PLease help, will give brainliest

In △HJK, m∠H=52∘, m∠K=73∘, and JK=14 yards.What is HK? Enter your answer as a decimal in the box. Round only your final answer to the nearest tenth.

(a) To determine the convergence or divergence of the sequence given by n = n * e^n * cos(n), we can apply the Limit Test. We'll find the limit as n approaches infinity:
lim (n→∞) [n * e^n * cos(n)]
As n becomes very large, e^n grows faster than any polynomial term (n, in this case), making the product n * e^n very large as well. Since cos(n) oscillates between -1 and 1, the product of these terms also oscillates and does not settle down to a specific value.
Therefore, the limit does not exist, and the sequence is divergent.
(b) To analyze the convergence of the sequence given by a_n = n^3 / 5, we again apply the Limit Test:
lim (n→∞) [n^3 / 5]
As n approaches infinity, the numerator (n^3) grows much faster than the constant denominator (5). This means the ratio becomes larger and larger without settling down to a specific value.
Thus, the limit does not exist, and the sequence is divergent.

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The area of the trapezoid with parallel sides of 2 and 8 and a height of 8 is 30 square units.

[IMAGE 1]

To find the area of a trapezoid, we recall the formula:

Area = (1/2) * (a + b) * h

where a and b are the lengths of the parallel sides,  

h is the height of the trapezoid.

From the graph, the parallel sides have lengths of 2 and 8, and the height is 8. i.e:

a = point(y₁, y₂)

a = point(0, -2) = 2 (that is length covered by side a)

b = point(y₁, y₂)

b = point(-4, 4) = 8

h = point(x₁, x₂)

h = point(-2, -8) = 6

Substituting the values into the formula:

Area = (1/2) * (2 + 8) * 6

    = (1/2) * 10 * 6

    = 5 * 6

    = 30

[IMAGE 2]

Since XW is parallel to YZ, then:

∠XWY = ∠WYZ = 2x

Recall that, the sum of angles in a triangle is equal 180°, then

∠YXW + ∠XWY + ∠XYW = 180°

From the image, we can see that ∠XYW is a right-angle, that means

∠XYW = 90°

Substitute the values into the equation above:

Recall:

∠YXW + ∠XWY + ∠XYW = 180

3x - 5° + 2x + 90 = 180

5x + 85 = 180

5x = 180 - 85

5x = 95

x = 95/5

x = 19

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To find the parametric equations for the tangent line, we need to find the derivative of the given parametric equations and evaluate it at the specified point:

x'(t) = 2t, y'(t) = 1/(t^2 + 15), z'(t) = 1

x'(4) = 8, y'(4) = 1/31, z'(4) = 1

So the direction vector of the tangent line is <8, 1/31, 1>.

To find a point on the tangent line, we can use the given point (4, ln(16), 1) as it lies on the curve.

Therefore, the parametric equations for the tangent line are:

x(t) = 4 + 8t
y(t) = ln(16) + (1/31)t
z(t) = 1 + t

Note that we can also write the parametric equations in vector form as:

r(t) = <4, ln(16), 1> + t<8, 1/31, 1>
To find the parametric equations for the tangent line to the curve at the specified point (4, ln(16), 1), we need to find the derivative of x(t), y(t), and z(t) with respect to the parameter t, and then evaluate these derivatives at the point corresponding to the given parameter value.

Given parametric equations:
x(t) = t^2 + 15
y(t) = ln(t^2 + 15)
z(t) = t

First, find the derivatives:
dx/dt = 2t
dy/dt = (1/(t^2 + 15)) * (2t)
dz/dt = 1

Now, find the value of t at the specified point. Since x = 4 and x(t) = t^2 + 15, we can solve for t:
4 = t^2 + 15
t^2 = -11
Since there's no real value of t that satisfies this equation, it seems there's an error in the given point or equations. Please verify the given information and try again.

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The scale factor used to create the image is given as follows:

k = 1.5.

A dilation can be defined as a transformation that multiplies the distance between every point in an object and a fixed point, called the center of dilation, by a constant factor called the scale factor.

The length of segment AB is given as follows:

AB = -6 - (-14) = 14 - 6 = 8.

The length of segment A'B' is given as follows:

A'B' = -9 - (-21) = 21 - 9 =12.

Hence the scale factor is given as follows:

k = 12/8

k = 1.5.

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Answer:

1.5

Step-by-step explanation:

We can substitute the value of T to get the final answer: [4Frac (pi/2)^2/3 - 5((pi/2)^4/4)]


To solve this problem, we need to use the fundamental theorem of calculus, which states that the definite integral of a function f(x) over an interval [a, b] can be evaluated by finding an antiderivative F(x) of f(x) and then subtracting F(a) from F(b).

In this case, we are given that Fx = Frac X23 is an antiderivative of fx. Therefore, we can write:
∫T 4fx - 5x^3 dx = [4F(x) - 5(x^4/4)]T

To evaluate this expression, we need to substitute T for x in the above expression and then subtract the result of substituting 0 for x. We get:
[4F(T) - 5(T^4/4)] - [4F(0) - 5(0^4/4)]

Since Fx = Frac X23, we have:
F(T) = Frac T23 and F(0) = Frac 023 = 0

Therefore, the expression simplifies to:
[4Frac T23 - 5(T^4/4)]

Finally, we can substitute the value of T to get the final answer:
[4Frac (pi/2)^2/3 - 5((pi/2)^4/4)]

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The middle line of the wave is 1.

The amplitude of the wave is 3.

The period of the wave is  180⁰.

The middle line a wave is the equilibrium or zero line, represents the average value or baseline of the wave.

From the wave graph, midline = 1

The amplitude of the wave is the maximum displacement of the wave;

amplitude = 3

The period of a wave is the tike taken for the wave to make one complete oscillation.

One complete oscillation = ( 225⁰ - 45⁰ )

One complete oscillation = 180⁰

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A real-world scenario that can be represented by the expression -4 1/2(2/5) is when it comes to calculating how much money one owes after applying discounts.

Lets consider that you're purchasing something worth $4.50 from your favorite store that has just announced on offering a big sale with a discount of about 40% (represented by the numeric fraction  2/5).

Let convert -4 1/2 which is a mixed number to an improper fraction:

-9/2

Multiply the improper fraction by the discount:

[tex]\frac{-9}{2} * \frac{2}{5}[/tex]

[tex]= \frac{-9}{10}[/tex]

Convert back to mixed number:

-0.9

Therefore, you'll owe $0.90 after applying the 40% discount to the $4.50 item.

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According to the statement the values of θ that satisfy sinθ = sin(-2/3π) over the interval [0, 2π] are θ = 2π/3 and θ = 5π/3.

To solve this equation, we need to use the periodicity of the sine function. The sine function has a period of 2π, which means that the values of sinθ repeat every 2π radians.
Given sinθ = sin(-2/3π), we can use the identity that sin(-x) = -sin(x) to rewrite the equation as sinθ = -sin(2/3π).
We can now use the unit circle or a calculator to find the values of sin(2/3π), which is equal to √3/2.
So, we have sinθ = -√3/2. To find the values of θ that satisfy this equation over the interval [0, 2π], we need to look at the unit circle or the sine graph and find where the sine function takes on the value of -√3/2.
We can see that the sine function is negative in the second and third quadrants, and it equals -√3/2 at two points in these quadrants: π/3 + 2πn and 2π/3 + 2πn, where n is an integer.
Since we are only interested in the values of θ over the interval [0, 2π], we need to eliminate any values of θ that fall outside of this interval.
The smaller value of θ that satisfies sinθ = -√3/2 is π - π/3 = 2π/3. The larger value of θ is 2π - π/3 = 5π/3. Both of these values fall within the interval [0, 2π].
Therefore, the values of θ that satisfy sinθ = sin(-2/3π) over the interval [0, 2π] are θ = 2π/3 and θ = 5π/3.

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Thus, the solution to the differential equation ′()=5() with the initial condition (0)=⟨4,4,4⟩ is ()=⟨4,4,4⟩.

To solve the differential equation ′()=5(), we first need to recognize that it is a first-order linear homogeneous equation. This means that we can solve it using separation of variables and integration.

Let's start by separating the variables:
′() = 5()
′()/() = 5

Now we can integrate both sides:
ln() = 5 + C

where C is the constant of integration. To find C, we need to use the initial condition (0)=⟨4,4,4⟩:
ln(4) = 5 + C
C = ln(4) - 5

Substituting this back into our equation, we get:
ln() = 5 + ln(4) - 5
ln() = ln(4)

Taking the exponential of both sides, we get:
() = 4

So the solution to the differential equation ′()=5() with the initial condition (0)=⟨4,4,4⟩ is ()=⟨4,4,4⟩.

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The limit of the ratio is infinity, the series diverges for all values of x except x = 0 and the radius of convergence is r = 0.

To find the radius of convergence of the series, we can use the ratio test.

The ratio of the (n+1)th and the nth term of the series is:

|(x(n+1)) / (x(n))| = ((n+1)^3) / (5(n+1))

We take the limit of this ratio as n approaches infinity:

lim |(x(n+1)) / (x(n))| = lim (((n+1)^3) / (5(n+1))) = lim ((n^3 + 3n^2 + 3n + 1) / (5n)) = ∞

Since the limit of the ratio is infinity, the series diverges for all values of x except x = 0. Hence, the radius of convergence is r = 0.

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[tex]t=5.825\sqrt[3]{\cfrac{w}{p}} ~~ \begin{cases} w=3,590\\ t=13.4 \end{cases}\implies 13.4=5.825\sqrt[3]{\cfrac{3590}{p}} \\\\\\ \cfrac{13.4}{5.825}=\sqrt[3]{\cfrac{3590}{p}}\implies \left( \cfrac{13.4}{5.825} \right)^3=\cfrac{3590}{p}\implies \cfrac{13.4^3}{5.825^3}=\cfrac{3590}{p} \\\\\\ 13.4^3p=(3590)5.825^3\implies p=\cfrac{(3590)5.825^3}{13.4^3}\implies p\approx 290~hp[/tex]

well, clearly Natasha rules!!

now 3) is simply asking on getting a couple of "w" and "p" and getting their time or "t".

In your assignment related to 'Radical Equations', you are dealing with equations that contain radicals with variables in the radicand. You solve them by isolating the radical on one side and then squaring both sides of the equation. Finally, you need to check the solution(s) by substituting back into the original equation.

In the given assignment, the topic is Radical Equations, which is an essential area of study in high school mathematics. Radical equations are equations that contain radicals with variables in the radicand. Solving such equations involves isolating the radical on one side of the equation and then squaring both sides.

Solving Radical Equations

Here are general steps to solve radical equations:

   Check your solution(s) by substituting them into the original equation to ensure they work.

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After calculation, it can be seen that the length of SU is a) 4√7.

In this question, we have to find out the length of the side SU of the triangle. We can see that there is a line passing through Angle T making a perpendicular to SU, which divides the triangle into two parts.

From this, it can also be concluded that the perpendicular T divides the side SU into half, so we will just find the length of one part of side SU and multiply it by 2.

We will look into the right triangle. This is a right angled triangle and the length of perpendicular is given as 6 and of hypotenuse is given as 8, so we will apply the Pythagoras theorem to find the side SU.

Base² = Hypotenuse² - Perpendicular²

Base² = 8² - 6²

Base² = 64 - 36

Base² = 28

Base = [tex]\sqrt{28}[/tex]

Base = 2√7

Now, the length of SU = base × 2

= 2√7 × 2

= 4√7

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The open interval of convergence for the function f(x) = cos(x) with Taylor series centered at x = 2π is equal to (-∞, ∞).

To find the Taylor series centered at x = 2π for the function f(x) = cos(x),

Use the Maclaurin series expansion of the cosine function.

The Maclaurin series expansion for cos(x) is,

cos(x) = Σ (-1)ⁿ × (x²ⁿ) / (2n)!

Let us find the first five nonzero terms of the Taylor series expansion,

n = 0

(-1)⁰ × (x²⁰) / (20)!

= 1 / 0!

= 1

n = 1

(-1)¹ × (x²¹) / (21)!

= -x² / 2!

n = 2

(-1)² × (x²²) / (22)!

= x⁴ / 4!

n = 3

(-1)³ × (x²³) / (23)!

= -x⁶ / 6!

n = 4

(-1)⁴ × (x²⁴) / (24)!

= x⁸ / 8!

So, the first five nonzero terms of the Taylor series centered at x = 2π for f(x) = cos(x) are,

f(x) = 1 - (x - 2π)² / 2! + (x - 2π)⁴ / 4! - (x - 2π)⁶ / 6! + (x - 2π)⁸ / 8!

Now let us determine the open interval of convergence for this Taylor series.

The Maclaurin series expansion of cos(x) converges for all values of x.

Therefore, the open interval of convergence for the given Taylor series centered at x = 2π is equal to (-∞, ∞).

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The acute angle between the two lines is approximately 81.87 degrees.

To find the acute angle between two lines, we first need to find the slopes of the two lines.

The given lines are:

9x - y = 7 ----(1)

x + 5y = 25 ----(2)

Solving equation (1) for y, we get:

y = 9x - 7

So the slope of the first line is 9.

Solving equation (2) for y, we get:

y = (25 - x)/5

So the slope of the second line is -1/5.

Now we can find the acute angle θ between the two lines using the formula:

θ = |arctan((m2 - m1)/(1 + m1m2))|

where m1 and m2 are the slopes of the two lines.

Plugging in the values, we get:

θ = |arctan((-1/5 - 9)/(1 + (9)(-1/5)))|

= |arctan((-46/5)/(-8/5))|

= |arctan(23/4)|

Using a calculator, we get:

θ ≈ 81.87 degrees

Therefore, the acute angle between the two lines is approximately 81.87 degrees.

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The correct option is: Player B is the most consistent, with an IQR of 2.5.

To determine the best measure of variability for the data, we need to consider the type of data we are dealing with. Since we are looking at the number of runs earned by each player, which is numerical data, the best measure of variability would be either the interquartile range (IQR) or the range.

To calculate the IQR for each player, we need to first find the median (middle number) of the data. Then we find the median of the lower half (Q1) and the median of the upper half (Q3) of the data. The IQR is the difference between Q3 and Q1.

For Player A:

Median = 3

Q1 = median of {1, 2, 2, 2, 3} = 2

Q3 = median of {3, 3, 4, 8} = 3.5

IQR = Q3 - Q1 = 3.5 - 2 = 1.5

For Player B:

Median = 2

Q1 = median of {1, 1, 2, 2} = 1.5

Q3 = median of {2, 4, 6} = 4

IQR = Q3 - Q1 = 4 - 1.5 = 2.5

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The length and width of the rectangle are 20 cm and 5 cm respectively.

Let's assume the length of the rectangle is x cm.

According to the given information, the width of the rectangle is 55 cm less than three times its length. So, the width can be expressed as:

Width = 3x - 55

Area = Length x Width.

Thus: Area = x * (3x - 55) = 100

[tex]3x^2 - 55x - 100 = 0[/tex]

Using the quadratic formula

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 3, b = -55, and c = -100.

x = (-(-55) ± √((-55)^2 - 4 * 3 * -100)) / (2 * 3)

= (55 ± √(3025 + 1200)) / 6

= (55 ± √4225) / 6

= (55 ± 65) / 6

x = (55 + 65) / 6 = 120 / 6 = 20 OR

x = (55 - 65) / 6 = -10 / 6 = -5/3

Therefore, the length of the rectangle is 20 cm.

Width = 3x - 55

= 3 x 20 - 55

= 60 - 55

= 5

Therefore, the dimensions of the rectangle are length = 20 cm and width = 5 cm.

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As the level of significance increases, the probability of making a type II error decreases.

What is probability?

Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, with 0 representing an impossible event and 1 representing a certain event. The probability of an event is calculated by dividing the number of ways the event can occur by the total number of possible outcomes.

If the level of significance of a hypothesis test is raised from 0.05 to 0.1, the probability of a type II error will decrease.

Type II error occurs when we fail to reject a null hypothesis that is actually false. It is the probability of accepting a false null hypothesis. By increasing the level of significance, we are making it easier to reject the null hypothesis, which in turn decreases the probability of accepting a false null hypothesis.

Hence, as the level of significance increases, the probability of making a type II error decreases.

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The required equation of the line perpendicular to line y= -1/2x-5 that passes through the given point (2,7) is y = 2x + 3.

The given line has a slope of -1/2 when we compare it standard equation of line y =mx+c.

Since we want a line that is perpendicular to this line, we need to find the negative reciprocal of the slope of the given line y= -1/2x-5.
The negative reciprocal of -1/2 is 2.
So, the slope of the line we want is 2.

Using the point-slope form of a line, we can write the equation of the line as:

y - y₁ = m(x - x₁)

where m is the slope and (x₁, y₁) is the given point of the line.

substitute the values, we get:

y - 7 = 2(x - 2)

y = 2x + 3

Therefore, the equation of the line perpendicular to y= -1/2x-5 that passes through the point (2,7) is y = 2x + 3.

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The characteristic polynomial of a 2×2 matrix a is λ^2 - tr(a)λ + det(a), where tr(a) is the trace and det(a) is the determinant of a.

To prove the given statement, let's consider a 2×2 matrix a with entries a11, a12, a21, and a22. The characteristic polynomial is defined as det(a - λI), where I is the identity matrix.
Expanding the determinant, we have:

det(a - λI) = (a11 - λ)(a22 - λ) - a21a12
= λ^2 - (a11 + a22)λ + a11a22 - a21a12

Comparing this with λ^2 - tr(a)λ + det(a), we observe that the term (a11 + a22) is the trace of a, tr(a), and the term a11a22 - a21a12 is the determinant of a, det(a). Thus, the characteristic polynomial is given by λ^2 - tr(a)λ + det(a).

In summary, the characteristic polynomial of a 2×2 matrix a is λ^2 - tr(a)λ + det(a), where tr(a) is the trace and det(a) is the determinant of a.


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Gross National Income (GNI) is the total income earned by a country's residents, including income earned abroad.

It is a measure of a country's economic performance and is used to compare the wealth of different countries. GNI is calculated by adding up all the income earned by residents, including wages, profits, and investment income, and adding in any income earned by residents from abroad, while subtracting any income earned by foreigners in the country.

To calculate GNI, a country's statistical agency collects data on the income earned by its residents and income earned abroad. For example, if a country's residents earn a total of $1 billion in wages, $500 million in profits, and $200 million in investment income, while earning an additional $300 million from abroad, the country's GNI would be $2 billion ($1 billion + $500 million + $200 million + $300 million).

GNI is an important measure of a country's economic performance, as it reflects the overall wealth of a country and its residents. It is often used in conjunction with other economic indicators, such as Gross Domestic Product (GDP), to evaluate a country's economic development and standard of living. However, it is important to note that GNI may not reflect the distribution of income within a country, as it measures total income rather than individual incomes.

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The correlation coefficient between two variables measures the strength and direction of the linear relationship between them. In this case, we are given that the correlation coefficient between 2X + 1 and 3Y + 4 is to be determined.

To solve this problem, we can use the following formula for the correlation coefficient:

rho = Cov(X,Y) / (SD(X) * SD(Y))

where Cov(X,Y) is the covariance between X and Y, and SD(X) and SD(Y) are the standard deviations of X and Y, respectively.

Now, let's apply this formula to 2X + 1 and 3Y + 4.

Cov(2X+1, 3Y+4) = Cov(2X, 3Y) = 6Cov(X,Y)

because the constants 1 and 4 do not affect the covariance.

SD(2X+1) = 2SD(X), and SD(3Y+4) = 3SD(Y), so

SD(2X+1) * SD(3Y+4) = 6SD(X) * SD(Y)

Putting these results together, we get:

rho = Cov(2X+1, 3Y+4) / (SD(2X+1) * SD(3Y+4))
= (6Cov(X,Y)) / (2SD(X) * 3SD(Y))
= (2Cov(X,Y)) / (SD(X) * SD(Y))

Thus, we see that the correlation coefficient between 2X+1 and 3Y+4 is two times the correlation coefficient between X and Y.

Therefore, the correct answer is (c) 6 Corr(X+1, Y+4).

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After executing the pseudocode, the values of x, y, and z will be: x = 9, y = 0, and z = 6.

The first line of the pseudocode sets z equal to the current value of x, which is 6. So z now has the value 6.

The second line of the pseudocode sets x equal to the current value of y, which is 9. So x now has the value 9.

The third line of the pseudocode sets y equal to the current value of z, which is 6. So y now has the value 6.

Therefore, after executing the pseudocode, the values of x, y, and z are: x = 9, y = 6, and z = 6. However, we can simplify this further by noticing that the third line of the pseudocode sets y equal to the value of z, which is now equal to x. So we can rewrite the values as: x = 9, y = 6, and z = x. And since x is now equal to 9, the final values are: x = 9, y = 6, and z = 9.

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If function "z = x² - xy + 4y²" and (x, y) changes from interval (1, -1) to (1.03, -0.95), then the value of dz is 11.46, and Δz is 0.46.

The "multi-variable" function z = f(x,y) is given to be : x² - xy + 4y²;

Differentiating the function "z" with respect to "x",

We get,

dz/dx = 2x - y + 0

dz/dx = 2x - y,     ...equation(1)

Differentiating the function "z" with respect to "y",

We get,

dz/dy = 0 - x.1 + 8y,

dz/dy = 8y - x,      ...equation(2)

So, the "total-derivative" of "z" can be written as :

dz = (2x - y)dx + (8y - x)dy,

Given that "z" changes from (1, -1) to (1.03, -0.95);

So, we substitute, (x,y) as (1, -1), and (dx,dy) = (1.03, -0.95),

We get,

dz = (2(-1)-1)(1.03 + 1) + (8(-9) -1)(-0.95 -1),

dz = (-3)(2.03) + (-9)(-1.95),

dz = -6.09 + 17.55,

dz = 11.46.

Now, we compute Δz,

The z-value corresponding to (1,-1),

z₁ = (1)² - (1)(-1) + 4(-1)² = -2, and

The z-value corresponding to (1.03, -0.95),

z₂ = (1.03)² - (1.03)(-0.95) + 4(-0.95)² = -1.57.

So, Δz = z₂ - z₁ = -1.57 -(-2) = 0.46.

Therefore, the value of dz is 11.46, and Δz is 0.46.

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The given question is incomplete, the complete question is

If the function z = x² - xy + 4y² and (x, y) changes from (1, -1) to (1.03, -0.95), Compare the values of dz and Δz.

Answer: 11.9

Step-by-step explanation:you have to corss multiply

We have given that, The sum of interior angles formed by the sides of a of pentagon is 540°.

★ According To The Question:-

[tex] \sf \longrightarrow \: Sum \: of \: all \: angles = 540 \\ [/tex]

[tex] \sf \longrightarrow \: \angle A + \angle B + \angle C + \angle D +\angle E \: = 540 \\ [/tex]

[tex] \sf \longrightarrow \: (130) + (x - 5) + (x + 30) +75 +(x - 35) \: = 540 \\ [/tex]

[tex] \sf \longrightarrow \: 130 + x - 5 + x + 30 +75 +x - 35 \: = 540 \\ [/tex]

[tex] \sf \longrightarrow \: 130 - 5+ 30+75 - 35+x + x +x \: = 540 \\ [/tex]

[tex] \sf \longrightarrow \: 130 - 5+ 30+75 - 35+3x \: = 540 \\ [/tex]

[tex] \sf \longrightarrow \: 125+ 30+75 - 35+3x \: = 540 \\ [/tex]

[tex] \sf \longrightarrow \: 155+75 - 35+3x \: = 540 \\ [/tex]

[tex] \sf \longrightarrow \: 230 - 35+3x \: = 540 \\ [/tex]

[tex] \sf \longrightarrow \: 195+3x \: = 540 \\ [/tex]

[tex] \sf \longrightarrow \: 3x \: = 540 - 195\\ [/tex]

[tex] \sf \longrightarrow \: 3x \: = 345\\ [/tex]

[tex] \sf \longrightarrow \: x \: = \frac{ 345}{3}\\ [/tex]

[tex] \sf \longrightarrow \: x \: = 115 \degree\\ [/tex]

________________________________________

★ Angle B :-

→ x - 5 °

→ 115 - 5

→ 115 - 5

→ 110°

Therefore Measure of angle B is 110°

Therefore, the eigenvalue of matrix A is λ = 1.64615.

To find the eigenvalues of matrix A = [1 -6; 12 -7], we need to solve the characteristic equation det(A - λI) = 0, where λ is the eigenvalue and I is the identity matrix.

Let's calculate the determinant of A - λI:

A - λI = [1 - 6; 12 - 7] - λ[1 0; 0 1]

= [1 - λ -6; 12 - λ -7]

Now, calculate the determinant:

det(A - λI) = (1 - λ)(-7 - (-6*12)) - (-6)(-7)

= (1 - λ)(-7 + 72) + 42

= (1 - λ)(65) + 42

= 65 - 65λ + 42

= 107 - 65λ

Setting the determinant equal to zero and solving for λ:

107 - 65λ = 0

-65λ = -107

λ = -107 / -65

λ = 1.64615

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The value of ordered pair which represent the 20 pounds of beans is,

⇒  (20, 16).

Since, The question is for which ordered pair represents the cost of 20 pounds of beans.

since our x-axis represents pounds of beans.

When we find 20, we can trace up to see which point corresponds with an x-value of 20.

It is like a y-value of 16 is the answer

Hence, this represents the cost of 20 pounds of beans.

So, The value of ordered pair which represent the 20 pounds of beans is,

⇒  (20, 16).

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a) The region enclosed by the circle is x^2 + y^2 = 4 in the first  quadrant.

In polar coordinates, the equation of the circle becomes r^2 = 4, and the region is bounded by 0 ≤ r ≤ 2 and 0 ≤ θ ≤ π/2. Therefore, the integral of an arbitrary function f(x,y) over this region is:

∫∫ f(x,y) dA = ∫₀^(π/2) ∫₀² f(r cos θ, r sin θ) r dr dθ

b) The region bounded by the curves y = x^2 and y = 2x - x^2.

In rectangular coordinates, the region is bounded by x^2 ≤ y ≤ 2x - x^2 and 0 ≤ x ≤ 2. Therefore, the integral of an arbitrary function f(x,y) over this region is:

∫∫ f(x,y) dA = ∫₀² ∫x²^(2x - x²) f(x, y) dy dx

Alternatively, we can use polar coordinates to express the  region as the region enclosed by the curves r sin θ = (r cos θ)^2 and r sin θ = 2r cos θ - (r cos θ)^2 in the first quadrant. Solving for r in terms of θ, we get:

r = sin θ / cos^2 θ and r = 2 cos θ - sin θ / cos^2 θ

Therefore, the integral of an arbitrary function f(x,y) over this region is:

∫∫ f(x,y) dA = ∫₀^(π/4) ∫sin θ / cos^2 θ^(2 cos θ - sin θ / cos^2 θ) f(r cos θ, r sin θ) r dr dθ

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The properties used in the given steps include the multiplication property of equality, addition property of equality, subtraction property of equality, and division property of equality.

The steps taken to solve the system of equations and the properties used are as follows:

1.Step 1: 2a + 7b = 0; 3a − 5b = 31

No specific property is used.

Step 2: 10a + 35b = 0; 21a − 35b = 217

Multiplication property of equality: Both sides of the equations are multiplied by 5 and 7 to eliminate coefficients and simplify the expressions.

Step 3: 31a = 217

Addition property of equality

Step 4: a = 7

Division property of equality as both sides are divided by 31 to solve for a.

2. Step 1: 2(7) + 7b = 0

Simplify: the expression is simplified by multiplying 2 and 7 to obtain 14.

Step 2: 14 + 7b = 0

Simplify

Step 3: 7b = −14

Simplify: the equation is simplified by subtracting 14 from both sides.

Step 4: b = −2

Division property of equality: both sides of the equation are divided by the coefficient of 'b' (7) to solve for 'b'.

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The value of -2.5 is the maximum associative strength in the given Rescorla-Wagner equation.

In the Rescorla-Wagner model, ∆vi represents the change in associative strength of a particular conditioned stimulus (CS) after a single trial of conditioning. The formula for computing ∆vi involves the learning rate (α) and the prediction error (δ). In the given equation, the prediction error is 10.00 - 0.00 = 10.00. The learning rate is 0.25. When we multiply these two values, we get 2.50. Since the prediction error is negative, the change in associative strength will also be negative. Therefore, the maximum associative strength will be the negative of 2.50, which is -2.5. This means that the CS is maximally associated with the unconditioned stimulus (US) after the conditioning trial.

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