Let z = 3+ bi and w = a + bi where a, b E R. Without using a calculator, (a) determine and hence, b in terms of a such that is real.

The cross product of

[tex](i+j) \times k[/tex]= 0.

To calculate the cross product of (i+j) and k,

Step 1: Assign unit vectors to the given vectors:

(i+j) = i + j + 0k

k = 0i + 0j + k

Step 2: Apply the cross product formula:

(i+j) × k = (i × 0i) + (i × 0j) + (i × k) + (j × 0i) + (j × 0j) + (j × k) + (0k × 0i) + (0k × 0j) + (0k × k)

Step 3: Simplify the cross product using the properties of the cross product:

(i × 0i) = (j × 0j) = (0k × 0i) = (0k × 0j) = 0

(i × k) = - (k × i)

(j × k) = - (k × j)

(k × k) = 0

Step 4: Substitute the simplified cross products into the formula:

(i+j) × k = 0 + 0 + (i × k) + 0 + 0 + (j × k) + 0 + 0 + 0

= 0 + 0 + (i × k) + 0 + 0 + (j × k) + 0 + 0 + 0

= (i × k) + (j × k)

Step 5: Calculate the cross products:

(i × k) = (0 - 0)k - (0 - 0)j = 0k - 0j = 0k

(j × k) = (0 - 0)i - (0 - 0)k = 0i - 0k = 0i

Step 6: Substitute the calculated cross products into the formula:

(i+j) × k = 0k + 0i

= 0k + 0

= 0

Therefore k=0.

Learn more about cross product here:

https://brainly.com/question/14708608

#SPJ4

The solution to the differential equation is [tex]y = e^{-\frac{x^2}{2} + \ln(50)}[/tex]

The value of y is decreasing for x > 0.

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

dy/dx = -xy

Rewrite as

dy/y = -x dx

Differentiate both sides

So, we have

ln|y| = -x²/2 + c

Take the exponent of both sides

[tex]|y| = e^{-\frac{x^2}{2} + c}[/tex]

Assume y > 0

So, we have

[tex]y = e^{-\frac{x^2}{2} + c}[/tex]

When x = 0, y = 50

So, we have

[tex]50 = e^{-\frac{0^2}{2} + c}[/tex]

Evaluate

c = ln(50)

So, we have

[tex]y = e^{-\frac{x^2}{2} + \ln(50)}[/tex]

Recall that dy/dx = -xy.

This means that if dy/dx < 0, then y is decreasing.

This means that, y is decreasing for x > 0.

Read more about differential equation at

https://brainly.com/question/1164377

#SPJ4

After considering the given data we conclude that
a) r = -2 + i and r = -2 - i value generated after solving the auxiliary equation
b)[tex]y_c = c_1e^{(-2t)} cos(t) + c_2e^{(-2t)} sin(t)[/tex] complementary solution
c) general solution to the nonhomogeneous equation is [tex]y = y_c + Y_p = c_1e^{(-2t)} cos(t) + c_2e^{(-2t)} sin(t) + (1/26)e^{(-21)} .[/tex]
d) initial value problem is [tex]y(t) = e^{(-2t)} (cos(t) + 2sin(t))/2 + (1/26)e^{(-21)} .[/tex]

To determine the position of the body at time t, we can apply the given differential equation to solve for y(t). To do this, we can follow the steps below:
(a) the characteristic (auxiliary) equation is
The characteristic equation is obtained by setting the coefficients of the differential equation to zero, which gives us the equation  [tex]r^2 + 4r + 5 = 0.[/tex]Solving this quadratic equation, we get r = -2 + i and r = -2 - i.
(b) evaluate the complementary solution, yc, to the corresponding homogeneous equation, [tex]y" + 4y' + 5y = 0.[/tex]
The complementary solution is given by[tex]y_c = c_1e^{(-2t)} cos(t) + c_2e^{(-2t)} sin(t),[/tex]where [tex]c_1[/tex] and [tex]c_2[/tex] are constants determined by the initial conditions.
(c) Calculate a particular solution, Yp, to the nonhomogeneous differential equation, [tex]Y" + 4y' + 5y = e^{(-21)}[/tex]. Hence state the general solution to the nonhomogeneous equation as [tex]y = y_c + y_p.[/tex]
To describe a particular solution, we can use the method of undetermined coefficients and guess a solution of the form [tex]Yp = Ae^{(-21)}[/tex], where A is a constant to be determined.
Staging this into the differential equation, we get A = 1/26. Therefore, the particular solution is [tex]Y_p = (1/26)e^{(-21)}[/tex]. The general solution to the nonhomogeneous equation is [tex]y = y_c + Y_p = c_1e^{(-2t)} cos(t) + c_2e^{(-2t)} sin(t) + (1/26)e^{(-21)}[/tex]
(d) Solve the initial value problem if the initial position of the body is 1 m and its initial velocity is zero.
Applying the initial conditions, we can solve for the constants [tex]c_1[/tex] and [tex]c_2[/tex] in the complementary solution. Since the initial velocity is zero, we have [tex]y'(0) = -2c_1 + c_2 = 0[/tex], which gives us [tex]c_2 = 2c_1.[/tex]
Applying the initial position, we have [tex]y(0) = c_1 = 1[/tex], which gives us [tex]c_2 = 2.[/tex]Therefore, the solution to the initial value problem is [tex]y(t) = e^{(-2t)} (cos(t) + 2sin(t))/2 + (1/26)e^{(-21)}[/tex]
To learn more about differential equation
https://brainly.com/question/28099315
#SPJ4

The statistic provides an estimate of the parameter based on the information obtained from the sample.

In this scenario, the population refers to the entire group of customers of the local store.

The sample is a subset of this population and consists of the 82 customers who were randomly surveyed.

The variable of interest is the age of the customers, as the store owner wants to determine the mean age.

Age is a continuous numerical variable since it represents a quantitative measurement.

The parameter is the unknown mean age of all customers of the local store, while the statistic is the calculated mean age of the 82 surveyed customers.

Learn more about parameter based

brainly.com/question/31824653

#SPJ11

Answer: 56

Step-by-step explanation: According to AIA(Alternate Interior Angles Congruency) they are equal. So just solve the equation that they're both equal to each other and you get x=-6. Plug this back in to find the angle measure, 56! Hope this helped!

Answer:

x = -6

56 degrees

Step-by-step explanation:

Those two angles that are labeled are equal for a reason that I forgot (but trust me they are). So, -8x+8 = -6x+20

just solve the equation to get x = -6

after plugging in -6 for x, you get that the angles that are labeled are 56 degrees.

Answer:

x = -3 and x = 2

Step-by-step explanation:

The expression for the quadratic equation is as follows:

(x+3)(x-2)

For (x+3)(x-2) to equal 0, either (x+3) or (x-2) must be equal to 0. So,

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

It implies,

x = -3 and x = 2

So, the values of x are -3 and 2.

Answer:

58

Step-by-step explanation:

Answer: 58°

Step-by-step explanation:

We know it's a right triangle, one of them must be 90°. The sum of the three interior angles are 180°, so we know the sum of the other two angles is 180 - 90 = 90°.

If one of the angles is 32°, another angle is 90 - 32 = 58°

This formula calculates the value corresponding to a 90% confidence level using the T.INV function. The first argument, 0.95, represents 1 minus the desired confidence level (0.05 for a 90% confidence level). The second argument, `COUNT(A2:A89)-1`, calculates the degrees of freedom by subtracting 1 from the count of data points.

To find the value from the appropriate probability table to construct a 90% confidence interval in cell B8, you can use the T.INV function in Excel. Here's how you can do it:

1. In cell B8, enter the formula:

  ```

  =T.INV(0.95, COUNT(A2:A89)-1)

  ```

  This formula calculates the value corresponding to a 90% confidence level using the T.INV function. The first argument, 0.95, represents 1 minus the desired confidence level (0.05 for a 90% confidence level). The second argument, `COUNT(A2:A89)-1`, calculates the degrees of freedom by subtracting 1 from the count of data points.

Make sure to adjust the cell references in the formula based on the actual location of your data in the spreadsheet.

Note: The T.INV function returns the value from the Student's t-distribution, which is used for small sample sizes or when the population standard deviation is unknown.

If you have a large sample size (usually considered more than 30), you can use the Z.INV function instead to calculate the value from the standard normal distribution.

Visit here to learn more about confidence level brainly.com/question/22851322

#SPJ11

Answer:

7:00 = 6:60

Step-by-step explanation:

6:60 - 3:30 = 3 hours and 30 minutes

Answer:

As per the conversion in the question, £40.50 = €54

Step-by-step explanation:

Here, we want to make a conversion;

from the question;

£3 = €4

£40.50 = €x

Thus;

3 * x = 4 * 40.5

x = (4 * 40.5)/3

x = €54

The function  [tex]f(x) = x^3 / (x^2 - 49)[/tex]has a vertical asymptote at x = 7 and x = -7. There are no slant asymptotes. To find the vertical asymptotes of the function f(x), we need to identify the values of x for which the denominator becomes zero, as division by zero is undefined.

In this case, the denominator is (x² - 49). To determine when it equals zero, we set it equal to zero and solve for x:

x² - 49 = 0

This equation can be factored as the difference of squares:

(x - 7)(x + 7) = 0

Setting each factor equal to zero, we find that x = 7 and x = -7. Therefore, the function has vertical asymptotes at x = 7 and x = -7.

On the other hand, to determine if there are any slant asymptotes, we need to check if the degree of the numerator (x³) is greater than the degree of the denominator [tex](x^2 - 49)[/tex]. In this case, the degree of the numerator is 3, while the degree of the denominator is 2. Since the degree of the numerator is greater, there are no slant asymptotes for this function.

In summary, the function [tex]f(x) = x^3 / (x^2 - 49)[/tex] has vertical asymptotes at x = 7 and x = -7. There are no slant asymptotes.

Learn more about denominator here: https://brainly.com/question/15007690

#SPJ11

Answer:

See explanations below

Step-by-step explanation:

evaluate sin 60. cos 30 sin +sin 30 .cos 60 what is the value of sin(30-60) what can you conclude​

According to the trigonometry identity

Sin 30  = 1/2

Sin 60 = √3/2

Cos 30 = √3/2

Cos 60 = 1/2

sin 60. cos 30 +sin 30 .cos 60

= √3/2(√3/2) + 1/2(1/2)

= √9/4 + 1/4

= 3/4 + 1/4

= 4/4

= 1

sin(30-60) = sin30cos60 - cos30sin60

sin(30-60) =1/2(1/2) - √3/2(√3/2)

sin(30-60) = 1/4 -  √9/4

sin(30-60) = 1/4 - 3/4

sin(30-60) = (1-3)/4

sin(30-60) = -2/4

sin(30-60)  = -1/2

hence the former fraction gives a positive values while the later gives a negative

21 is the right answer.

180-138=42°

and since it is given in the question that the triangle is isoceles ,so half of 42 is 21°

Answer:

the last three answers is what i think

Step-by-step explanation:

The true statements from the sinusoidal graph are:

From the table, we have the following parameters:

So, the radius (r) of the wheel is:

[tex]r = \frac{64 - 4}{2}[/tex]

This gives

[tex]r = \frac{60}{2}[/tex]

[tex]r = 30[/tex]

Also, from the table, the lowest point of the wheel is 4 ft;

The Ferris wheel is at this lowest point at 7.5 seconds and 17.5 seconds

Hence, the true statement is (c) and (d)

Read more about sinusoidal functions at:

https://brainly.com/question/2410297

Answer: good luck

Step-by-step explanation:

Glacial HVAC, Inc. had a relative market share of 54.39% in Fulton County in 2010, based on the sale of 3,295 air conditioning units, compared to their largest competitor, Estes Heating and Air.

To calculate the relative market share, we need to divide Glacial HVAC's sales by the total market sales. The formula for relative market share is:

Relative Market Share = Glacial HVAC's Sales / Total Market Sales.

In this case, Glacial HVAC sold 3,295 units, and their largest competitor, Estes Heating and Air, sold 2,759 units. So the total market sales would be the sum of these two figures, which is 3,295 + 2,759 = 6,054 units.

Now, we can calculate the relative market share:

Relative Market Share = 3,295 / 6,054 * 100%.

Evaluating the expression, we find that Glacial HVAC's relative market share is approximately 54.39% when rounded to two decimal places.

This means that Glacial HVAC had a market share of 54.39% in Fulton County in 2010, indicating their position in the market relative to their largest competitor.

Learn more about decimals here:

https://brainly.com/question/30958821

#SPJ11

Answer:

y - 6

Step-by-step explanation:

y is the number and  it is less than 6 so 6 has to be negative:

y - 6

Answer:

1/4 is x and 1 is b

Step-by-step explanation:

Answer:

Hey I know this isn’t an answer but your bf Jordan told me to tell you he is looking for you and he needs to talk to you. Sorry again for wasting ur time but he kinda worried

Step-by-step explanation:

[tex]\boxed{\large{\bold{\blue{ANSWER~:) }}}}[/tex]

Here,

The two lines l and m are parallel,

cut by a transversat line t

So The two angles are supplementary each other.

we know that,

The sum of two supplementary angles are 180°

According to the question,

Therefore,

The value of x is 8°

1. is a central angle, therefore, the arc will have the same measurement as the angle. Set the equation:

Arc = Central Angle

2x - 7 = 47

Isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS.

First, add 7 to both sides of the equation:

2x - 7 (+7) = 47 (+7)

2x = 47 + 7

2x = 54

Next, divide 2 from both sides of the equation:

(2x)/2 = (54)/2

x = 54/2 = 27

27 is your answer.

2. is a inscribed angle, meaning that the angle measurement will be half of the arc. Set the equation:

212 = 2(13x - 24)

Isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS.

First, divide 2 from both sides of the equation:

(212)/2 = (2(13x - 24))/2

106 = 13x - 24

Next, add 24 to both sides of the equation:

106 (+24) = 13x - 24 (+24)

106 + 24 = 13x

130 = 13x

Finally, divide 13 from both sides of the equation:

(130)/13 = (13x)/13

x = 130/13

x = 10

10 is your answer.

3. is a central angle. The arc and the angle will, therefore, have the same measurement:

137 = 3x + 5

First, subtract 5 from both sides of the equation:

137 (-5) = 3x + 5 (-5)

137 - 5 = 3x

132 = 3x

Next, divide 3 from both sides of the equation to isolate the variable, x:

(132)/3 = (3x)/3

x = 132/3 = 44

44 is your answer.

4. is a inscribed angle. The arc will be twice the measurement of the angle.

86 = 2(2x + 3)

First, isolate the variable x by dividing 2 from both sides of the equation.

(86)/2 = (2(2x + 3)/2

43 = 2x + 3

Next, subtract 3 from both sides of the equation:

43 (-3)  = 2x + 3 (-3)

40 = 2x

Finally, divide 2 from both sides of the equation:

(40)/2 = (2x)/2

x = 40/2 = 20

20 is your answer.

~

Exercise 1: When tossing 4 identical and distinct coins, the number of macrostates and microstates are given below:MoleculesMacrostatesMicrostates4 coins16 states2^4=16Microstates: The number of ways in which the particles can be distributed among different energy levels is referred to as microstates. Macrostates: The number of ways in which the total energy of the system can be divided into different energy levels is referred to as macrostates. The distribution is represented in the following table: Distribution Microstates (W) Macrostates (Ω)TTTT1111HHHHT4C4,216HHHH3C4,715

Exercise 2:When distributing two distinct particles among three cells, the number of macrostates and microstates are as follows: Molecules Macrostates Microstates 2 particles10 states3^2=9Microstates: The number of ways in which the particles can be distributed among different energy levels is referred to as microstates. Macrostates: The number of ways in which the total energy of the system can be divided into different energy levels is referred to as macrostates. The distribution is represented in the following table: Distribution Microstates (W) Macrostates (Ω)2 in 11C21,23 in 11C31,33 in 11C32,310 in total 9.

Know more about microstates:

https://brainly.com/question/32556718

#SPJ11

Answer:

x=0

Step-by-step explanation:

have a nice day or evening

Answer:

See Explanation

Step-by-step explanation:

The question has missing details, as the scale factor is not given. However, I will give a general explanation on how to calculate the area and perimeter of a dilated shape (triangle).

The following assumptions, apply:

(1) Scale factor of 1/2 from the blue to the yellow triangle.

(2) The dimension of the blue triangle are:

[tex]Base = x[/tex]

[tex]Sides= y\ and\ z[/tex]

[tex]Height= h[/tex]

First, calculate the dimensions of the yellow triangle.

The dimension will be the product of the scale factor and the dimensions of the blue triangle.

So, we have:

[tex]Base = \frac{1}{2} * x = \frac{1}{2}x[/tex]

[tex]Sides = \frac{1}{2}y\ and\ \frac{1}{2}z[/tex]

[tex]Height = \frac{1}{2}h[/tex]

The perimeter of the blue triangle is:

[tex]P_1 =Base + Sides[/tex]

[tex]P_1 = x + y + z[/tex]

The perimeter of the yellow triangle is:

[tex]P_2 =Base + Sides[/tex]

[tex]P_2 = \frac{1}{2}x + \frac{1}{2}y + \frac{1}{2}z[/tex]

Factorize

[tex]P_2 = \frac{1}{2}[x + y + z][/tex]

Recall that: [tex]P_1 = x + y + z[/tex]

So:

[tex]P_2 = \frac{1}{2}*P_1[/tex]

This implies that the perimeter of the yellow triangle is a product of the scale factor and the perimeter of the blue triangle.

The area of the blue triangle is:

[tex]A_1 = \frac{1}{2}* Base * Height[/tex]

[tex]A_1 = \frac{1}{2} * x* h[/tex]

[tex]A_1 = \frac{1}{2} xh[/tex]

The area of the yellow triangle is:

[tex]A_2 = \frac{1}{2} * Base * Height[/tex]

[tex]A_2 = \frac{1}{2}* (\frac{1}{2}x) * (\frac{1}{2}h)[/tex]

Rewrite as:

[tex]A_2 = \frac{1}{2}* \frac{1}{2} [\frac{1}{2}x h][/tex]

[tex]A_2 = (\frac{1}{2})^2 *[\frac{1}{2}x h][/tex]

Recall that:[tex]A_1 = \frac{1}{2} xh[/tex]

So:

[tex]A_2 = (\frac{1}{2})^2 *A_1[/tex]

This implies that the area of the yellow triangle is a product of the square of the scale factor and the area of the blue triangle.

There are no points in C for which the function g(z) is holomorphic.

The functions given are :

f(z) = z² and g(z) = x² - y² + 2xy (where z = x + iy)

We need to determine the points in C for which the functions are holomorphic.

(a) To check whether f(z) = z² is holomorphic or not, we will verify the Cauchy-Riemann equations (CRE) which are:

u x = v y and v x = - u y

Let us assume that f(z) = u(x, y) + iv(x, y)

Substituting in f(z) = z², we have f(z) = (x + iy)²= x² + 2ixy - y²

Now comparing with u(x, y) + iv(x, y), we get :

u(x, y) = x² - y² and v(x, y) = 2xy

Now applying the CRE, we get :

u x = 2xv

y = 2xu

y = - 2yv

x = 2y

We can see that both the CRE are satisfied.

Hence, f(z) = z² is holomorphic for all values of z in C.

(b) Similarly, for g(z) = x² - y² + 2xy (where z = x + iy), we have g(z) = u(x, y) + iv(x, y)

Substituting in g(z) = x² - y² + 2xy, we have g(z) = x² - y² + 2ixy

Now comparing with u(x, y) + iv(x, y), we get :

u(x, y) = x² - y² and v(x, y) = 2xy

Now applying the CRE, we get :

u x = 2xv

y = 2xu

y = - 2yv

x = 2x

Since the CRE are not satisfied, g(z) = x² - y² + 2xy (where z = x + iy) is not holomorphic at any point in C.

Therefore, we can say that there are no points in C for which the function g(z) is holomorphic.

To learn more about functions visit : https://brainly.com/question/11624077

#SPJ11

The probability which is in a random sample of 133 people from State College, more than 50 are not college students is 0.0000313 (rounded to 4 decimal places).

Given that in State College, Pennsylvania, the proportion of college students in the population is 71.1%.

We need to find the probability that in a random sample of 133 people from State College, more than 50 are not college students.

We need to round the final answer to at least 4 decimal places and intermediate z-value calculations to 2 decimal places.

The proportion of college students in State College, Pennsylvania is 71.1%.

Therefore, the proportion of non-college students in State College, Pennsylvania is 100% - 71.1% = 28.9%.

Let X be the number of non-college students in a sample of 133 people from State College, Pennsylvania.

As the sample is random, X follows the binomial distribution with parameters n = 133 and p = 0.289.

The probability of getting more than 50 non-college students can be obtained using the normal distribution approximation to the binomial distribution.

Using the normal distribution approximation, we can convert the binomial distribution to a standard normal distribution using the following formula: Z = (X - np) / sqrt(npq)

Where q = 1 - p is the proportion of college students in the population, and np = 133 x 0.289

= 38.397 and npq = 133 x 0.289 x 0.711 = 9.728.

The probability of getting more than 50 non-college students is : P(X > 50)

= P(Z > (50 - 38.397) / sqrt(9.728))

= P(Z > 3.92)

= 0.0000313 (rounded to 4 decimal places).

For similar question on random sample.

brainly.com/question/29348144

#SPJ4

The total number of ways is: {Total number of ways} =[1120]

Let's say we have 7 candies and 14 stickers that need to be distributed among 4 children in such a way that each child gets at least 1 candy and more stickers than candies. So, let's divide the candies first. Then we can use the formula of stars and bars to distribute the remaining stickers.

Let's assume that the candies have already been divided into 4 groups such that each group contains at least 1 candy.

Then, the total number of ways of dividing 7 candies among 4 children is given by 3 stars and 4 bars (one less than the number of children). For example, the following diagram shows one possible distribution of the candies:
Each star represents one candy, and the bars represent the separations between the groups.

For example, the above diagram represents the distribution of candies as follows:

Child 1: 1 candy

Child 2: 2 candies

Child 3: 1 candy

Child 4: 3 candies

Now, we need to distribute the 14 stickers among the 4 children in such a way that each child gets more stickers than candies. We can do this by using the formula of stars and bars again. This time, we need to distribute 14 stickers among 4 children such that each child gets more than 1 candy. Let's represent the candies by stars and the stickers by bars, then we need to distribute 14 bars among 3 stars and 4 bars.

Here, the first three bars represent the stickers for the first child, the next two bars represent the stickers for the second child, the next bar represents the stickers for the third child, and the remaining eight bars represent the stickers for the fourth child.

So, the total number of ways of distributing 7 candies and 14 stickers among 4 children such that each child gets at least 1 candy and more stickers than candies is given by the product of the number of ways of distributing the candies and the number of ways of distributing the stickers.

Therefore, the total number of ways is: {Total number of ways} =[1120]

learn more about number of ways here:

https://brainly.com/question/30649502

#SPJ11

Answer:

the answer is B

Step-by-step explanation:

because you just put 100 underneath

Answer:

B, 71/100

Step-by-step explanation:

To do this, just put each fraction into a calculator, and whichever fraction results in the decimal you're looking for will be your answer.

Answer:$15 per yard

Step-by-step explanation: $825 divided by 55 = $15