solves the problem on a semicircular region is given by G(x, xo) = მ (x-xo)(cos(θ)sin(θ)).

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The equation has one repeated irrational number solution. thus Option A is correct.

A quadratic equation is the second-order degree algebraic expression in a variable. the standard form of this expression is  ax² + bx + c = 0 where a. b are coefficients and x is the variable and c is a constant.

The given equation is the quadratic equation by

[tex]-4x^2+ 24x-36=0[/tex]

We need to determine the number and type of solutions.

Noted that If the discriminant equals zero, the equation has one repeated rational number solution.

Since we know that If the discriminant is positive, then the equation has two real solutions.

And If the discriminant is negative, then the equation has two imaginary solutions.

To Calculate the discriminant of the equation [tex]-4x^2+ 24x-36=0[/tex] by substituting a= -4, b= 24  and c= -36 into the discriminant formula:

Since the discriminant equals zero, it follows that the equation has one repeated irrational number solution x = 3

The equation has one repeated irrational number solution.

Option A is correct.

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The two plans will cost the same when Yolanda has driven 233 miles distance. That is, $70 + (0.30 x 233) = $233 = 0.80 x 233.

To find out how many miles Yolanda needs to drive for the two plans to cost the same, we need to set up an equation. The equation should calculate the total cost of each plan. For the first plan, the total cost is the initial fee plus the cost of the miles driven. For the second plan, the total cost is just the cost of the miles driven. We can then set the two equations equal to each other and solve for the number of miles distance driven. The final result is that Yolanda needs to drive 233 miles for the two plans to cost the same.

Plan 1: C = F + (D × CPM)

Plan 2: C = (D × CPM)

Plan 1 = Plan 2

F + (D × CPM) = (D × CPM)

F = 0

D × CPM = D × CPM

D = 233

Therefore, Yolanda needs to drive 233 miles for the two plans to cost the same.

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The number of litres of water the tank can hold is 48 litres

Volume is defined as the space occupied within the boundaries of an object in three-dimensional space. It is also known as the capacity of the object.

Volume can be measured in cubic units, litres and gallons.

If the rate of pump is 8minutes per minute, therefore in 15 minutes, 15×8 = 120 litres is in the tank.

120 litres is 5/2 full

the volume of the tank will be 120×2/5

= 240/5

= 48litres

Therefore the volume of the tank is 48litres

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Answer:

6y

Step-by-step explanation:

What can you remove form the left side that makes this a false equation?

It's 6y. So place 6y on the right side.

6y + 13 = 6y + 8

Subtract 6y from both sides.

13 = 8

Since 13 = 8 is a false statement, the equation 6y + 13 = 6y + 8 has no solution.

Answer: 6y

Step-by-step explanation:

i did the iready and got it right

its 6y

Answer:

[tex]\log_{10}\sqrt{a} -\dfrac{3}{4}=\dfrac{1}{2}\log_{10}a-\dfrac{3}{4}[/tex]

Step-by-step explanation:

Given:

[tex]x=\dfrac{\sqrt{10a\sqrt{0.1}}}{10}[/tex]

Therefore:

[tex]\implies \log_{10}x=\log_{10}\left(\dfrac{\sqrt{10a\sqrt{0.1}}}{10} \right)[/tex]

[tex]\textsf{Apply the log quotient law}: \quad \log_a \left(\dfrac{x}{y}\right)=\log_ax - \log_ay[/tex]

[tex]\implies \log_{10}\sqrt{10a\sqrt{0.1}}-\log_{10}10[/tex]

[tex]\textsf{Apply log law}: \quad \log_aa=1[/tex]

[tex]\implies \log_{10}\sqrt{10a\sqrt{0.1}}-1[/tex]

[tex]\textsf{Apply radical rule: $\sqrt{ab}=\sqrt{a}\sqrt{b}$, \quad assuming $a \geq 0,\;b\geq 0$}[/tex]

[tex]\implies \log_{10}\left(\sqrt{10}\sqrt{a}\sqrt{\sqrt{0.1}}}\right)-1[/tex]

[tex]\textsf{Apply the log product law}: \quad \log_axy=\log_ax + \log_ay[/tex]

[tex]\implies \log_{10}\sqrt{10} + \log_{10}\sqrt{a} + \log_{10}\sqrt{\sqrt{0.1}}}\right)-1[/tex]

Rewrite the radicals:

[tex]\implies \log_{10} \left(10\right)^{\frac{1}{2}} + \log_{10}\sqrt{a} + \log_{10}\left(0.1\right)^{\frac{1}{4}\right)-1[/tex]

[tex]\textsf{Apply the log power law}: \quad \log_ax^n=n\log_ax[/tex]

[tex]\implies \dfrac{1}{2}\log_{10} 10 + \log_{10}\sqrt{a} + \dfrac{1}{4}\log_{10}0.1-1[/tex]

[tex]\textsf{Apply log laws}: \quad \log_aa=1 \quad \textsf{and} \quad \log_a\dfrac{1}{a}=-1[/tex]

[tex]\implies \dfrac{1}{2}(1) + \log_{10}\sqrt{a} + \dfrac{1}{4}(-1)-1[/tex]

[tex]\implies \dfrac{1}{2}+ \log_{10}\sqrt{a} -\dfrac{1}{4}-1[/tex]

[tex]\implies \log_{10}\sqrt{a} -\dfrac{3}{4}[/tex]

If you want to simplify further:

[tex]\implies \log_{10}\left(a\right)^{\frac{1}{2}} -\dfrac{3}{4}[/tex]

[tex]\implies \dfrac{1}{2}\log_{10}a-\dfrac{3}{4}[/tex]

Answer:

56

Step-by-step explanation: 10a+10 0.1 -=

The probability that a randomly selected bike is green given that the bike has road bike tires is 0.66

Given :

Nellie is shopping for a new bicycle. She is most interested in color and type of tires.

Probability :

Probability means possibility. It is a branch of mathematics that deals with the occurrence of a random event .

From the table given :

Total number of 6 road bikes = 4

Total road bikes = 2 + 4 = 6

Probability = 4 / 6

= 2 / 3

= 0.66

Hence the  probability that a randomly selected bike is green given that the bike has road bike tires is 0.66 .

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The correct option is D , as the number of shares issued is 80000.

What is stock ?

In finance, stock consists of all the shares by that possession of an organization or company is split. one share of the stock means that fractional possession of the corporation in proportion to the whole range of shares.

Main body:

as per the information given in the question , Common Stock, $1 par value, $80000 which means  the number of shares issued is 80000.

hence ,correct option is D as the number of shares issued is 80000.

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The point of intersection (x,y) is  = (2,23)

Replace every x with 2 and simplify

y = 7x+9

y = 7*2 + 9

y = 14 + 9

y = 23 is the y coordinate

The vertical line x = 2 meets the diagonal line y = 7x+9 at the location (x,y) = (2,23)

What are Coordinates ?

The position of a point or points on a graph or grid is shown by its coordinates. The origin is defined as the point (0, 0). The coordinates are two units to the right in the x-direction and four units up in the y-direction according to the point (2, 4).

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Given inputs A, B, C, D, E, design a CMOS circuit to implement the logic function: Y(not) = A(B+C) + D + E has  5-c-mos, 5-pmos total 10 mos transistors

this can be realize by takeing three c-mos in parallel which gives ABC ,two c-mos are series which is in serires with

another c-mos whic gives D(A+B), now connect ABC and D(A+B) in series

the out put will be (ABC + D(A+B) )'

so we require total 6-mos and 6-pmos total 12mos transistors

F = AC + BD

pull down

this can be realize by takeing two c-mos in series which gives AB ,two cmos are in series

which whic gives BD, now connect AC and BD in parallel

pull up

this can be realize by takeing two c-mos in parallel which gives Ac ,two c-mos are in parallel

which whic gives BD, now connect AC and BD in series

the output is (AC+BD)'

to avoid the complement we have to connect the output to c-mos inverter then we get AC+BD

so we require 5-nmos, 5-pmos total 10 mos transistors

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As per the binomial distribution, the value of var(Y) is 0.00008

What is meant by binomial distribution?

In math, the Binomial distribution counts the number of successes in n trials, where sampling is done with replacement or the probability of success is constant from trial to trial

Here we have given that,  X be a random variable representing the number of rolls of a fair die until we see the first 6. Next, choose with replacement a sample of size X from an urn with 5 red and 4 green balls. Let Y be the number of green balls in the sample.

And we need to find the value of var(Y).

Here we know that the value of n = 6 and the value of x is 4 and the probability is 0.05.

Then as per the binomial distribution formula, it can be calculated as,

By using a calculator, you can enter trials=6, p=0.05, and X=4 into a binomial probability distribution function (PDF). If doing this by hand, apply the binomial probability formula:

=> [6!/4!(6-4)!] x 0.05⁴ x (1- 0.05)⁶⁻⁴

=> 0.00008

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The least squares line , y = mx + b form from given data is y = ( -32/100 )x + ( -39/100).

We have an equation of line is

y(x) = mx + b --(1)

given data, (-2, 1), (2, 1),(-1,-1), (0,-2), (2,-2) and (-2, 1)

plugging each point on above equation we get, 1 = -2m + b ; 1 = 2m + b ; -1 = -m + b ; -2 = 0m + b ;

-2 = 2m +b ; 1 = -2m + b

The above equations can be written in matrix form as,

[ -2 1 ; 2 1 ; -1 1 ; 0 1 ; 2 1 ; -2 1] [ m ; b]

= [ 1 ; 1 ; -1 ; -2 ; -2 ; 1]

AX = B

X = (Aᵀ A)⁻¹ AᵀB where Aᵀ is transpose of matrix A and (Aᵀ A)⁻¹ is inverse of (Aᵀ A).

AᵀA = [ -2 2 -1 0 2 -2 ; 1 1 1 1 1] [ -2 1 ; 2 1 ; -1 1 ; 0 1 ; 2 1 ; -2 1] = [ 17 -1 ; -1 6 ]

so, X = [ 17 -1 ; -1 6 ] ^-1 [ -2 2 -1 0 2 -2 ; 1 1 1 1 1][ 1 ; 1 ; -1 ; -2 ; -2 ; 1]

let [ 17 -1 ; -1 6 ] = P

[ 17 -1 ; -1 6 ]⁻¹ = adj.P/det P

det P = 17×6 - 1 = 101

P⁻¹ = 1/101 [ 6 1 ; 1 17 ]

Now, X = 1/101 [ 6 1 ; 1 17 ] [ -2 2 -1 0 2 -2 ; 1 1 1 1 1][ 1 ; 1 ; -1 ; -2 ; -2 ; 1]

=> X = 1/101[ 6 1 ; 1 17 ] [ -5 ; -2 ]

=> X = 1/101 [ -32 ; -39] = [ -32/100 ; -39/100 ]

=> [ m ; b ] = [ -32/100 ; -39/100 ]

=> m = -32/100 , b = -39/100

then, y = mx + b

=> y = ( -32/100 )x + ( -39/100)

Hence, the required equation is

y = ( -32/100 )x + ( -39/100).

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Therefore , [tex](\frac{1}{6})^{2}[/tex] is the probability of getting 5 of same number on all the dice.

The probability of winning is actually closer to 1/3 (25/72). Carnival game operators want you to think that a game is fair like I did, so that you will play. If you knew that the probability of winning a game was only about 1/3, most likely you would not waste your money.

Mathematical descriptions of the likelihood that an event will occur or that a claim will be true are called probabilities.

Here,

[tex](\frac{1}{6})^{2}[/tex] is the probability of getting 5 of same number on all the dice.

Therefore , [tex](\frac{1}{6})^{2}[/tex] is the probability of getting 5 of same number on all the dice.

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The function given by f(x) = 1/1000×e^(-x/1000) is the expression that represents the function that 10 or fewer of the bulbs survive to time 1200.

Given, the lifetime X of a certain brand of sixty-watt light bulb is exponentially distributed with population mean 1000 hours.

That is, 5.231(e/1000)f(x) x > 0. I

f 30 light bulbs having this lifetime distribution are placed on test.

we have to find the probability that 10 or fewer of these light bulbs survive to time 1200.

as, we have to find the expression for the function f(x) so that 10 or fewer of the given light bulbs survive to time 1200.

f(x) = 1/1000×e^(-x/1000)

So, the function given by f(x) = 1/1000×e^(-x/1000) is the expression that represents the function that 10 or fewer of the bulbs survive to time 1200.

Hence, the function given by f(x) = 1/1000×e^(-x/1000) is the expression that represents the function that 10 or fewer of the bulbs survive to time 1200.

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Answer:

Adults: 134

Senior: 241

Step-by-step explanation:

Known

Number of adult tickets sold: x

Revenue from adult tickets: $11x

Number of senior tickets sold: y

Revenue from senior tickets: $8y

From the problem, we found the equation [tex]y = 2x - 27[/tex] and 3,402 = 11x + 8y

Combining the two equation:

[tex]3,402 = 11x + 8(2x-27)\\3,402 = 11x + 16x - 216\\3,402 + 216 = 27x\\27x = 3,618[/tex]

Divided both side by 27

[tex]x = 134[/tex]

Since

[tex]y = 2x - 27[/tex]

Inserting the value of x,

[tex]y = 2(134) -27\\y = 268 - 27\\y = 241[/tex]

So, there are 134 adult tickets sold and 241 senior tickets sold.

The 92.5% of confidence interval for the percentage of adults is

(-1.176 ,  1.342).

as per given in the question,

The total number of adults (n) is 649

The number of smokers (x) = 54

the probability of getting a smoker is P =x/n

=> P = 54/649

=> 0.083

not getting a smoker is

1 - P = 1-0.083

=> 0.916

At 92.5% confidence level the Z critical value is ,

[tex]\alpha[/tex] = 1 - 92.5%

=> 1 - 0.925

=> 0.075

[tex]\alpha[/tex] / 2 = 0.075/ 2

=> 0.0375

now,

for the two tail test

Z[tex]\alpha[/tex]/2

= Z(0.0375)

= 1.7805

Margin of error

E = [tex]Z\alpha / 2 * \sqrt{( p * (1 - p) / n)}[/tex]

= 1.780 [tex]\times (\sqrt{((0.083 \times 0.916 ) / 649)}[/tex]

= 1.2596

A 92.5 % confidence interval for population proportion p is ,

P- E < P < P + E

0.083- 1.259 < P < 0.083 + 1.259

-1.176 < p < 1.342

(-1.176 ,  1.342) is the interval

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The likelihood or the probability is 121, or 8.33%.

What is probability?

Simply put, probability measures how probable something is to occur. We can discuss the probabilities of various outcomes, or how likely they are, whenever we are unsure of how an event will turn out. Statistics is the study of events subject to probability.

The likelihood that a population parameter will be less than a given value when the null hypothesis is true is expressed as the p-value, also known as a probability value or a measure of significance, for a particular statistical model.

We must make a table for easier comprehension in order to determine the P(x=4);

Refer to the attachment for table:

There are three different ways that P(x=4) can occur based on the table. Out of 36 possible results (6 x 6 = 36), they are (1,3), (2,2), and (3,1). As a result, the chance of the event is 363; it is then reduced to 121, or 8.33%.

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Answer:

Domain: 0<=x<=500

Range: 0<=y<=250

Step-by-step explanation:

I'm not sure if this is what you asked for. She spend 2 minutes reading 1 page, there are 250 pages, therefore: 250*2=500

the range is simply the total pages

Hope this help!

Answer:

<CYZ = 60 degrees

Step-by-step explanation:

CY = CZ  ==> both segments are radiuses of the circle since they both have

                     a CenterPoint at Point C and a point touching the edge of the

                     circle

CY = YZ  ==> Since CY=CZ and YZ=CZ, CY also equal YZ. In addition, this

                     means that CY, YZ, and CZ are all radiuses of the circle

Hence, CY = YZ = CZ.

Since all sides of the triangle are equal, that means that all angles of the triangle are equal as well:

<CYZ + <CZY + <YCZ = 180  ==> 180 is the total degrees in a triangle

<CYZ + <CYZ + <CYZ = 180  ==> all angles have the same measure

3 * <CYZ = 180  ==> solve for <CYZ

(3 * <CYZ)/3 = 180/3 ==> divide by 3 on both sides to isolate <CYZ

<CYZ = 60 degrees

Answer: [tex]\angle CYZ=60^{\circ}[/tex]

Step-by-step explanation:

[tex]\overline{CZ}=\overline{CY}[/tex] (radii of a circle)

[tex]\implies \overline{YZ}=\overline{CZ}=\overline{CY}[/tex]

[tex]\therefore \angle CYZ=60^{\circ}[/tex] (equilateral triangle)

Answer:

Step-by-step explanation:

The sum of (2.25 x 10^19) and (6.7 x 10^17) is (2.317 x 10^19).

To find the sum of these two numbers, we first need to line up the decimal points and add the numbers as we would with any other addition problem:

2.25 x 10^19

+ 6.7 x 10^17

Then, we can carry the 1 and add the numbers:

2.25 x 10^19

6.7 x 10^17

2.317 x 10^19

The sum of these two numbers is 2.317 x 10^19, which is equal to choice B.

Null Hypothesis: H0: There is an equal distribution of contralateral and ipsilateral tumors in the population.Ha: There is not an equal distribution of contralateral and ipsilateral tumors in the population.

1) Calculate the observed value:

The observed value is 26 out of 41 cases.

2) Calculate the expected value:

The expected value is 20.5 out of 41 cases.

3) Calculate the test statistic:

The test statistic is (26-20.5)/sqrt(20.5) = 2.44

4) State the critical value:

The critical value for a two-sided test with a significance level of 0.05 is 1.96.

5) The test statistic (2.44) is greater than the critical value (1.96), so we reject the null hypothesis and conclude that there is not an equal distribution of contralateral and ipsilateral tumors in the population.

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The  program for a  given rectangular matrix m,  if an element of m is equal to 0, sets that element's entire row and column to 0

def zeroify_matrix(m):  

rows = len(m)  

cols = len(m[0])  

rows_to_zeroify = []  

cols_to_zeroify = []  

for i in range(rows):

  for j in range(cols):

  if m[i][j] == 0:  

 rows_to_zeroify.append(i)  

cols_to_zeroify.append(j)  

for i in range(rows):  

for j in range(cols):

  if i in rows_to_zeroify or j in cols_to_zeroify:

   m[i][j] = 0  

return m

It is coded in python for the given problem of recreating a rectangular matrix, for a single element to be 0, the whole row nad column will contain zero elements.

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Answer:

Rate= I*100/P*T

R= 1120 * 100/2000*7

R= 112000/14000

R= 8%

Answer:

42

Step-by-step explanation:

To simplify the expression -7x3x(-2)x(-5)x(-1), we need to apply the rules of exponentiation and multiplication.

First, we note that the terms -7, -2, -5, and -1 are all negative numbers, which means that they will be multiplied together to give a positive result.

Next, we apply the rule that says that any term raised to the power of 0 is equal to 1. Since -1 raised to the 0 power is 1, this means that the expression -7x3x(-2)x(-5)x(-1) is equivalent to -7x3x(-2)x(-5)x1.

Next, we apply the rule that says that any term raised to the power of 1 is equal to itself. Since 3 raised to the 1 power is 3, and 5 raised to the 1 power is 5, this means that the expression -7x3x(-2)x(-5)x1 is equivalent to -7x3x(-2)x5x1.

Finally, we apply the rule that says that the product of any number and its multiplicative inverse (or reciprocal) is equal to 1. Since the reciprocal of 5 is 1/5, this means that the expression -7x3x(-2)x5x1 is equivalent to -7x3x(-2)x(1/5)x1.

We can now use the rules of multiplication to simplify the expression further. Since any number multiplied by 1 is equal to itself, this means that the expression -7x3x(-2)x(1/5)x1 is equivalent to -7x3x(-2)x(1/5).

Next, we use the rule that says that the product of two negative numbers is a positive number. This means that the expression -7x3x(-2)x(1/5) is equivalent to 7x3x2x(1/5).

Finally, we can apply the rules of exponentiation and multiplication to evaluate the expression. Since any number raised to the power of 1 is equal to itself, this means that the expression 7x3x2x(1/5) is equivalent to 7x3x2. We can then multiply these terms together to get the final result:

7x3x2 = 42

Therefore, the simplified expression is equal to 42.

Answer: Rise over Run

Step-by-step explanation: Take the change in the x coordinates, and divide by the change in y coordinates.

For example, to find slope when given 2 coordinates:

(2,5)     (5, 10)

The change in the x coordinates is 3, because from 2-->5 it moves 3 places to the right. (run)

The change in the y coordinates is 5, because from 5-->10 it moves 5 places upwards. (rise)

Use rise/run.

Rise= 5 (change in y coordinate)

Run = 3 (change in x coordinate)

5/3 is the slope.

On questions like #6, the slope is -3. That is because the equation for slope is y=mx+b.

m is the coefficient of the x, so -3 is the slope.

Likewise, the slope of #2 is 7. The slope of #10 is 3/4.

If the number of solutions to the equation that satisfy the given condition is the number of ways to place all the items into the categories. Then, the answer is 210 .

In the question ,

it is given that ,

number 22 is divided into 22 individual items ,

the three categories in which these items are placed are x₁, x₂ and x₃ .

x₁ + x₂ + x₃ = 22    .....equation(1)

(x₁ - 1) + (x₂ - 1) + (x₃ - 1) = 22 - 3 = 19

let a = x₁ - 1 , b = x₂ - 1  and c = x₃ - 1 .

Hence , a + b + c = 19    ....equation(2)

Now , because the number of solution of equation(1) excluding zero will be equal to the number of solutions of equation(2) including the zero .

So , we subtracted 1 from x₁, x₂ and x₃ respectively .

Now , we have to divide 19 into 3 parts where any part can be zero also or any part is a whole number .

For Example : let us have 19 apples , which must be divided into 3 parts by placing 2 J's among 19 A's .

Now we have reduced the problem to such that we have to find the permutation of 19 A's and 2 J's that is 21 characters in which every permutation is a solution .

So , the  number of ways of arranging 21 characters in which 19 are same(A)  and 2 are same (J)

= 21!/(2! * 19!)

= 21*10 = 210

It means there are 210 non negative integral solutions of a+b+c = 19 .

It means total positive solutions of x₁ + x₂ + x₃ = 22  is 210 .

Therefore , the number of solution to the equation is 210 .  

The given question is incomplete , the complete question is

Start by thinking of the number 22 as divided into 22 individual items and the variables x₁, x₂, and x₃ as three categories into which these items are placed. since each [tex]x_{i}[/tex] is a positive integer, start by placing one item in each category and distribute the remaining items among the categories. The number of solutions to the equation that satisfy the given condition is the number of ways to place all the items into the categories. Thus, the answer is ______  .

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The rate of change of the average price of the new car is $274.50 per year.

The rate of change of the average price of the new car is a function of the change in average price over the years and the change in the number of years.

Rate of change = change in the average price / change in the number of years

Rate of change = (price in 2002 - price in 1994)  / difference in the number of years

Rate of change = (19,126 - 16,930) / (2002 - 1994)

Rate of change = 2196 / 8

Rate of change = $274.50

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