Let α ∈ R and consider the differential equation dy dy dc ay , subject to the initial condition y(0) = 1.
(a) Show that y = ex ear is the solution of the Cauchy problem.
(b) Find a solution expressed as a Maclaurin series for the equation. Right away,
Using the Existence and Uniqueness Theorem, conclude that = BA n=0 -xn n!

The exact differential equation of (yety +7x) dx + (xey - 4)dy = 0 should be solved in order to get the answer in the form F(x, y) = C where F(x, y) has no constant term.

F(x, y) = =0=c с =.Explanation:An exact differential equation of the form M(x, y) dx + N(x, y) dy = 0 is exact when its partial derivatives are such that ∂M/∂y = ∂N/∂x is satisfied.Therefore, the equation (yety +7x) dx + (xey - 4)dy = 0 is an exact differential equation as the partial derivatives of the functions are:Mx = 7 and Ny = xe^y, and thus Mx = Ny.The next step is to find the function F(x, y), which satisfies the condition ∂F/∂x = M and ∂F/∂y = N.

The integral of M with respect to x is:F(x, y) = ∫Mdx + C1F(x, y) = 7x + C1And the integral of N with respect to y is:F(x, y) = ∫Ndy + C2F(x, y) = xey - 4y + C2To solve for C2, equate the values of F(x, y) from both equations7x + C1 = xey - 4y + C2Thus, the final answer of the exact differential equation (yety +7x) dx + (xey - 4)dy = 0 in the form F(x, y) = C where F(x, y) has no constant term. F(x, y) = =0=c с = isF(x, y) = yety + 7x - xey + 4y = 0.

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The solution to the Bernoulli equation y' - ⅟ₓ y = 4 / (xy)² involves transforming it into a linear equation through a suitable substitution. By substituting u = y^(1-1/x), we obtain a linear equation in terms of u. Solving this linear equation and reverting the substitution yields the solution for y.

To solve the Bernoulli equation y' - ⅟ₓ y = 4 / (xy)², we can use a substitution to transform it into a linear equation. Let's substitute u = y^(1-1/x). Taking the derivative of u with respect to x using the chain rule, we have du/dx = (1-1/x)y^(-1/x) * y'. Rearranging this equation, we get y' = x(1-1/x)u^(x/(x-1)) * du/dx.

Substituting these expressions for y' and y into the original Bernoulli equation, we have x(1-1/x)u^(x/(x-1)) * du/dx - ⅟ₓ u = 4 / (xy)². Simplifying further, we have (1-1/x)u^(x/(x-1)) * du/dx - ⅟ₓ u = 4 / x³y².

Now, let's multiply the entire equation by x³ to eliminate the denominators. This gives us (1-1/x)(x³u^(x/(x-1))) * du/dx - u = 4 / y².

We can now see that the equation is linear in terms of u. By solving this linear equation, we obtain the value of u. Finally, reverting the substitution u = y^(1-1/x), we can find the solution for y.

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The equation of the tangent line at (-2, 1) is (b) y = -8x - 15

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

y = (x² - 3)²

Expand

y = (x² - 3)(x² - 3)

Evaluate the products

So, we have

y = x⁴ - 3x² - 3x² + 9

Evaluate

y = x⁴ - 6x² + 9

Calculate the slope of the line by differentiating the function

So, we have

dy/dx = 4x³ - 12x

The point of contact is given as

(x, y) = (-2, 1)

This means that x = -2

So, we have

dy/dx = 4(-2)³ - 12(-2) = -8

The equation of the tangent line can then be calculated using

y = dy/dx * x + c

So, we have

y =  -8x + c

Using the points, we have

-8 * -2 + c = 1

Evaluate

16 + c = 1

So, we have

c = 1 - 16

Evaluate

c = -15

So, the equation becomes

y = -8x - 15

Hence, the equation of the tangent line is y = -8x - 15

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Therefore, the correct inequality representing Jed's situation is: d. 1.29t ≤ m + 17.26.

Let's analyze the given options:

10x + 1 < 9x:

Subtracting 9x from both sides gives x + 1 < 0, which simplifies to x < -1. This inequality represents the condition where x is less than -1.

10x < 19x - 2:

Subtracting 10x from both sides gives 0 < 9x - 2. Adding 2 to both sides gives 2 < 9x, which simplifies to 2/9 < x. This inequality represents the condition where x is greater than 2/9.

9x + 1 > 10x:

Subtracting 10x from both sides gives -x + 1 > 0, which simplifies to x < 1. This inequality represents the condition where x is less than 1.

Now, let's analyze the inequality representing Jed's situation:

Lett be his monthly spending limit and m represent the total number of songs downloaded.

The given information states that Jed can download 25 songs per month for $14.99, and additional songs cost $1.29 each. The total cost t can be represented as:

t = 14.99 + 1.29m

Since Jed's monthly spending limit is denoted by Lett, we have the inequality:

1.29m ≤ Lett - 14.99

Comparing the options provided:

a. 1.29m t + 10.01: This option does not represent the correct relationship between 1.29m and t.

b. 1.29m ≤ t + 17.26: This option does not correctly reflect the cost of $14.99 for the initial 25 songs. It overestimates the cost by adding 17.26 instead of subtracting it.

c. 1.29t ≤ m + 10.01: This option incorrectly swaps the variables t and m, and it also does not represent the correct relationship between the cost and the number of songs.

d. 1.29t ≤ m + 17.26: This option correctly represents the relationship between the cost and the number of songs, with the appropriate values subtracted.

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The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical concept that provides a rough approximation of the spread of data in a normal distribution.

The following statements are true about the Empirical Rule:

It applies to all curves, bell-shaped curves and not bell-shaped curves: The Empirical Rule can be applied to any distribution, regardless of its shape. However, it provides a more accurate approximation for distributions that closely resemble a bell-shaped curve.

Approximately 68% of the population is within one standard deviation of the mean: According to the Empirical Rule, in a normal distribution, about 68% of the data falls within one standard deviation of the mean. This means that the majority of the observations are clustered around the average value.

Approximately 95% of the population is within two standard deviations of the mean: The Empirical Rule states that approximately 95% of the data falls within two standard deviations of the mean in a normal distribution. This suggests that the data is relatively concentrated within this range.

Approximately 99.7% of the population is within three standard deviations of the mean: The Empirical Rule states that nearly all (about 99.7%) of the data falls within three standard deviations of the mean in a normal distribution. This implies that the data is highly concentrated within this interval.

It can be used when working with normal distributions: The Empirical Rule is most commonly applied to normal distributions, as it provides a useful approximation of the data spread. However, it can also be applied to other distributions, although the accuracy may vary.

We are allowed to use it when working with standard normal distributions: The Empirical Rule can be used when working with standard normal distributions, where the mean is 0 and the standard deviation is 1. In this case, the percentages within the standard deviation intervals remain the same.

In summary, the Empirical Rule is a statistical guideline that provides an estimate of how data is distributed in a dataset, particularly in a normal distribution. It is applicable to various distributions, but its accuracy is highest for distributions that closely resemble a bell-shaped curve.

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The proportions of the adults who took the 1944 and recent surveys, which were total abstainers, are 0.380 and 0.33, respectively.

(a) Sample proportion for the 1944 survey is calculated as follows: From the 1100 adults surveyed, 418 indicated that they were total abstainers. Therefore, the sample proportion for the 1944 survey is calculated as follows:

p = 418/1100

p = 0.380
(b) Hypotheses:H0: The proportion of adults who abstain from alcohol is equal to 0.380.H1: The proportion of adults who abstain from alcohol is not equal to 0.380. Level of significance = α = 0.05. The test statistic: Z = (p - P) / sqrt [(PQ) / n]

Where: P = Proportion of adults who abstain from alcohol in the 1944 survey = 0.380, Q = 1 - P = 1 - 0.380 = 0.620

p = Proportion of adults who abstain from alcohol in the recent survey = 0.330 n = Total number of adults surveyed = 1100Substituting the values into the equation:

Z = (0.330 - 0.380) / sqrt [(0.380 x 0.620) / 1100]

Z = -2.413

Suppose the calculated Z-value is less than -1.96 or greater than +1.96. In that case, we reject the null hypothesis H0 at α = 0.05 level of significance and conclude that there is a significant difference in the proportion of adults who abstain from alcohol between the two surveys.

At α = 0.05 level of significance, the critical value is ±1.96. Since the calculated Z-value (-2.413) is less than -1.96, we reject the null hypothesis H0 at α = 0.05 significance level. Therefore, there is sufficient evidence to conclude that the proportion of adults who abstain from alcohol has changed between the two surveys.

The sample proportion for the 1944 survey is calculated as follows:

p = 418/1100

p = 0.380

The sample proportion for the recent survey is calculated as follows:

p = 363/1100

p = 0.330.

Therefore, the proportions of adults who took the 1944 and recent surveys, total abstainers, are 0.380 and 0.330, respectively. (Round to three decimal places as needed.

At α = 0.05 level of significance, the critical value is ±1.96. Since the calculated Z-value (-2.413) is less than -1.96, we reject the null hypothesis H0 at α = 0.05 significance level. Therefore, there is sufficient evidence to conclude that the proportion of adults who abstain from alcohol has changed between the two surveys.

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(a)The sample proportion for the 1944 survey is approximately 0.380, and for the recent survey, it is approximately 0.330.(b) The proportion of adults who totally abstain from alcohol has changed at the 0.05 level of significance. Therefore, based on the given data and the hypothesis test, there is evidence to suggest that the proportion of adults who totally abstain from alcohol has changed.

(a) To determine the sample proportion for each sample, we divide the number of total abstainers by the total number of adults surveyed.

For the 1944 survey:

Sample proportion = Number of total abstainers / Total number of adults surveyed

Sample proportion = 418 / 1100

Sample proportion ≈ 0.380 (rounded to three decimal places)

For the recent survey:

Sample proportion = Number of total abstainers / Total number of adults surveyed

Sample proportion = 363 / 1100

Sample proportion ≈ 0.330 (rounded to three decimal places)

The sample proportion for the 1944 survey is approximately 0.380, and for the recent survey, it is approximately 0.330.

(b) To determine if the proportion of adults who totally abstain from alcohol has changed, we can perform a hypothesis test. We can use the chi-square test for proportions to compare the two sample proportions.

The null hypothesis (H_(0)) is that there is no difference in the proportion of adults who totally abstain from alcohol between the two surveys.

The alternative hypothesis (H_(a)) is that there is a difference in the proportion of adults who totally abstain from alcohol between the two surveys.

Using the chi-square test for proportions, we can calculate the test statistic and compare it to the critical value at a significance level of 0.05.

If the test statistic is greater than the critical value, we reject the null hypothesis and conclude that the proportion has changed. Otherwise, if the test statistic is less than or equal to the critical value, we fail to reject the null hypothesis and conclude that the proportion has not changed.

Since we do not have information about the observed frequencies in each category, we cannot calculate the test statistic directly. However, we can compare the sample proportions using a normal approximation.

The test statistic can be calculated as follows:

z = (p_(1) - p_(2)) / (\sqrt((p × (1 - p)) × ((1 / n_(1)) + (1 / n_(2)))))

Where:

p_(1) = Sample proportion for the 1944 survey

p_(2) = Sample proportion for the recent survey

p = Pooled proportion ([(p_(1) × n_(1)) + (p_(2) × n_(2))] / [n_(1) + n_(2)])

n_(1) = Sample size for the 1944 survey

n_(2) = Sample size for the recent survey

Using the provided values:

p_(1) = 0.380

p_(2) = 0.330

n_(1) = 1100

n_(2) = 1100

Let's calculate the test statistic:

p = [(p_(1) × n_(1)) + (p_(2) × n_(2))] / [n_(1) + n_(2)]

= [(0.380 × 1100) + (0.330 × 1100)] / (1100 + 1100)

= (418 + 363) / 2200

≈ 0.377 (rounded to three decimal places)

z = (p_(1) - p_(2)) / (\sqrt((p × (1 - p)) × ((1 / n_(1)) + (1 / n_(2)))))

= (0.380 - 0.330) / (\sqrt((0.377 × (1 - 0.377)) × ((1 / 1100) + (1 / 1100))))

≈ 2.639 (rounded to three decimal places)

Using a significance level of 0.05, we can compare the test statistic to the critical value from the standard normal distribution. The critical value for a two-tailed test with a significance level of 0.05 is approximately ±1.96. Since the test statistic (2.639) is greater than the critical value ( (1.96), we reject the null hypothesis. We conclude that the proportion of adults who totally abstain from alcohol has changed at the 0.05 level of significance.

Therefore, based on the given data and the hypothesis test, there is evidence to suggest that the proportion of adults who totally abstain from alcohol has changed.

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The flux of the vector field F(x, y, z) = (3xy, 4(y² + e²²²), (z + sin(xy))) over the surface S of the solid E, bounded by the parabolic cylinder z = 4-x², and the planes z = 0, y = 0, y + x = 2, is calculated as follows.

Firstly, we need to find the outward unit normal vector to the surface S, denoted by n. Then, we evaluate the dot product of F and n over the surface S. Finally, we integrate this dot product over the surface S to obtain the flux of the vector field.

To calculate the outward unit normal vector n, we consider the surfaces that bound the solid E. These surfaces are given by z = 4-x², z = 0, y = 0, and y + x = 2. By taking the gradient of the surfaces and normalizing the resulting vectors, we determine the outward unit normal vector for each surface.

Next, we evaluate the dot product of the vector field F and the outward unit normal vector n at each point on the surface S. This gives us the flux density at each point. Then, we integrate the flux density over the surface S using a suitable parameterization of the surface.

The final result is the total flux of the vector field F over the surface S, which represents the amount of flow through the surface. The specific numerical value of the flux depends on the exact parameterization of the surface and the limits of integration used in the calculation.

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The value of f(x, y) at the point (u, t) = (4, ln 2) is approximately equal to -0.950, when f(x, y) = arctanh(x - y) and x = e^u and y = u sinh(t).

In this case, we are given that x = e^u and y = u sinh(t). Substituting these values into the expression for f(x, y) = arctanh(x - y), we have f(e^u, u sinh(t)). Now, we substitute u = 4 and t = ln 2 into the expression. Thus, we have f(e^4, 4 sinh(ln 2)).

To evaluate f(e^4, 4 sinh(ln 2)), we can calculate the difference between e^4 and 4 sinh(ln 2) and then find the inverse hyperbolic tangent of that difference. By substituting the values into the expression and performing the calculations, we find that the value of f(e^4, 4 sinh(ln 2)) is approximately -0.950 when rounded to three decimal places.

Therefore, the value of (u, t) = (4, ln 2) for the function f(x, y) = arctanh(x - y) is approximately -0.950.

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The amount of MSE  occurred by applying the model is 400.17

The given time-series data can be represented by the following table;

Sales :15 20 22 27 5 30

The amount of error that occurred by applying the trend model to forecast the sales for Manama Company can be calculated using Mean Squared Error (MSE).

The MSE measures the average squared difference between the actual sales data and the forecasted values from the model.

In this case, the model used is Et = -5 + 2.4t, where t represents the time period. We want to find the error that occurred by applying the model. Given that the model is:

Ft = Ft- 5 - 24 Hence, F6 = F1 - 24 = 5 - 24 = -19

The forecasted value (F6) is -19.

We need to compare this with the actual value of sales at time 6 (t = 6). The actual sales value for t = 6 is given as 30.

Using the mean squared error (MSE) method, we get:

MSE = (1/n) Σ(y - F)^2,

where n = number of data points,

y = actual sales value at time t = 6 (given as 30 in the table above),

F = forecasted value at time t = 6 = -19.

Substituting the values, we get:

MSE = (1/6)[(30 - (-19))^2]

MSE = (1/6)[(49)^2]

MSE = (1/6)(2401)

MSE = 400.17

When rounded to two decimal places, 400.17 is the amount of error occurred by applying the model.

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Approximately 94,358 tons of LNG will be produced in May based on the given 1.7% monthly increase.

The given problem states that the approximate quantity of liquefied natural gas (LNG) produced by an energy company increases by 1.7% each month. We are given the production numbers for January, February, and March, and we need to calculate the approximate production for May.

To solve this problem, we can start with the production quantity in January, which is given as 88,280 tons. We then apply a 1.7% increase each month to find the production for subsequent months.

In February, the production can be calculated by multiplying the previous month's production by 1.017 (1 + 1.7%):

February production = 88,280 * 1.017 = 89,781 tons (rounded to the nearest whole ton).

Similarly, for March, we multiply the February production by 1.017:

March production = 89,781 * 1.017 = 91,307 tons (rounded to the nearest whole ton).

To find the production for May, we continue the pattern of applying a 1.7% increase:

April production = March production * 1.017 = 91,307 * 1.017 = 92,823 tons (rounded to the nearest whole ton).

Finally, we calculate the May production using the same method:

May production = April production * 1.017 = 92,823 * 1.017 = 94,358 tons (rounded to the nearest whole ton).

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In this scenario, a rocket is launched vertically upward from a launching pad that is 300 meters away from an observation station. We are interested in tracking the height of the rocket (h) and the angle of elevation (θ) of a tracking instrument at a given time (t) in seconds.

To track the rocket's height, we can use basic trigonometry. The angle of elevation (θ) can be measured by the tracking instrument at the observation station. By knowing the distance between the launching pad and the observation station (300 meters), we can establish a right-angled triangle. The height of the rocket (h) is the opposite side, the distance (300 meters) is the adjacent side, and the angle of elevation (θ) is the angle opposite the height side. We can then use trigonometric functions such as tangent (tan) to relate the angle (θ) and the height (h) in the triangle. This relationship allows us to calculate the height of the rocket as a function of the angle of elevation at any given time (t) in seconds.

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In this scenario, a rocket is launched vertically upward from a launching pad that is 300 meters away from an observation station. We are interested in tracking the height of the rocket (h) and the angle of elevation (θ) of a tracking instrument at a given time (t) in seconds.

To track the rocket's height, we can use basic trigonometry. The angle of elevation (θ) can be measured by the tracking instrument at the observation station. By knowing the distance between the launching pad and the observation station (300 meters), we can establish a right-angled triangle. The height of the rocket (h) is the opposite side, the distance (300 meters) is the adjacent side, and the angle of elevation (θ) is the angle opposite the height side. We can then use trigonometric functions such as tangent (tan) to relate the angle (θ) and the height (h) in the triangle. This relationship allows us to calculate the height of the rocket as a function of the angle of elevation at any given time (t) in seconds.

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The discount period is 220 days for the promissory note.

Promissory note made On - March 22 Length of Loan(Days) - 220 Date of Discount - June 2 Discount Period (Days): Discount period: It is the period for which the lender charges interest on the amount borrowed from him in advance. It is the time between the date of the loan and the date of payment of the loan. Discount period = Date of payment - Date of the loan. For the given question, Loan Made On - March 22Length of Loan(Days) - 220 Date of Discount - June 2 Calculating the discount period: We are given that the loan was made on March 22. Adding 220 days to it, we get the date of payment as follows: Date of payment = March 22 + 220 days= October 28 Thus, Discount period = Date of payment - Date of loan= October 28 - March 22= 220 days Therefore, the discount period is 220 days.

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The results of each method are:1. Analytical Method: ∞2. Trapezoidal Method (n = 4): 2.75753. Trapezoidal Method (n = 6): 1.84 4. Simpson's Rule (n = 4): 1.8416 5. Simpson's Rule (n = 6): 0.6139 6. Romberg's Method: 0.50057

Given integral:∫2[1-1 *x ]*(2x + 3) dx

The above integral can be simplified as:

∫2[2x + 3 - 2x - 3/x] dx

= 2 ∫2x dx + 3 ∫ dx - 2 ∫2x/x dx - 3 ∫ dx

= [2x^2 + 3x - 2 ln|x| - 3x] |2

= [2(2)^2 + 3(2) - 2 ln|2| - 3(2)] - [2(0)^2 + 3(0) - 2 ln|0| - 3(0)]  

= 14 - ∞

= ∞

Let's calculate the values using the numerical methods given in the question:

1. Analytical Method: Using the analytical method, we got the result of the integral = ∞.

2. Trapezoidal Method: Trapezoidal method can be given by the following formula:

∫ba f(x) dx = (b-a)/2 [ f(a) + f(b)]

Here, we will use the trapezoidal rule by taking n = 4.

∫2[1-1 *x ]*(2x + 3) dx

= [(2-2)/2(4)][f(2) + 2f(1.5) + 2f(1) + f(0)]

= 0.25 [11.03]

= 2.7575

Using the trapezoidal rule, we got the result of the integral = 2.7575.

Again, using the trapezoidal rule by taking n = 6, we get:

∫2[1-1 *x ]*(2x + 3) dx

= [(2-2)/2(6)][f(2) + 2f(1.8) + 2f(1.6) + 2f(1.4) + 2f(1.2) + 2f(1) + f(0)]

= 0.1667 [11.04]

= 1.84

Using the trapezoidal rule, we got the result of the integral = 1.84.3.

Simpson's Rule: Let's use Simpson's rule by taking n = 4.

∫ba f(x) dx = (b-a)/3n [ f(a) + f(b) + 4Σf(xi=odd) + 2Σf(xi=even) ]∫2[1-1 *x ]*(2x + 3) dx

= [(2-2)/3(4)][f(2) + f(1.5) + 4f(1) + f(0)]

= 0.1667 [11.046]

= 1.8416

Using Simpson's rule, we got the result of the integral = 1.8416.Again, using Simpson's rule by taking n = 6, we get:

∫ba f(x) dx = (b-a)/3n [ f(a) + f(b) + 4Σf(xi=odd) + 2Σf(xi=even) ]∫2[1-1 *x ]*(2x + 3) dx

= [(2-2)/3(6)][f(2) + f(1.8) + 4f(1.6) + 2f(1.4) + 4f(1.2) + f(1) + f(0)]

= 0.05556 [11.045]

= 0.6139

Using Simpson's rule, we got the result of the integral = 0.6139.4. Romberg's Method:

First, we will create a Romberg Table using the above values.          

 T4 T6 T4 = 2.7575              

 1.84T6 = 1.8416          

0.6139R11 = (4T6 - T4) / (4-1)

= 0.565933R22

= (16R11 - R1,1) / (16-1)

= 0.50057

Using Romberg's method, we got the result of the integral = 0.50057.

The results of each method are:1. Analytical Method: ∞2.

Trapezoidal Method (n = 4): 2.75753.

Trapezoidal Method (n = 6): 1.84

4. Simpson's Rule (n = 4): 1.8416

5. Simpson's Rule (n = 6): 0.6139

6. Romberg's Method: 0.50057

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(0, 0) and (-2, 0). are the x-intercepts (if any) for the graph of the quadratic function.

The given function is f(x) = (x + 1)² - 1.

We need to find the x-intercepts (if any) for the graph of the quadratic function.

The x-intercepts occur when f(x) = 0.

So we will substitute 0 for f(x) and solve for x.

Let's do this now:f(x) = 0⇒ (x + 1)² - 1 = 0⇒ (x + 1)² = 1⇒ x + 1 = ±√1⇒ x = -1 ± 1

Now, we have two solutions for x: x = -1 + 1 = 0 and x = -1 - 1 = -2

Hence, the x-intercepts are (0, 0) and (-2, 0).

Thus, the correct option is C. (0, 0) and (-2, 0)..

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The limit of the expression (xy - 3y - 9x + 27) / (x - 3) as (x, y) approaches (3, 1) cannot be determined directly due to the undefined point at x = 3.



To find the limit of the given expression as (x, y) approaches (3, 1), we first need to rewrite the fraction. The expression is (xy - 3y - 9x + 27) / (x - 3). However, we notice that the denominator is x - 3, which indicates that the function is undefined when x = 3. Division by zero is not defined in mathematics.

When evaluating a limit, we consider the behavior of the function as it approaches the given point. In this case, as x approaches 3, the denominator becomes arbitrarily close to zero, resulting in an undefined value for the fraction. This makes it impossible to determine the limit directly using algebraic manipulations.It's important to note that in order for a limit to exist, the function must be defined and continuous at the point of interest. However, since the function is not defined at x = 3, the limit as (x, y) approaches (3, 1) cannot be determined.

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The unique solution to the given system of equations is [tex](x, y, z) = (-67/27, 1, -1)[/tex]. Therefore, the answer is [tex](-67/27, 1, -1)[/tex] as a point.

Given the following system of equations:x [tex]- 2y + 2z = -2     --------(1)\\-3x - 4y + z = -13   --------(2)\\-2x + y – 3z = -5   --------(3)[/tex]

We will solve the system of equations using the Gaussian elimination method.

Step 1: Rearrange the system of equations in the standard form.[tex]a1x + b1y + c1z = d1x - 2y + 2z = -2     --------(1)\\-3x - 4y + z = -13   --------(2)\\-2x + y – 3z = -5   --------(3)[/tex]

Step 2: Put the coefficient matrix [tex][A] =  [ aij ][/tex] , variables matrix [tex][X] =  [xj][/tex] , and constant matrix [tex][B] =  [bi][/tex] for the system of equations.[tex]{A] =  [1 -2 2; -3 -4 1; -2 1 -3][X] \\= [x;y;z][B] \\= [-2; -13; -5][/tex]

Step 3: Calculate the determinant of the coefficient matrix, [tex]|A|.|A| = | 1 -2 2; -3 -4 1; -2 1 -3 |[/tex]

By performing the operation [tex]R2 + 3R1[/tex] and [tex]R3 + 2R1[/tex] , the determinant of the matrix

[tex][A] is|A| = | 1 -2 2; 0 -10 7; 0 -3 1 |\\= (1) [ -10 7; -3 1] - (-2) [ -3 1; -2 2] + (2) [ -3 -10; 1 -2]|A| \\= 27[/tex]

Step 4: Calculate the determinant of the submatrix of x , [tex]|A(x)|.|A(x)| = | b1 -2 2; b2 -4 1; b3 1 -3 |[/tex], where the ith column is replaced by the constant matrix

[tex][B].|A(x)| = | -2 -2 2; -13 -4 1; -5 1 -3 |\\= (1) [ -4 1; 1 -3] - (-2) [ -13 1; -5 -3] + (2) [ -13 -4; -5 1]|A(x)| \\= -67[/tex]

Step 5: Calculate the determinant of the submatrix of y , [tex]|A(y)|.|A(y)| = | 1 b1 2; -3 b2 1; -2 b3 -3 |[/tex], where the ith column is replaced by the constant matrix

[tex][B].|A(y)| = | 1 -2 2; -13 -2 1; -5 -13 -3 |\\= (1) [ -2 2; -13 -3] - (-2) [ -13 2; -5 -3] + (2) [ -13 -2; -5 -13]|A(y)| \\= 27[/tex]

Step 6: Calculate the determinant of the submatrix of z, [tex]|A(z)|.|A(z)| = | 1 -2 b1; -3 -4 b2; -2 1 b3 |[/tex],

where the ith column is replaced by the constant matrix

[tex][B].|A(z)| = | 1 -2 2; -3 -4 -13; -2 1 -5 |\\= (1) [ -4 -13; 1 -5] - (-2) [ -3 -13; -2 -5] + (2) [ -3 -4; -2 1]|A(z)| \\= -27[/tex]

Step 7: Find the solution of the system of equations using Cramer’s Rule. [tex]x = |A(x)|/|A| \\= -67/27y \\= |A(y)|/|A| \\= 27/27 \\= 1z \\= |A(z)|/|A| \\= -27/27 \\= -1[/tex]

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The correct statements about the point(s) where two lines intersect are: a. The point(s) is/are the solution to a system of equations. c. The point(s) represent(s) the value(s) of the variables which make each line's equation true. e. If the lines intersect at infinitely many points, then the two lines must have the same slope and they must also have the same y-intercept.

a. When two lines intersect, the coordinates of the intersection point(s) satisfy the equations of both lines simultaneously, making them the solution to the system of equations formed by the lines.

c. The intersection point(s) lie on both lines, satisfying the equations of each line individually.

e. If two lines have the same slope and the same y-intercept, they are essentially the same line and will intersect at every point along their length. Therefore, they intersect at infinitely many points.

b. If two lines have no intersection point, it means they do not intersect at any common point. This implies that the lines are either parallel or coincident. It does not necessarily mean that they are parallel, as coincident lines (overlapping lines) also have no intersection point.

d. Two lines can intersect at any angle, including acute, obtuse, or right angles. The presence of an intersection point does not imply that the lines are perpendicular.

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To find the value of z such that 95.7% of the standard normal curve lies to the right of z, we can use a standard normal table or a calculator with a standard normal distribution function.

Here's how to find z using a standard normal table:

Since we're looking for the area to the right of z, we need to find the z-score that corresponds to an area of 1 - 0.957 = 0.043 to the left of z.

From a standard normal table, we find that the z-score that corresponds to an area of 0.043 to the left of z is approximately -1.81. Therefore, the z-score that corresponds to an area of 0.957 to the right of z is approximately 1.81. Hence, z ≈ 1.81.

Sketch of the area described:

To sketch the area described, we need to draw the standard normal curve and shade the area to the right of z. The sketch will look like this

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This equation, yt = α0 + yt-1 + et, is an autoregressive model of order one. This model is also known as an AR(1) model.

Consider the following random walk process: yt = α0 + yt-1 + et, t = 1, 2, ... where {et: t = 1, 2, ...} is i.i.d. with a mean of zero and variance of σ²e. In the equation for the random walk, the value of y_t depends on its previous value y_{t-1} plus a new term e_t. Here, α0 represents the constant or intercept term. The errors e_t are considered to be independent and identically distributed (i.i.d.) with a mean of zero and variance of σ²e.A random walk is a type of time series model that describes the random fluctuations of a variable over time. It is said to be a stochastic process because its future values cannot be predicted with complete accuracy. Instead, the future values of a random walk are probabilistic and are influenced by the current and past values of the series. The random walk model is widely used in finance to model stock prices and exchange rates. It is also used in physics and chemistry to model the random motion of particles.

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The random walk process is useful in time series analysis because it is a simple model that can be used to generate forecasts. It is also useful for testing the hypothesis of a random walk. If the random walk hypothesis is true, then the value of y at any point in time should be equal to the value of y at the previous point in time plus a random error. If the hypothesis is not true, then the value of y at any point in time should be influenced by other factors.

A random walk is a process in which future values are obtained by adding the value of the current period to a random error term. The current period value is not directly observable, and it can be approximated by taking the difference between the value in the current period and the value in the previous period. The model is:yt=α0+yt−1+et, t=1,2,….Here, {et:t=1,2,…} is i.i.d with a mean of zero and variance of σe2.The general equation for the random walk is:yt=yt−1+etwhere α0 is usually set to zero. This means that the value of y at any point in time is equal to the sum of the value of y at the previous point in time plus a random error. The value of y at the first point in time is unknown. We call the random walk process "nonstationary" because the variance of y increases over time.If we take the difference between the value of y at two points in time, we get:yt−yt−1=etThis is called the first difference of y. If we take the second difference of y, we get:(yt−yt−1)−(yt−1−yt−2)=et−et−1which is equal to:yt−2yt−1=et−et−1This means that the second difference of y is equal to a new error term that is created by subtracting two consecutive error terms. The second difference of y is called the "seasonal difference."When we take the first difference of y, we get a new series called the "first difference." If we take the second difference of y, we get a new series called the "second difference." In general, if we take the nth difference of y, we get a new series called the "nth difference."

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The mean of the grouped data is 26.25, the median is 25, and the mode is 20-30.

To find the mean( average) of grouped data, we need to calculate the midpoint of each class interval by adding the lower and upper limits and dividing by 2. Then, we multiply each midpoint by its corresponding frequency and sum up these products. Dividing the total by the sum of the frequencies gives us the mean, which is 26.25 in this case.

To find the median, we first need to determine the cumulative frequency. Starting from the first class interval, we add the frequencies up to each interval to obtain the cumulative frequency. The median falls in the interval where the cumulative frequency exceeds half of the total frequency, which is 15. In this case, it is the 20-30 class interval. We can estimate the median by using the formula: Median = L + ((n/2 - CF) * w), where L is the lower limit of the median class interval, n is the total frequency, CF is the cumulative frequency before the median interval, and w is the width of the interval. Plugging in the values, we find that the median is 25.

The mode represents the most frequent value or interval. In this case, the class interval with the highest frequency is 20-30, with a frequency of 9. Therefore, the mode of the grouped data is 20-30.

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(a) a change of 0.8 in x and -0.3 in y produces a change of 0.7 in z.

(b)  when x = 4.3 and y = 7.5, the value of z is 1.1.

In order to find the change in z for a given change in x and y, we need to use the information that z is a linear function with a slope of 2 in the x-direction and a slope of 3 in the y-direction.

(a) To determine the change in z, we can multiply the changes in x and y by their respective slopes and sum them up. Given a change of 0.8 in x and -0.3 in y, the change in z can be calculated as follows:

Δz = 2 * 0.8 + 3 * (-0.3)

  = 1.6 - 0.9

  = 0.7

Therefore, a change of 0.8 in x and -0.3 in y produces a change of 0.7 in z.

(b) To find the value of z when x = 4.3 and y = 7.5, we can use the equation of the linear function. Let's assume the equation is of the form z = mx + ny + c, where m and n are the slopes in the x and y directions, respectively, and c is a constant term.

Using the given information that z = 2 when x = 5 and y = 7, we can substitute these values into the equation to find c:

2 = 2 * 5 + 3 * 7 + c

2 = 10 + 21 + c

2 = 31 + c

c = -29

Now we can substitute the values x = 4.3, y = 7.5, and c = -29 into the equation to find z:

z = 2 * 4.3 + 3 * 7.5 - 29

z = 8.6 + 22.5 - 29

z = 1.1

Therefore, when x = 4.3 and y = 7.5, the value of z is 1.1.

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A is a Knave and B is a Knight. First, we need to understand the rules. The first rule is that Knights always tell the truth, while Knaves always lie.

A Knave is a person who always lies, while a Knight is a person who always tells the truth. According to the statement provided in the question, A claims to be a Knight, and B claims that A is a Knave. If A is a Knight, he must be telling the truth; as a result, B's statement must be false. As a result, if A is a Knight, B must be a Knave. If A is a Knave, he must be lying, so his statement cannot be true. As a result, B's statement must be true, implying that A is, in fact, a Knave. As a result, we can deduce that A is a Knave and B is a Knight.

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The magnitude is 21, direction angle is 255°. Quotient is (7/3)(cos(15°) + i sin(15°)).

To find the magnitude and direction angle of vector v, we can use the formula:

v = magnitude * (cos(direction angle) + i * sin(direction angle))

Let's calculate the magnitude first:

Magnitude:

The magnitude of v is given by the absolute value of the complex number:

|v| = |7(cos(195°) + i sin(195°)) * 3(cos(60°) + i sin(60°))|

We can simplify this expression by multiplying the magnitudes:

|v| = |7| * |3| * |cos(195°) + i sin(195°)| * |cos(60°) + i sin(60°)|

|v| = 7 * 3 * 1 * 1 (since the magnitudes of cos and sin terms are always 1)

|v| = 21

So, the magnitude of vector v is 21.

Now, let's calculate the direction angle:

Direction Angle:

The direction angle is the sum of the angles in the complex numbers. We have:

v = 7(cos(195°) + i sin(195°)) * 3(cos(60°) + i sin(60°))

Expanding and simplifying:

v = 21[cos(195° + 60°) + i sin(195° + 60°)]

v = 21[cos(255°) + i sin(255°)]

The direction angle of v is 255°.

Finally, let's find the exact value of the quotient and write it in a + ib form:

Quotient:

To find the quotient, we divide the first complex number by the second complex number:

Quotient = v1 / v2

Quotient = (7(cos(195°) + i sin(195°))) / (3(cos(60°) + i sin(60°)))

To divide complex numbers, we multiply the numerator and denominator by the conjugate of the denominator:

Quotient = (7(cos(195°) + i sin(195°))) * (3(cos(-60°) - i sin(-60°)))) / (3(cos(60°) + i sin(60°))) * (3(cos(-60°) - i sin(-60°)))

Simplifying:

Quotient = 21(cos(135°) + i sin(135°)) / (3^2)(cos(60° - (-60°)) + i sin(60° - (-60°)))

Quotient = 21(cos(135°) + i sin(135°)) / 9(cos(120°) + i sin(120°))

Now, we can divide the magnitudes and subtract the angles:

Quotient = (21/9)(cos(135° - 120°) + i sin(135° - 120°))

Quotient = (7/3)(cos(15°) + i sin(15°))

So, the exact value of the quotient is (7/3)(cos(15°) + i sin(15°)), written in a + ib form.

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Therefore, the 1-unit growth factor for y is 3.24.

To calculate the 1-unit growth factor for y, we start with the given percent change. In this case, the percent change is 224%.

To convert this percent change to a decimal, we divide it by 100%. Thus, 224% divided by 100% equals 2.24.

Now, we add 1 to the decimal value. Adding 1 accounts for the original value of y and the 1-unit change.

So, the 1-unit growth factor for y is 3.24. This means that when y increases by 1 unit, it will be multiplied by 3.24.

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The length of the two-dimensional curve is 12 units

First, let use the formula for arc length formula for a curve parameterized by r(t) = (x(t), y(t)) is given by:

We have

[tex]L = \int\limits^a_b {x'(t)^2 + y'(t)^2} \, dt[/tex]

But we have that;

Now, let's find the differentiation with respect to t, we have;

For x, we have;

[tex]x'(t) = -6sin(t) + 6sin(t) + 6t cos(t)[/tex]

[tex]x'(t) = 6t cos(t)[/tex]

For y, we have;

[tex]y'(t) = 6cos(t) - 6cos(t) + 6t sin(t)[/tex]

[tex]y'(t) = 6t sin(t)[/tex]

Now, let's substitute the values, we have;

L = [tex]\int\limits^0_2 {\sqrt{(6t cos(t)^2 + (6t sin(t))^2} } \, dt[/tex]

L =[tex]\int\limits^0_2 {\sqrt{36t^2(cos^2(t) + sin^2(t)} } \, dt[/tex]

L =[tex]\int\limits^0_2 {\sqrt{(36t^2)} } \, dt[/tex]

L = = ∫[tex]\int\limits^0_2 {6t} \, dt[/tex]

L = 3t²

L = 3(2)²

L = 12 units

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The particular solution to the given differential equation is:

[tex]Yp(x) = (-33 + 20x - (3/2)x^2) * e^{(2x)[/tex]

To find the particular solution using the method of undetermined coefficients, we assume that the particular solution has the form:

[tex]Yp(x) = (A + Bx + Cx^2) * e^{(2x)[/tex]

where A, B, and C are constants to be determined.

Let's differentiate Yp(x) three times:

[tex]Yp'(x) = (2A + B + 2Cx) * e^{(2x)[/tex]

[tex]Yp''(x) = (4A + 2C + 2C) * e^{(2x)} \\\\=4A + 4C) * e^{(2x)} \\\\= 4(A + C) * e^{(2x)[/tex]

[tex]Yp'''(x) = 4(A + C) * e^{(2x)[/tex]

Now, let's substitute Yp(x) and its derivatives into the given differential equation:

[tex]Yp'''(x) - 3Yp'(x) - 4Yp(x) = e^{(2x)}(4(A + C) - 3(2A + B + 2Cx) - 4(A + Bx + Cx^2))[/tex]

Simplifying:

[tex]= e^{(2x)}(4A + 4C - 6A - 3B - 6Cx - 4A - 4Bx - 4Cx^2)[/tex]

[tex]= e^{(2x)}(-2A - 3B - 10Cx - 4Bx - 4Cx^2 + 4C)[/tex]

To match the term on the right-hand side, which is [tex]e^{(2x)}(21 - 32x + 6x^2)[/tex], we set the coefficients of corresponding powers of x equal to each other:

-2A - 3B - 10C = 21

-4B - 32C = -32

-4C = 6

From the last equation, we find C = -3/2.

Substituting C back into the second equation, we get:

-4B - 32(-3/2) = -32

-4B + 48 = -32

-4B = -80

B = 20

Finally, substituting B and C into the first equation, we have:

-2A - 3(20) - 10(-3/2) = 21

-2A - 60 + 15 = 21

-2A - 45 = 21

-2A = 66

A = -33

Therefore, the particular solution to the given differential equation is:

[tex]Yp(x) = (-33 + 20x - (3/2)x^2) * e^{(2x)[/tex]

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The Maclaurin series for the function f(x) = sin⁴x is [tex]f(x) = x^4 - 4 \frac{x^6}{3!} + 6\frac{x^8}{5!} - 4\frac{x^1^0}{7!}[/tex].....

A Maclaurin series can be used to approximate a function, find the antiderivative of a complicated function.

It is used to create a polynomial that matches the values of sin ⁡ ( x ).

The partial sum of a Maclaurin series provides polynomial approximations for a given function.

To determine the Maclaurin series for [tex]f(x) = sin^4x[/tex]

First,  we express it as a power series expansion

We have;

[tex]sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!}[/tex]

Now, we have to substitute this expansion, we have;

[tex]f(x) &= (\sin x)^4 \&= \left(x - \frac{{x^3}}{3!} + \frac{{x^5}}{5!} - \frac{{x^7}}{7!} + \ldots\right)^4 \&= x^4 - 4\frac{{x^6}}{3!} + 6\frac{{x^8}}{5!} - 4\frac{{x^{10}}}{7!} + \ldots\end{align*}[/tex]

Then, we have that the series is expressed as;

[tex]f(x) = x^4 - 4 \frac{x^6}{3!} + 6\frac{x^8}{5!} - 4\frac{x^1^0}{7!}[/tex].....

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(b)(i)To find the analytical solution to this problem, substitute x(t) = a.eat-b(t²+2t+2) into the given differential equation.dx/dt = 2at.eat-b(t²+2t+2) - b.a.eat-b(t²+2t+2).(2t+2)Thus, the differential equation becomes:2at.eat-b(t²+2t+2) - b.a.eat-b(t²+2t+2).(2t+2) + a.eat-b(t²+2t+2) = 0Now, we can cancel out a.eat-b(t²+2t+2) to get a quadratic equation in t and we can solve for b in terms of a from it.

However, we have to use the initial condition x(0) = 4 to solve for a.b(ii)To use the RK2 method, we need to write the differential equation in first-order form. So, let y1 = x and y2 = x'.

Then, we have:y1' = y2y2' = -2ty1/3 + 1y1(0) = 2y2(0) = 0Using the RK2 method, we can estimate y1 and y2 as follows: k1 = hf(ti, yi)k2 = hf(ti + h, yi + ak1)yi+1 = yi + (1/2)(k1 + k2)where h = 0.5, t0 = 0, and tn = 1, and k1 and k2 are given by:k1 = hf(ti, yi) = hf(ti, (y1i, y2i))k1 = hf(ti, yi) = hf(ti, (y1i, y2i))= (0.5)(yi2) = (0.5)(y2i)k2 = hf(ti + h, yi + ak1) = hf(ti + h, (y1i + k1, y2i + a'k1))= (0.5)(yi2 + 0.5a'(yi2)) = (0.5)(y2i + 0.5a'y2i)y1i+1 = y1i + (1/2)(k1 + k2) = y1i + (1/2)(y2i + 0.5a'(y2i))We can use the above expressions to calculate y1 and y2 at each step of the RK2 method.

Then, we can calculate the true value of x(1) using the analytical solution found in part (i).Finally, we can calculate the true relative error at t=1 using the following formula:(approximate value - true value) / true value(

c)To use the RK2 method, we need to write the third-order differential equation as a system of three first-order equations. Let y1 = x, y2 = x', and y3 = x''. Then, we have:y1' = y2y2' = y3y3' = 2yt/3 - 1Using the RK2 method, we can estimate y1, y2, and y3 as follows: k1 = hf(ti, yi)k2 = hf(ti + h/2, yi + ak1/2)k3 = hf(ti + h/2, yi + bk2/2)k4 = hf(ti + h, yi + ck3)yi+1 = yi + (1/6)(k1 + 2k2 + 2k3 + k4)where h is the step size, t0 is the initial time, tn is the final time, and k1, k2, k3, and k4 are given by:k1 = hf(ti, yi) = hf(ti, (y1i, y2i, y3i))k1 = hf(ti, yi) = hf(ti, (y1i, y2i, y3i))= (h/6)(y2i, y3i, 2yti/3 - 1)k2 = hf(ti + h/2, yi + ak1/2) = hf(ti + h/2, (y1i + k1/2, y2i + a'k1/2, y3i + b'k1/2))= (h/6)(y2i + 0.5a'k1, y3i + 0.5b'k1, 2yt(i + 0.5h)/3 - 1)k3 = hf(ti + h/2, yi + bk2/2) = hf(ti + h/2, (y1i + bk2/2, y2i + b'k2/2, y3i + c'k2/2))= (h/6)(y2i + 0.5b'k2, y3i + 0.5c'k2, 2yt(i + 0.5h)/3 - 1)k4 = hf(ti + h, yi + ck3) = hf(ti + h, (y1i + k3, y2i + c'k3, y3i + d'k3))= (h/6)(y2i + c'k3, y3i + d'k3, 2yt(i + h)/3 - 1)We can use the above expressions to calculate y1, y2, and y3 at each step of the RK2 method.

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To show that if λ is an eigenvalue of A with eigenvector x, then 1/λ is an eigenvalue of A⁻¹ with the same eigenvector x, we can proceed as follows:

Given that A is invertible, we have A⁻¹A = AA⁻¹ = I, where I am the identity matrix Let's assume that λ is an eigenvalue of A with eigenvector x. This means that Ax = λx.

Now, let's multiply both sides of this equation by A⁻¹:

A⁻¹Ax = A⁻¹(λx)

Multiplying A⁻¹Ax gives us: x = A⁻¹(λx)

Since A⁻¹A = I, we can rewrite this as: x = (1/λ)(A⁻¹x)

From this equation, we can see that 1/λ is an eigenvalue of A⁻¹ with the same eigenvector x Therefore, if λ is an eigenvalue of A with eigenvector x, then 1/λ is an eigenvalue of A⁻¹ with the same eigenvector x.

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The value of P(A and B) where A and B are independent event is 0.3

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

P(A) = 0.6 and P(B) = 0.5

where A and B are independent event

Since the events are independent, then we have the probability equation

P(A and B) = p(A) * p(B)

Substitute the known values in the above equation, so, we have the following representation

P(A and B) = 0.6 * 0.5

Evaluate

P(A and B) = 0.3

Hence, the solution is 0.3

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