Draw the image of the given rotation of the preimage

The simplified expression is -22 degrees (rounded to the nearest degree).

What is Trigonometry?

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, and the functions that describe those relationships. It has applications in fields such as engineering, physics, astronomy, and navigation.

We can use the following trigonometric identity to simplify the expression:

arctan(x) + arctan(y) = arctan[(x+y) / (1-xy)]

In this case, we can substitute x = 5 and y = 6 to get:

arctan 5 + arctan 6 = arctan[(5+6) / (1 - 5*6)]

Simplifying the denominator, we get:

arctan 5 + arctan 6 = arctan(11/-29)

To find the degree measure of this angle, we can use a calculator to evaluate the inverse tangent of -11/29 and convert the result to degrees.

The result is approximately -22 degrees (rounded to the nearest degree).

Therefore, the simplified expression is:

arctan 5 + arctan 6 = -22 degrees (rounded to the nearest degree).

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$$x = \frac{1}{21} \begin{bmatrix} 4b_1 + 2b_2 \\ 5b_1 - b_2 \end{bmatrix}.$$

To compute the least squared error solution to \( A x=b \), we will use the formula \(x = (A^T A)^{-1} A^T b\).

In this case, \(A = \begin{bmatrix} 4 & 1 \\ 5 & 1 \\ 6 & 1 \end{bmatrix}\), so \(A^T = \begin{bmatrix} 4 & 5 & 6 \\ 1 & 1 & 1 \end{bmatrix}\). Therefore, we have that

$$(A^T A)^{-1} A^T b = \left(\begin{bmatrix} 4 & 5 & 6 \\ 1 & 1 & 1 \end{bmatrix} \begin{bmatrix} 4 & 1 \\ 5 & 1 \\ 6 & 1 \end{bmatrix}\right)^{-1} \begin{bmatrix} 4 & 5 & 6 \\ 1 & 1 & 1 \end{bmatrix} b$$

From here, we can compute the inverse of the matrix \(A^T A\) to get

$$(A^T A)^{-1} = \frac{1}{21}\begin{bmatrix} -1 & 2 \\ 2 & -1 \end{bmatrix}$$

Substituting this into the equation above and multiplying through, we get

$$x = \frac{1}{21}\begin{bmatrix} -1 & 2 \\ 2 & -1 \end{bmatrix} \begin{bmatrix} 4 & 5 & 6 \\ 1 & 1 & 1 \end{bmatrix} b = \frac{1}{21} \begin{bmatrix} 4b_1 + 2b_2 \\ 5b_1 - b_2 \end{bmatrix}$$

Therefore, the least squared error solution to \( A x=b \) is $$x = \frac{1}{21} \begin{bmatrix} 4b_1 + 2b_2 \\ 5b_1 - b_2 \end{bmatrix}.$$

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Answer:

it is 3/4

Step-by-step explanation:

....................

Fully  parenthesize form of arithmetic operation are -

1. For the expression x + y * z - x, you can fully parenthesize it as follows: ((x + (y * z)) - x)
2. For the expression y - z * x / w + z, you can fully parenthesize it as follows: (((y - ((z * x) / w)) + z)
3. For the expression (z - w + x) / z * w, you can fully parenthesize it as follows: (((z - w) + x) / z) * w
4. For the expression (x * w / z) - (x - y) * y, you can fully parenthesize it as follows: (((x * w) / z) - ((x - y) * y))
5. For the expression z - (w + x ) - y * z - y, you can fully parenthesize it as follows: (((z - (w + x)) - (y * z)) - y)
6. For the expression (x - y + z) * z / z, you can fully parenthesize it as follows: (((x - y) + z) * z) / z

Given arithmetic expressions: 1. x + y * z - x2. y - z * x / w + z3. (z - w + x) /z * w4. (x * w / z) - (x - y) * y5. z - (w + x ) - y * z - y6. (x - y + z) * z / z. To apply fully parenthesize form in solving the arithmetic operation, the given arithmetic expressions are:1. ((x) + ((y) * (z))) - (x)2. ((y) - (((z) * (x)) / (w))) + (z)3. ((((z) - (w)) + (x)) / (z)) * (w)4. ((((x) * (w)) / (z)) - ((x) - (y))) * (y)5. (z) - (((w) + (x)) - ((y) * (z))) - (y)6. (((x) - (y) + (z)) * (z)) / (z)

Therefore, the fully parenthesized form in solving the arithmetic operation for the given expressions is shown as above.

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An estimated 46 students in both surveys combined chose in-line skating as their favorite.

To estimate how many students in both surveys combined chose in-line skating as their favorite, we can use the given percentage and the total number of students surveyed.
First, we need to calculate the number of students who chose in-line skating in the first survey. We can do this by multiplying the percentage by the total number of students surveyed:
23% × 100 = 23
So, 23 students in the first survey chose in-line skating as their favorite.
Next, we need to add this number to the number of students who chose in-line skating in the second survey. Since we don't have information about the second survey, we can assume that the same percentage of students chose in-line skating:
23% × 100 = 23
Finally, we can add the two numbers together to get the total number of students who chose in-line skating in both surveys:
23 + 23 = 46

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The values of x that make the equation equals 0 are -3 and 3

The equation given from the question is represented as

(6x^2 - 54)/(5x^2 - 20) = 0

The format of the above equation is that of a radical equation

For the expression to equal to 0, the numerator must be 0

So, we have

6x^2 - 54 = 0

Evaluate the like terms

6x^2 = 54

Divide both sides by 6

x^2 = 9

take the square root of both sides

x = -3 and x = 3

Hence, the solutions are -3 and 3

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Answer:

C) 100

Step-by-step explanation:

[tex]3.5c - 1.5d > 50[/tex]

[tex]3.5c - 1.5(200) > 50[/tex]

[tex]3.5c - 300 > 50[/tex]

[tex]3.5c > 350[/tex]

[tex]c > 100[/tex]

Answer: As the two cylinders have the same height, the cylinder with the greater radius will have the greater volume. Therefore, the statement "The cylinder with radius has a greater volume than the cylinder with radius /2" is true.

Step-by-step explanation:

Answer: x-14

Step-by-step explanation:

Answer:

a,b,c,d

Step-by-step explanation:

all are rational except efg

a) To solve for the unknowns using Gaussian elimination, follow these steps:
1. Multiply the first equation by -1 and add it to the second equation to get x2 = 7 + x3.
2. Multiply the first equation by -3 and add it to the third equation to get x3 = 3.
3. Substitute x3 = 3 into the first equation to get x1 = 2.

b) To solve for the unknowns using Gauss-Jordan elimination, follow these steps:
1. Multiply the first equation by -1 and add it to the second equation to get x2 = 7 + x3.
2. Multiply the first equation by -3 and add it to the third equation to get x3 = 3.
3. Substitute x3 = 3 into the second equation to get x2 = 4.
4. Substitute x3 = 3 and x2 = 4 into the first equation to get x1 = 2.

c) To solve for the unknowns using LU decomposition, follow these steps:
1. Compute the LU decomposition of the coefficient matrix A.
2. Solve Ly = b using forward substitution to get y = [1, 4, 3]^T.
3. Solve Ux = y using backward substitution to get x = [2, 4, 3]^T.

d) To solve for the unknowns using matrix-inverse based solution, follow these steps:
1. Compute the inverse of the coefficient matrix A.
2. Multiply the inverse of A with b to get x = [2, 4, 3]^T.

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The value of the scale factor is 3. And the value of the variables 'v' and 'w' will be 36 and 96, respectively.

Dilation is the process of increasing the size of an item without affecting its form. Depending on the scale factor, the object's size can be raised or lowered.

The scale factor is given as,

SF = 162 / 54

SF = 3

Then the equation is given as,

v / (72 + v) = 1/3

Simplify the equation, then we have

3v = 72 + v

2v = 72

v = 36

Then the other equation is given as,

48 / (48 + w) = 1/3

144 = 48 + w

w = 96

The value of the scale factor is 3. And the value of the variables 'v' and 'w' will be 36 and 96, respectively.

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It will take 11 days to sell 3872 liters of fuel at the same rate as 1760 liters sold in 5 days.

We can use the following proportion to solve the problem:

1760 liters / 5 days = 3872 liters / x days

Where x is the number of days it will take to sell 3872 liters of fuel.

We can use the unitary method to solve this problem.

Let's start by finding out how much fuel is sold in one day. To do this, we need to divide the total amount of fuel sold (1760 liters) by the number of days it was sold for (5 days):

1760 liters ÷ 5 days = 352 liters per day

Now we can use this rate to find out how many days it will take to sell 3872 liters of fuel:

3872 liters ÷ 352 liters per day = 11 days (rounded to the nearest whole number)

Therefore, it will take 11 days to sell 3872 liters of fuel at the same rate as 1760 liters sold in 5 days.

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The x-values that are excluded from the solution set of the rational equation are those that would make the denominator of any of the fractions equal to zero. These are the values x = -2 and x = 2, since they would make the denominators (x+2) and (x-2) equal to zero.

To solve the equation for x, we can first find a common denominator for all the fractions and then combine them into one fraction:

(2)/(x+2)=(3)/(x-2)-(7)/((x+2)(x-2))

Multiply both sides of the equation by the common denominator (x+2)(x-2) to eliminate the fractions

:2(x+2)(x-2) = 3(x+2)(x-2) - 7

Distribute the terms:

2x^2 - 8 = 3x^2 - 12x + 12 - 7

Combine like terms and rearrange the equation:

x^2 - 12x + 13 = 0

Use the quadratic formula to find the values of x:

x = (-b ± √(b^2 - 4ac))/(2a) = (-(-12) ± √((-12)^2 - 4(1)(13)))/(2(1)) = (12 ± √(144 - 52))/(2) = (12 ± √92)/(2) = (12 ± 2√23)/(2) = 6 ± √23

Therefore, the solutions for x are 6 + √23 and 6 - √23.

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The fraction of a day that the average fire burns is given as follows:

100%.

The fraction of a day that the average fire burns is obtained applying the proportions in the context of the problem.

For the log cabin, the fraction is given as follows:

1/4.

The average fire burns four times this length, hence the length is given as follows:

4 x 1/4 = 4/4 = 1 = 100% of a day.

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The equation of line in the point-slope form is y - 2 = -1(x + 2)

The equation of a straight line is a relationship between x and y coordinates, The point-slope form of the equation of a straight line is,

y-y₁ = m(x-x₁), where m is the slope of the line.

Given that,

A graph having straight line,

and it can be seen in the graph it is passing through (-2, 2) & (-1, 1)

Slope m = (y₂-y₁)/(x₂-x₁)

              = (2 - 1)/ (-2 -(-1))

              =  1/-1

              =   -1

So now taking point (-2, 2) and slope is -1

The point slope form of the equation is:

y - 2 =  -1 (x - (-2))

y - 2 = -1(x + 2)

Hence, the equation is y - 2 = -1(x + 2)

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The solution for the inequality x² + 36 > 12x is {x| x € R and x ≠ 6) . Option D

From the information given, we have the inequality;

x² + 36 > 12x

Now, a quadratic expression;

x² - 12x + 36

Solve the quadratic expression by multiplying the coefficient of x squared by the constant value 36.

Then, find the pair factors of the product that would add up to give the coefficient of x, -12.

Substitute the factors, we have ;

x² - 6x - 6x + 36

Group in pairs, we have;

(x² - 6x) - (6x + 36)

factor the common terms

x(x - 6) - 6( x - 6)

Then, we have;

x - 6> 0

x > 6

And, x -6 > 0

x > 6

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The scale factor from the original pool's dimensions to the new pool's dimensions is approximately 3.682.

The volume of a rectangular prism (such as a swimming pool) is given by the formula V = lwh, where l is the length, w is the width, and h is the height.

Let x be the scale factor from the original pool's dimensions to the new pool's dimensions. Then, the dimensions of the new pool can be expressed as lx, wx, and hx, where l, w, and h are the dimensions of the original pool.

We know that the original pool can hold 60,000 L of water, so we can set up the equation:

60,000 = lwh

Substituting in lx for l, wx for w, and hx for h, we get:

60,000 =[tex](lx)(wx)(hx)[/tex]

We also know that the new pool can hold 3,039,180 L of water, so we can set up another equation:

3,039,180 = [tex](lx)(wx)(hx)[/tex]

The result of dividing the second equation by the first equation is:

3,039,180/60,000 = [tex](lx)(wx)(hx)/(lx)(wx)(hx)[/tex]

Simplifying, we get:

50.653 =[tex]x^3[/tex]

The result of taking the cube root of both sides is:

x ≈ 3.682

Therefore, the scale factor from the original pool's dimensions to the new pool's dimensions is approximately 3.682.

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A) The probability of at least one red ball being obtained is 1 - the probability of no red balls being obtained.

The probability of no red balls being obtained is (7/17)*(6/16)*(5/15)*(4/14)*(3/13) = 0.0096. Therefore, the probability of at least one red ball being obtained is 1 - 0.0096 = 0.9904.

B) The probability of two red balls and three black balls being obtained is (10/17)*(9/16)*(7/15)*(6/14)*(5/13) = 0.0909. The probability of one red ball and four black balls being obtained is (10/17)*(7/16)*(6/15)*(5/14)*(4/13) = 0.0216.
C) The probability of getting k red balls and (5-k) black balls is (10 choose k)*(7 choose (5-k))/(17 choose 5). This can be calculated for each value of k from 0 to 5 and summed to find the total probability.

For example, when k=0, the probability is (10 choose 0)*(7 choose 5)/(17 choose 5) = 0.0096. When k=1, the probability is (10 choose 1)*(7 choose 4)/(17 choose 5) = 0.0909. The total probability is the sum of these individual probabilities.

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$\lambda - s$ is an eigenvalue of W – s$\lambda$ and $\mathbf{v}$ is the corresponding eigenvector.

Linear Algebra: Let W $\in$ $\mathbb{R}^{n \times n}$, s $\in$ $\mathbb{R}$, and $\lambda$ an eigenvalue of W. To prove that $\lambda$ – s is an eigenvalue of W – s$\lambda$, we will use the definition of eigenvalues and eigenvectors:

An eigenvalue $\lambda$ of a square matrix A $\in$ $\mathbb{R}^{n \times n}$ is a scalar such that there exists a nonzero vector $\mathbf{v} \in \mathbb{R}^n$ for which the following equation holds:

A$\mathbf{v}$ = $\lambda \mathbf{v}$

Therefore, we can rearrange the equation to show that W – s$\lambda$ $\mathbf{v}$ = $(\lambda -s) \mathbf{v}$, which implies that $\lambda - s$ is an eigenvalue of W – s$\lambda$ and $\mathbf{v}$ is the corresponding eigenvector.

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Answer:

0.38

Step-by-step explanation:

To determine the sample size needed to estimate the proportion of supporters with a margin of error of 2.3% and a 90% confidence level,

we can use the formula:
n = (Z^2 * p * (1-p)) / E^2
Where:
- n is the sample size
- Z is the Z-score for the desired confidence level (for 90% confidence, Z = 1.645)
- p is the estimated proportion of supporters (if we have no prior knowledge, we can use 0.5 as a conservative estimate)
- E is the desired margin of error (2.3% or 0.023)

Plugging in the values, we get:
n = (1.645^2 * 0.5 * (1-0.5)) / 0.023^2
n = 752.3
Since we cannot have a fraction of a person, we round up to the nearest whole number to get a sample size of 753.

Therefore, a sample size of 753 is needed to estimate the proportion of supporters with a margin of error of 2.3% and a 90% confidence level.

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The firm should accept the project with an Internal Rate of Return (IRR) of 13%, as it is greater than their WACC of 8.5%.

To find the Weighted Average Cost of Capital (WACC) for the firm, we need to calculate the cost of debt, cost of preferred stock, and cost of equity:

Cost of Debt: 8% coupon bond / $1100 price = 7.2727% cost of debt

Cost of Preferred Stock: $5 dividend / $50 price = 10% cost of preferred stock

Cost of Equity: Risk-free rate + (Beta * Market risk premium) = 4% + (1.5 * 8%) = 14% cost of equity

WACC: (2 Mil * 7.2727%) + (4 Mil * 10%) + (6 Mil * 14%) / (2 Mil + 4 Mil + 6 Mil) = 8.5% WACC

The firm should accept the project with an Internal Rate of Return (IRR) of 13%, as it is greater than their WACC of 8.5%.

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Answer:

a. x = 11; mAB = 34°

b. x = 25.5; mAB = 46°

c. x = 17; mAB = 46°

d. x = 11; mAB = 17

Step-by-step explanation:

A non-Abelian group is a group in which the order of the elements matters, meaning that the order that they are written in changes the outcome of the operation.

An automorphism is a transformation that preserves the group structure and is an isomorphism fromGto itself. SinceGis a non-Abelian group, its elements are not all the same, meaning that there is a difference between some elements.

This difference can then be used to generate an automorphism that is not the identity. For example, an automorphism in a groupGcould exchange two elements,g1andg2, and then evaluate the group operation on the exchange.

This automorphism does not leave the group unchanged, and therefore is not the identity. In conclusion, a non-Abelian groupGhas an automorphism that is not the identity because its elements are not all the same and can be exchanged to generate a new automorphism.

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Answer:

Full answer: 0.26385224274406332453825857519789 Rounded to the nearest hundredth: 0.26 Rounded to the nearest tenth: 0.3

Step-by-step explanation:

Answer:

Not a function

Step-by-step explanation:

For this to be a function there can not be 2 of the same x values.

Our points for this are (−2, −5), (−1, −3), (−2, 6), (5, 7)

Your x value is the number on the left side of the point=

When we look at our points we see that -2 is repeating for (-2,-5) and (-2,6)

If we plot these points on a graph and use the vertical line test we see that our line passes thru the 2 points.

So this is not a function