The Parallelogram Law states that ||a+b||2+||a-b||2=2||a||2+2||b||2.
a) Give a geometric interpretation of the ParallelogramLaw.
b) Prove the Parallelogram Law. (Hint: Use theTriangle Inequality)

Answer:

$35.75

Step-by-step explanation:

=18.75+12.50+(2 x 7.25) = $35.75

The value of x that makes A + xB perpendicular to C is -2.5. By setting the dot product of A + xB and C equal to zero, we can solve for x and determine the required value.



To determine the value of x such that A + xB is perpendicular to C, we need to ensure that the dot product of A + xB and C is zero.

Let's calculate the dot product:

(A + xB) · C = (i + 2j + 3k + x(0i + 2j + k)) · (4ij)

Expanding the dot product:

= i · 4ij + 2j · 4ij + 3k · 4ij + x(0i · 4ij + 2j · 4ij + k · 4ij)

= 0 + 8j^2 + 12k^2 + 8xj^2

Since i · j = j · k = i · k = 0, and j · j = 1, k · k = 1, we can simplify:

= 0 + 8(1) + 12(1) + 8x(1)

= 8 + 12 + 8x

= 20 + 8x

To ensure that the dot product is zero, we set it equal to zero:

20 + 8x = 0

Solving for x, we get:

8x = -20

x = -20/8

x = -2.5

Therefore, when x = -2.5, the vector A + xB will be perpendicular to C.

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Histograms of the given data: a) bar width $20, b) bar width $10, c) relative frequency with bar width $10.

a) To create a histogram with a bar width of $20 for the given data, we group the data into intervals of $20 and count the frequency of values within each interval. Here is the histogram:

|$20-$39|******

|$40-$59|************

|$60-$79|*********

|$80-$99|*

Note: The asterisks (*) represent the frequency of values within each interval.

b) To create a histogram with a bar width of $10, we group the data into intervals of $10 and count the frequency within each interval. Here is the histogram:

|$10-$19|*

|$20-$29|***

|$30-$39|*****

|$40-$49|******

|$50-$59|*******

|$60-$69|****

|$70-$79|**

|$80-$89|*

|$90-$99|

c) To create a relative frequency histogram with a bar width of $10, we calculate the proportion of values within each interval by dividing the frequency by the total number of samples (20 in this case). Here is the relative frequency histogram:

|$10-$19|0.05

|$20-$29|0.15

|$30-$39|0.20

|$40-$49|0.20

|$50-$59|0.20

|$60-$69|0.15

|$70-$79|0.10

|$80-$89|0.05

|$90-$99|

Note: The values represent the proportion (relative frequency) of values within each interval.

Remember, the histograms provide visual representations of the distribution of the data, allowing you to observe the concentration of values within different ranges.

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The probability that at least three have eliminated jobs during the last year is 1.2 %

This is a binomial probability problem, where the probability of success is p = 0.13 (the proportion of businesses that have eliminated jobs), and the number of trials is n = 5 (the number of businesses selected at random).

To find the probability that at least three of the businesses have eliminated jobs, we need to find the probability of three, four, or five successes. We can calculate this using the binomial probability formula or a binomial probability table:

P(X ≥ 3) = P(X = 3) + P(X = 4) + P(X = 5)

Using the binomial probability formula, we can find the probability of each individual outcome and then add them up:

P(X = k) = (n choose k) * p^k * (1 - p)^(n-k)

where (n choose k) is the binomial coefficient, which represents the number of ways to choose k successes from n trials.

P(X = 3) = (5 choose 3) * 0.13^3 * 0.87^2 = 0.0115

P(X = 4) = (5 choose 4) * 0.13^4 * 0.87^1 = 0.0004

P(X = 5) = (5 choose 5) * 0.13^5 * 0.87^0 = 0.00001

Therefore, the probability that at least three of the businesses have eliminated jobs during the last year is:

P(X ≥ 3) = 0.0115 + 0.0004 + 0.00001 = 0.0119

So the probability is approximately 0.012 or 1.2%.

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This probability that a randomly selected student from the classroom doesn't really play an item is , which equals 0.5 or 50%.

We have,

Probability is the possibility of something occurring to occur, to clarify. We may talk about the possibility with one result, or the likelihood of several outcomes, when we don't understand how an occurrence will turn out. Biostatistics seems to be the study of things with a probability distribution.

Is the ace a playing card?

The number one is known as the ace and is denoted by the letter A in the majority of Western card games. The ace scores highest, surpassing even the king, in games predicated on the supremacy of one level over the other, such as the majority of trick-taking games.

The total of a number of masculine and female pupils involved in sports  represents the total amount of pupils that don't play an instrument.

The sum of the four numbers in the table,  , corresponds to the total amount of pupils in the class, or 60 .

So, 0.5 or 50% is the likelihood that a randomly selected student from of the class doesn't really play an instrument.

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In Experiment 1, we need to determine if the mean oral condition measurement after 6 weeks (TOTALCW6) is different based on the treatment group (TRT).

To analyze this data, we need to use a statistical test that can compare the means of two or more groups. One possible option is the independent two sample t-test, which can compare the means of two groups. However, since there are multiple treatment groups in this experiment, a better option would be ANOVA (Analysis of Variance). ANOVA can compare the means of three or more groups, making it a suitable choice for our analysis. ANOVA can also test whether the means of different groups are significantly different from each other or not. Thus, we can conclude that ANOVA is the appropriate statistical test to use in Experiment 1.

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The Euclidean norm of the residuals is 0.2916

To find the Euclidean norm of the residuals, we first need to calculate the residuals for each equation in the system. The residual of an equation is the difference between the left-hand side (LHS) and the right-hand side (RHS) of the equation.

Given the system of equations:

x1 - x2 = 2 (Equation 1)

-2x1 + 5x2 = -1 (Equation 2)

Let's calculate the residuals:

Residual 1 = LHS of Equation 1 - RHS of Equation 1

= (x1 - x2) - 2

Residual 2 = LHS of Equation 2 - RHS of Equation 2

= (-2x1 + 5x2) - (-1)

Now, substitute the numerical values x1 = 2.96 and x2 = 1.04 into the residuals:

Residual 1 = (2.96 - 1.04) - 2

= 1.92 - 2

= -0.08

Residual 2 = (-2 * 2.96 + 5 * 1.04) - (-1)

= (-5.92 + 5.20) - (-1)

= -0.72 + 1

= 0.28

The Euclidean norm of the residuals is calculated by taking the square root of the sum of the squares of the residuals:

Euclidean norm = sqrt((Residual 1)^2 + (Residual 2)^2)

= sqrt((-0.08)^2 + (0.28)^2)

= sqrt(0.0064 + 0.0784)

= sqrt(0.0848)

≈ 0.2916

Therefore, the Euclidean norm of the residuals, rounded to four decimal places, is approximately 0.2916.

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The absolute value of the error in the instructor's estimate is 0.644.

To find the absolute value of the error in the instructor's estimate, we need to calculate the difference between the sample mean and the population mean.

Given:

Population mean (μ) = 3.15

Sample mean ([tex]\bar{X}[/tex]) = (3.89 + 4.00 + 3.85 + 3.77 + 3.81 + 3.43 + 3.28 + 3.27 + 3.56 + 3.92) / 10

= 36.78/10

= 3.678

Absolute value of the error = |[tex]\bar{X}[/tex] - μ|

|[tex]\bar{X}[/tex] - μ| = |3.678 - 3.15| = 0.528

Therefore, the absolute value of the error in the instructor's estimate is 0.644.

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The coefficient of the term containing y^8 in the expansion of [(x/2)-4y]^9 is -126.

To find the coefficient of the term containing y^8, we can use the Binomial Theorem. According to the Binomial Theorem, the expansion of (a + b)^n can be written as:

(a + b)^n = C(n,0) * a^n * b^0 + C(n,1) * a^(n-1) * b^1 + C(n,2) * a^(n-2) * b^2 + ... + C(n,k) * a^(n-k) * b^k + ... + C(n,n) * a^0 * b^n

where C(n,k) is the binomial coefficient given by C(n,k) = n! / (k! * (n-k)!).

In our case, a = x/2 and b = -4y. Plugging these values into the formula, we have:

[(x/2)-4y]^9 = C(9,0) * (x/2)^9 * (-4y)^0 + C(9,1) * (x/2)^8 * (-4y)^1 + C(9,2) * (x/2)^7 * (-4y)^2 + ... + C(9,8) * (x/2)^(9-8) * (-4y)^8 + C(9,9) * (x/2)^0 * (-4y)^9

The term containing y^8 is C(9,8) * (x/2)^(9-8) * (-4y)^8 = C(9,8) * (x/2) * (-4y)^8.

The binomial coefficient C(9,8) is equal to 9, and the term (x/2) * (-4y)^8 simplifies to (-4)^8 * (x/2) * y^8 = 65536 * (x/2) * y^8.

Therefore, the coefficient of the term containing y^8 is 65536 * (x/2) = 32768x.

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The x-coordinates of the points (x, y) on the curve where the tangent line is horizontal are x = 7 and x = 3.

To find the x-coordinates of the points where the tangent line to the curve is horizontal, we need to find the values of x that make the derivative of the function equal to zero.

Given the curve equation [tex]y=\frac{(x - 7)^6}{(x - 3)^7}[/tex], let's differentiate it with respect to x: [tex]y=\frac{(x - 7)^6}{(x - 3)^7}[/tex]

Taking the derivative of both sides: [tex]\frac{dy}{dx} = [\frac{(x - 7)^6}{(x - 3)^7}]''[/tex]

To simplify the expression, we can rewrite it as: [tex]\frac{dy}{dx} = (x - 7)^6 (x-3)^{-7}[/tex]

Now, let's set the derivative equal to zero: [tex]0=\frac{dy}{dx} = (x - 7)^6 (x-3)^{-7}[/tex]

Since we're looking for the x-coordinates, we need to solve the equation for x. This equation suggests that either the numerator [tex](x - 7)^6[/tex] should be zero or the denominator [tex](x - 3)^7[/tex] should be zero.

Setting the numerator equal to zero:

[tex](x - 7)^6 = 0[/tex]

Solving this equation yields:

x - 7 = 0

x = 7

Now, setting the denominator equal to zero:[tex](x - 3)^7 = 0[/tex]

Solving this equation yields:

x - 3 = 0

x = 3

Therefore, the x-coordinates of the points (x, y) on the curve where the tangent line is horizontal are x = 7 and x = 3.

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The product of 4.0x10^-2m and 8.1 is 3.24x10^-1m.

To express the product of 4.0x10^-2 m and 8.1, we can perform the multiplication and simplify the result. Let's go through the steps:

Step 1: Multiply the numbers: 4.0x10^-2 m * 8.1

To multiply these numbers, we multiply the decimal parts and add the exponents of 10. The calculation is as follows:

4.0 * 8.1 = 32.4

Next, we add the exponents:

10^-2 * 10^0 = 10^-2+0 = 10^-2

Step 2: Simplify the result

The result of the multiplication is 32.4 times 10 raised to the power of -2. We can write this as:

32.4 * 10^-2

When we have a number expressed in scientific notation, such as 4.0x10^-2 m, it means that we have a coefficient (4.0) multiplied by 10 raised to a certain power (-2 in this case). This notation is commonly used to represent very large or very small numbers in a concise and convenient manner.

In the context of measurements, the coefficient (4.0) represents the numerical value, and the exponent (-2) indicates the order of magnitude or scale. The base of 10 implies that the number is expressed in powers of 10.

Multiplying this value by 8.1 results in a product of 32.4. The exponent remains the same since multiplying by 10 does not change the scale of the number. Therefore, the final result is 32.4 times 10 raised to the power of -2.

Interpreting this in practical terms, the product of 4.0x10^-2 m and 8.1 is equivalent to 32.4 times 10 raised to the power of -2 meters. This can be understood as a small distance or length measurement due to the negative exponent. It signifies that the number is scaled down by a factor of 100 (10 raised to the power of 2), making it 100 times smaller than a meter.

Thus, the expression 32.4 * 10^-2 m represents the product of 4.0x10^-2 m and 8.1, where the value is 32.4 and the unit is meters, adjusted according to the appropriate scale denoted by the exponent.

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To find a vector that is perpendicular to another vector, we can take the cross product of the given vector and any non-zero vector. The resulting vector will be perpendicular to the original vector. In this case, we are given the vector 〈2, -1, 3〉, and we need to find a vector that is perpendicular to it.

To find a vector perpendicular to 〈2, -1, 3〉, we can take the cross product of this vector with any non-zero vector. The cross product of two vectors, say vector A and vector B, is a vector that is perpendicular to both A and B.

Let's choose a non-zero vector, say 〈1, 0, 0〉, and take the cross product with 〈2, -1, 3〉:

〈1, 0, 0〉 × 〈2, -1, 3〉

The result of the cross product will give us a vector that is perpendicular to both 〈2, -1, 3〉 and 〈1, 0, 0〉. We can calculate this cross product to find the desired vector.

The resulting vector will be perpendicular to 〈2, -1, 3〉. It's important to note that there are infinitely many vectors that are perpendicular to a given vector, as long as they are non-zero and not collinear with the original vector.

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According to the question we have Therefore, the solution to the equation 7^x+5= 6^x is x ≈ 27.3.

The given equation is 7^x+5= 6^x. We need to solve this equation for x. Here is the step-by-step explanation:7^x+5= 6^xLet's take ln on both sides: ln(7^x+5) = ln(6^x) .

Using log properties, we get :x ln(7) + 5ln(7) = x ln (6)

Now we can get x on one side by subtracting x ln(6) from both sides and factor x out: x ln(7) - x ln(6) = -5ln(7)x(ln(7) - ln(6)) = -5ln(7)x = (-5ln(7))/(ln(7) - ln(6)) .

We can use a calculator to simplify this: x ≈ 27.3 .

Therefore, the solution to the equation 7^x+5= 6^x is x ≈ 27.3.

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The determinant of 5a is equal to the determinant of a multiplied by 5 raised to the power of n

The determinant of a matrix is a scalar value that represents certain properties of the matrix.

In particular, the determinant of a square matrix is related to its invertibility and the scaling factor of its linear transformation.

For a square matrix A, if we multiply each element of A by a scalar k, the determinant of the resulting matrix kA is equal to the determinant of A raised to the power of the number of rows or columns in A:

[tex]det(kA) = (k^n) * det(A)[/tex]

Where n is the number of rows (or columns) in the matrix A.

In the given case, if a is an n × n matrix, the determinant of the matrix 5a would be:

[tex]det(5a) = (5^n) * det(a)[/tex]

So, the relationship between the determinant of an n x n matrix a and the determinant of 5a is that the determinant of 5a is equal to the determinant of a multiplied by 5 raised to the power of n.

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When PQ intersects SR at x = 36, the value of RTQ is 152.

Let's start by setting the equations of the lines PQ and SR equal to each other:

PQ: 4x + 8

SR: 224 - 2x

Since both lines intersect, we can equate them and solve for x:

4x + 8 = 224 - 2x

To solve this equation, we can combine like terms by adding 2x to both sides and subtracting 8 from both sides:

4x + 2x = 224 - 8

6x = 216

Dividing both sides of the equation by 6, we find:

x = 216 / 6

x = 36

Now that we have the value of x, we can substitute it back into either equation to find the corresponding value of RTQ. Let's use the equation of SR:

SR: 224 - 2x

Substituting x = 36, we have:

SR = 224 - 2(36)

SR = 224 - 72

SR = 152

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According to the information, the statement D best summarizes why Cassius is frustrated with Caesar.

In this passage, Cassius expresses his frustration with Caesar by highlighting Caesar's weaknesses and shortcomings. Cassius finds it unbelievable that someone with Caesar's feeble temper and physical vulnerabilities, such as his trembling during a fever and losing his color and luster, could rise to such power and be idolized by others.

Cassius is exasperated by the fact that someone with evident flaws and weaknesses has managed to achieve such dominance and acclaim. Therefore, Cassius's frustration stems from his disbelief that a man with Caesar's weaknesses can become so influential and hold such authority.

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Cassius' frustration with Caesar is derived from his disbelief that Caesar could achieve such a powerful position despite having significant weaknesses.

According to the given passage, it appears that Cassius experiences frustration with Caesar primarily due to Caesar's rise in power despite what Cassius perceives as unmistakable weaknesses. Cassius is incredulous that Caesar, a man whom he views as deeply flawed, can hold such a position of influence. This sentiment is reflected in his disbelief: 'Cassius cannot believe that a man with all of Caesar’s weaknesses can become so powerful.' Therefore, his frustration derives from his inability to reconcile Caesar's perceived flaws with his substantial power and influence.

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A wave has an amplitude of 2 cm (y-direction) and a frequency of 12 Hz, and the distance (x-direction) from a crest to the nearest trough is measured to be 5 cm. The velocity of the wave is 60 cm/s. The correct option is d. 60cm/s.

The given parameters are:

Amplitude, A = 2 cm

Frequency, f = 12 Hz

Wavelength, λ = distance between two nearest troughs or crests = 5 cm

We need to calculate the velocity of the wave. The formula to calculate the velocity of a wave is:

v = fλ

Where,

v = Velocity of the wave

f = frequency of the wave

λ = wavelength of the wave

Substituting the given values in the above formula, we get:

v = fλ

v = 12 Hz × 5 cm

v = 60 cm/s

Therefore, the velocity of the wave is 60 cm/s, which is option D.

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The sum of the given convergent series is equal to e^6 when x = 6.

The given series can be recognized as a Taylor series evaluated at x = 6, with the terms being the factorial of each successive natural number.

Let's break down the series:

1 + 6 + 622! + 633! + 644! + ⋯ + 6nn! + ⋯

Since the terms involve factorials, it resembles the exponential function series:

e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ⋯ + x^n/n! + ⋯

Comparing the two series, we can see that x = 6 in the given series corresponds to the exponent in the exponential function series.

Therefore, the sum of the given convergent series is equal to e^6 when x = 6.

In mathematical notation, the sum of the series is:

1 + 6 + 622! + 633! + 644! + ⋯ + 6nn! + ⋯ = e^6

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Equation (1) cannot be consistently estimated using Ordinary Least Square method due to Endogeneity.

Endogeneity occurs when there is a correlation between the explanatory variables and the error term in the regression equation.

In Equation (1), Qi represents the quantity demanded and supplied in equilibrium, which is determined by the equilibrium price (P) and income (Y). However, Equation (2) states that the equilibrium price (P) is determined by Qi itself. This creates a problem of endogeneity because there is a feedback loop between the dependent variable (Qi) and the independent variables (P and Y).

Hence, due to Endogeneity OLS cannot be used to consistently estimate equation(1).

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The experimental probability of Winston getting a hit on his next attempt is 1 (or 100%).

The experimental probability of getting a hit on Winston's next attempt can be calculated by dividing the number of successful outcomes (hits) by the total number of attempts (at bats).

In this case, since Winston got a hit on all 9 of his previous at bats, we can say that the probability of getting a hit is 100% or 1.

As a function, we can represent this probability as:

P(hit) = 1

This means that there is a 100% chance of Winston getting a hit on his next attempt, based on the information given.

It's important to note that experimental probability is based on observed outcomes and may not necessarily reflect the true underlying probability. In this case, if Winston has a perfect record of getting hits so far, it doesn't guarantee that he will always get a hit in the future.

Probability is often calculated based on a large number of trials to provide a more accurate estimate of the likelihood of an event occurring.

Additionally, it's essential to consider other factors such as the skill level of the player, the quality of the opposing team, and any changes in circumstances that might affect the outcome.

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The estimate of ∫e^x dx using rectangles with heights given by right-hand endpoints and four subintervals (n = 4) can be determined by evaluating the function at those endpoints and multiplying by the width of each rectangle.

Among the given options, the correct representation of the estimate is:

∫e^x dx is approximately (0.5)e^0.5 + (0.5)e^1 + (0.5)e^1.5 + (0.5)e^2.

This is because we divide the interval [0,2] into four subintervals of equal width, each with a width of 0.5. For the right-hand endpoint approximation, we evaluate the function e^x at those endpoints.

The height of each rectangle is given by e^x evaluated at the right-hand endpoint of each subinterval. The width of each rectangle is 0.5.

By multiplying the height and width of each rectangle and summing them up, we obtain the estimate of the integral.

Therefore, the correct representation is (0.5)e^0.5 + (0.5)e^1 + (0.5)e^1.5 + (0.5)e^2 as an estimate of ∫e^x dx using rectangles with right-hand endpoints and four subintervals.

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Answer: Vertics are 4 and 5

Step-by-step explanation: premirgen

Answer: Vertics are 4 and 5

The average value f_ave of the function f(θ) = 14 sec²(θ/2) on the interval [0, pi/2] is (28/pi).

To find the average value of a function f on a closed interval [a, b], we need to evaluate the definite integral of f(x) over that interval and divide it by the length of the interval (b - a).

In this case, the function is f(θ) = 14 sec²(θ/2) and the interval is [0, pi/2]. To calculate the average value, we integrate f(theta) from 0 to pi/2:

f_ave = (1/(pi/2 - 0)) * ∫[0, pi/2] 14 sec²(θ/2) d(θ).

Using the integral properties, we can simplify this expression:

f_ave = (2/pi) * ∫[0, pi/2] 14 sec²(θ/2) d(θ).

Evaluating the integral, we get:

f_ave = (2/pi) * [14 tan(θ/2)] [from 0 to pi/2]

     = (2/pi) * (14 tan(pi/4) - 14 tan(0))

     = (2/pi) * (14 - 0)

     = 28/pi.

Therefore, the average value of the function f(θ) = 14 sec²(θ/2) on the interval [0, pi/2] is (28/pi).

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An instance variable can be accessed from any instance method in the class.

An instance variable is a variable that is declared within the class but outside of any method and is accessible by all instance methods of the class. It is unique to each instance of the class and can hold different values for each instance. An instance variable is also known as a member variable.
In contrast, a local variable is a variable that is declared within a method and can only be accessed within that method. A static variable, on the other hand, is a variable that is shared by all instances of the class and can be accessed using the class name instead of an instance name.
It is important to understand the differences between these types of variables to effectively design and implement object-oriented programs.

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According to the statement Therefore, 1529 adults attended the game. There were 382 children and 1529 adults in attendance.

Let's use algebra to solve this problem. Let's call the number of children who attended the game "c" and the number of adults who attended the game "a".

The total number of people who attended the game is 1911, so c + a = 1911.

The total amount collected is $26,257, so 12c + 17a = 26257.Now we have two equations and two variables, so we can solve for "c" and "a".

We can start by solving the equation c + a = 1911

for one of the variables. Let's solve for "a": a = 1911 - c .

Now we can substitute this expression for "a" into the other equation:12c + 17a = 2625712c + 17(1911 - c) = 2625712c + 32487 - 17c = 262575c = 1910c = 382 .

Therefore, 382 children attended the game.

We can substitute this value into the equation we found for "a":a = 1911 - ca = 1911 - 382a = 1529 .

Therefore, 1529 adults attended the game. There were 382 children and 1529 adults in attendance.

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After finding x the nearest integer is 12.

Let us take ,

First integer be x .

Second integer be y .

Also let us consider x > y .

According to first Condition :-

⇒ x - y = 1.

⇒ x = y + 1. .............(i)

According to second Condition :-

⇒ x × y = 30 .

⇒ ( y + 1 )y = 30 . [ From (i) ]

⇒ y² + y = 30.

⇒ y² + y - 30 = 0 .

⇒ y² + 6y - 5y -30 = 0.

⇒ y ( y + 6 ) -5 ( y + 6 ) = 0 .

⇒ ( y + 6 ) ( y - 5 ) = 0 .

Here y can sustain both values 5 and minus 6 as y is an integer.

So , x = -6 , 5 .

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The line in slope intercept form is y=x-6.

From the given graph, (0, -6) and (6, 0).

The standard form of the slope intercept form is y=mx+c.

Slope (m) = (0+6)/(6-0)

= 6/6

= 1

Substitute m=1 and (x, y)=(0, -6) in y=mx+c, we get

-6=1(0)+c

c=-6

So, slope intercept form is y=x-6

Therefore, the line in slope intercept form is y=x-6.

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The correct answer is: the Maximum likelihood estimator of θ;The likelihood function is given by;

[tex]L(θ) = θ^n(1+x_1)...(1+x_n)^{-(1+θ)}[/ tex ] The log likelihood function is;[tex]l(θ) = n log(θ) - (1+θ)∑log(1+x_i)[/tex]Differentiating w.r.t θ and equating to 0;[tex]\frac{\partial l(θ)}{\partial θ} = \frac{n}{θ} - ∑log(1+x_i) - n = 0[/tex]Therefore, the Maximum likelihood estimator of θ is;[tex]\hat{θ} = \frac{n}{∑log(1+x_i) + n}[/tex](b)

A complete sufficient statistic for θ is a function of X₁,..., X, that contains all the information that is relevant to the determination of θ;

By factorizing the pdf f(x;θ),

we have;[tex]f(x;θ) = θ(1+x)^{-(1+θ)}[/tex]

Thus, the joint pdf is given by;[tex]f(x_1,...,x_n;θ) = θ^n(∏(1+x_i))^{-(1+θ)}[/tex]

Let Y = ∏(1+x_i)

;Hence, the joint pdf is given by

[tex]f(x_1,...,x_n;θ) = θ^nY^{-(1+θ)}[/tex]

Thus, a complete sufficient statistic for θ is Y.(c)

The main answer is the Cramer-Rao Lower Bound for 1/θ;Let X ~ f(x;θ), where f(x;θ) = {θ (1+x) ^-(1+θ) 0

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The equation of the plane passing through the point Oo and parallel to the y-z plane is x = x₀, where x₀ represents the x-coordinate of the point Oo.

To find the equation of the plane passing through the point Oo and parallel to the y-z plane, we need to determine the coefficients of the equation Ax + By + Cz + D = 0, where (A, B, C) represents the normal vector to the plane.

Since the plane is parallel to the y-z plane, it means that it is perpendicular to the x-axis. Therefore, the x-component of the normal vector is 1, and the y and z-components are both 0.

So, we have a normal vector N = (1, 0, 0).

Now, we need to find the value of D to complete the equation of the plane. Since the plane passes through the point Oo, we can substitute the coordinates of Oo (x₀, y₀, z₀) into the equation to solve for D.

Let's assume the coordinates of Oo are (x₀, y₀, z₀). Then we have:

1(x₀) + 0(y₀) + 0(z₀) + D = 0

x₀ + D = 0

D = -x₀

Therefore, the equation of the plane passing through Oo and parallel to the y-z plane is:

x - x₀ = 0

This can be simplified as:

x = x₀

So, the equation of the plane is x = x₀, where x₀ is the x-coordinate of the point Oo.

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The value of cos a and sin a are 0.5, 0.28 respectively.

From the figure,

We have the following information from the question:

The coordinates of c are (0. 96, 0. 28).

and, To find the value of cos a and sin a

Now, According to the question:

We have the square and inscribed a triangle .

From using the triangle to find the value of cos a and sin a.

Now, We know that:

Cos a = base/ hypotenuse

Sin a = Altitude/ base

Now, put the value in above formula :

Cos a= 0.5/1 = 1/2 = 0.5

Sin a= 0.28/1 = 0.28

Hence, The value of cos a and sin a are 0.5, 0.28 respectively.

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