Consider the second-order system of ODES.x"= - 45x + 36y, y"= 36x – 45y. = a. Compute the eigenvalues of the coefficient matrix A and find a corresponding eigenvector for each one. You must get all six entries correct to receive creditb. Calculate the natural frequencies w of the system and enter them as a comma separated list. c. Use the eigenvalue method to find the general solution to this system of differential equations. Use a1, az, bi, b2 to denote arbitrary constants, and enter them as a1, a2, b1,b2.

Answer:

3/4

Step-by-step explanation:

There are 6 options that are less than seven. there are 8 options in total. This means 6 out of eight are less than seven. this is 6/8. Simplify this and you get 3/4. the answer is 3/4.

y = [tex]2^x +e^4x - cos(e^3x)[/tex]  b. y =[tex]3e^2x[/tex] / √2x+1.

dy/dx = d/dx ([tex]2^x)[/tex] + d/dx ([tex]e^4x)[/tex] - d/dx [tex](cos(e^3x))[/tex]

= [tex]2^x[/tex]ln(2) + 4[tex]e^4x[/tex] + sin[tex](e^3x) (3e^3x)[/tex]

= [tex]2^x[/tex] ln(2) + 4[tex]e^4x[/tex] + [tex]3e^3x sin(e^3x)[/tex]

Therefore, the derivative of y with respect to x is

 [tex]2^x[/tex] ln(2) + 4[tex]e^4x[/tex] + [tex]3e^3x sin(e^3x)[/tex]

dy/dx = [3([tex]e^2x[/tex])(√(2x+1))' - (√(2x+1))(3[tex]e^2x[/tex])'] / (2x+1)]

= [3([tex]e^2x[/tex])/(2√(2x+1))) - (3[tex]e^2x[/tex])(1/[tex](2(2x+1/2)^(3/2)[/tex]))] / (2x+1)]

= [3[tex]e^2x([/tex][tex]2(2x+1/2)^(3/2)[/tex] - √(2x+1))] / [tex](2(2x+1/2)^(3/2)(2x+1)[/tex]

= [3[tex]e^2x[/tex](4x+2) - √(2x+1))] / [tex](2(2x+1/2)^(3/2)(2x+1)[/tex]

Therefore, the derivative of y with respect to x is

= [3[tex]e^2x[/tex](4x+2) - √(2x+1))] / [tex](2(2x+1/2)^(3/2)(2x+1)[/tex]

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Based on the given information, we know that the cost of a bag of chips is $52.20 more than the cost of both water bottles.

To solve this problem, we need to set up an equation based on the given information. Let's assume the cost of one water bottle is "w" and the cost of one bag of chips is "c". We also know that you picked up "d" dalions of milk bottles and "r" raise bags of Youttore.

Therefore, the equation will be:
d*w + r*c + 2*w + c = 280

Simplifying the equation, we get:
d*w + r*c + 3*w + c = 280

We also know that "c" is $52.20 more than the cost of both water bottles, so we can substitute that in the equation:
d*w + r*(w+52.20) + 3*w + (w+52.20) = 280

Simplifying again, we get:
d*w + r*w + 3*w + 53.20r + 52.20 = 280

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The quotient of the expression 5x²/7 ÷ 10x³/21 when evaluated is 3/(2x)

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

5x²/7 ÷ 10x³/21

Assume that no denominator has a value of 0, we have

5x²/7 ÷ 10x³/21 = 5x²/7 ÷ 10x³/(7 * 3)

Express as products

So, we have the following representation

5x²/7 ÷ 10x³/21 = 5x²/7 * (7 * 3)/10x³

When the factors are evaluated, we have

5x²/7 ÷ 10x³/21 = 5 * 3/10x

So, we have

5x²/7 ÷ 10x³/21 = 15/10x

This gives

5x²/7 ÷ 10x³/21 = 3/(2x)

Hence, the solution is 3/(2x)

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Complete question

Find the quotient. Assume that no denominator has a value of 0.

5x^2/7÷10x^3/21

The derivative of g(x) is (-4x²-3x+1)cos(1+2x) - (2x³ - 2x^2 + x)tcos(1+2x) + t(1+2x)sin(1+2x) + C, where C is a constant of integration.

The derivative is a mathematical concept that represents the rate at which a function changes. It is essentially the slope of the tangent line to the curve of the function at a given point.

Integration is the process of finding the integral of a function, which involves calculating the area under its curve. It is the reverse of differentiation and is used in calculus and mathematical analysis.

According to the given information:

To find the derivative of g(x), we first need to evaluate the integral:

g(x) = ∫[1, 2x+1] (1-2t)sin(t) dt

Using the product rule of differentiation, we have:

g'(x) = (d/dx) [∫[1, 2x+1] (1-2t)sin(t) dt]

= (2-2x)sin(2x+1) - ∫[1, 2x+1] 2sin(t) dt

Simplifying the second term, we get:

g'(x) = (2-2x)sin(2x+1) - 2[cos(2x+1) - cos(1)]

Therefore, the derivative of g(x) is g'(x) = (2-2x)sin(2x+1) - 2[cos(2x+1) - cos(1)].

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A probability is a number that reflects the chance or likelihood that a particular event will occur

a. The random variable is the number of meals purchased per week by a student.

b. Table of probability distribution:

Meals per Week Probability

10 0.15

14 0.45

18 0.30

21 0.10

c. P(X > 14) = P(X = 18) + P(X = 21) = 0.30 + 0.10 = 0.40

d. P(not purchasing 21 meals) = 1 - P(purchasing 21 meals) = 1 - 0.10 = 0.90

e. The average number of meals purchased per week can be calculated as the weighted mean of the number of meals and their respective probabilities:

μ = (10 x 0.15) + (14 x 0.45) + (18 x 0.30) + (21 x 0.10) = 14.7 meals per week.

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Since each insertion has an amortized time complexity of O(1), the overall time complexity for filling an array of size n is O(n).

When using a dynamically resizing array, the array's capacity is doubled with each insertion. The amortized time of each insertion can be analyzed using the accounting method. In this case, let's assign a cost of 3 for each insertion operation:

1 for the actual insertion and 2 as a "token" that will be used later when the array needs to be resized. Now, let's analyze how the tokens are used: - When the array is resized the first time, it has a capacity of

2. It has 2 tokens (1 for each of the 2 elements), which are enough to pay for copying those elements to the new array. - When the array is resized again (capacity = 4), it has 4 tokens (1 for each element).

Again, there are enough tokens to pay for copying the elements to the new array. - This pattern continues as the array keeps doubling in size. Since the total cost of each insertion is 3, the amortized time complexity of each insertion is O(1).

For the overall process of filling an array of size n, we can calculate the total time complexity as the sum of the cost of individual insertions.

Since each insertion has an amortized time complexity of O(1), the overall time complexity for filling an array of size n is O(n).

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The answers are:

B. Represents motion's speed and direction.

C. Can be shown by a vector arrow with dimensions and a direction.

A. Either a good or bad thing.

The path or direction that an object takes as it moves is referred to as its direction of motion. It can be expressed using terminology like up, down, left, right, forward, backward, or using compass directions like north, south, east, or west. It represents the orientation of an object's motion in space.

The following three responses correspond to velocity but not speed:

B. represents motion's speed and direction.

C. can be shown by a vector arrow with dimensions and a direction.

A. either a good or bad thing.

The definition of velocity is the rate and direction of an object's motion. As a result, it takes into account both the speed and direction of an object's motion. Since velocity is a vector quantity, an arrow that shows both its magnitude and direction can be used to symbolise it.

Contrarily, speed is defined as the amount (size) of an object's velocity, without taking into account its direction. The fact that speed is a scalar variable means that it only provides us with information about the magnitude of an object's motion, not its direction.

All three of the options—B, C, and A—discuss aspects of velocity that don't apply to speed. Option C shows that velocity can be represented by a vector arrow showing both size and direction, whereas option A shows that velocity can be positive or negative depending on the direction of motion. Option B shows that velocity includes both rate (magnitude) and direction. Options D and E apply to speed but not to velocity because they define scalar quantity attributes.

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The midsegment theorem and Thales theorem indicates that we get;

8. x = 35/4, y = 15

10. x = 6, y = 13/2

The midsegment theorem states that a segment that joins the midpoints of two of the sides of a triangle, is parallel to and half the length of the third side of the triangle.

8. The congruence markings in the diagram indicates that we get;

2·y + 6 = 3·y - 9

3·y - 2·y = 6 + 9 = 15

y = 15

The midsegment theorem indicates that we get;

2 × (x + 23) = 6·x + 11

2·x + 46 = 6·x + 11

6·x - 2·x = 4·x = 46 - 11 = 35

x = 35/4

10. The midsegment theorem indicates that we get;

2·x = 3·x - 6

3·x - 2·x = x = 6

x = 6

The Thales theorem, also known as the triangle proportionality theorem indicates that we get;

y = (2·x + 1)/2

y = (2 × 6 + 1)/2 = 13/2

y = 13/2

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Check the picture below.

[tex]\textit{area of a trapezoid}\\\\ A=\cfrac{h(a+b)}{2}~~ \begin{cases} h~~=height\\ a,b=\stackrel{parallel~sides}{bases~\hfill }\\[-0.5em] \hrulefill\\ a=3\\ b=11\\ h=5 \end{cases}\implies A=\cfrac{5(3+11)}{2}\implies A=35~units^2[/tex]

The conjugate of √8 - √9 is as follows:

(√8 + √9).

A pair of entities connected together is referred to as being conjugate. For instance, the two smileys—smiley and sad—are identical save from one set of characteristics that is essentially the complete opposite of the other. These smileys are identical, but you'll see if you look closely that they have the opposite facial expressions: one has a smile, and the other has a frown. Similar to this, the term "conjugate" in mathematics designates either the conjugate of a complex number or the conjugate of a surd when the number only undergoes a sign change with respect to a few constraints.

Here in the question,

The binomial is given as:

√8 - √9

The negative of this or when the operation sign is changed in the binomial, we get the conjugate as:

√8 + √9

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The derivative of y with respect to x is sinh⁻¹(x/2) + x / (2√(4 + x²)) - 2x.

To find the derivative of y with respect to x, we need to use the chain rule and the derivative of inverse hyperbolic sine function:

dy/dx = (d/dx) [x sinh⁻¹(x/2) - (4 + x²)]

First, we need to find the derivative of the first term, using the chain rule:

(d/dx) [x sinh⁻¹(x/2)] = sinh⁻¹(x/2) + x (d/dx) sinh⁻¹(x/2)

Now, we need to find the derivative of sinh⁻¹(x/2), which is given by:

(d/dx) sinh⁻¹(u) = 1 / √(1 + u²) * (du/dx)

where u = x/2, so du/dx = 1/2:

(d/dx) sinh⁻¹(x/2) = 1 / √(1 + (x/2)²) * (1/2)

Substituting this back into the first term, we get:

(d/dx) [x sinh⁻¹(x/2)] = sinh⁻¹(x/2) + x / (2 √(1 + (x/2)²))

Now, we can substitute this and the derivative of the second term into the expression for dy/dx:

dy/dx = sinh⁻¹(x/2) + x / (2 √(1 + (x/2)²)) - 2x

Simplifying this expression, we get:

dy/dx = sinh⁻¹(x/2) / 2 + x / (2 √(1 + (x/2)²)) - 2x

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The number of kcal that would be available if a client has just eaten a food consisting of 4 grams of protein, 18 grams of carbohydrate, and 1 gram of fat will be 97 kcal.

To calculate this, we need to multiply the number of grams of protein by 4 (because there are 4 kcal in 1 gram of protein), the number of grams of carbohydrate by 4 (because there are also 4 kcal in 1 gram of carbohydrate), and the number of grams of fat by 9 (because there are 9 kcal in 1 gram of fat).

So, for this food, we have:

4 grams of protein x 4 kcal/gram = 16 kcal from protein
18 grams of carbohydrate x 4 kcal/gram = 72 kcal from carbohydrate
1 gram of fat x 9 kcal/gram = 9 kcal from fat

Adding these up, we get:

16 kcal + 72 kcal + 9 kcal = 97 kcal

So, the total number of kcal in this food is 97 kcal.

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The standard deviation of the coating thickness for this process is approximately 0.0746.

To find the standard deviation of the coating thickness for this process, we can follow these steps:

Calculate the mean thickness:

The mean thickness is calculated by summing up all the thickness values and dividing by the number of values:

mean thickness = (0.1 + 0.14 + 0.18 + 0.16) / 4 = 0.15

Calculate the variance:

The variance of a uniform distribution is calculated as:

variance = (b - a)^2 / 12

where "a" is the minimum value of the distribution (in this case, 0.1), "b" is the maximum value of the distribution (in this case, 0.18), and the constant 12 comes from the formula for the variance of a uniform distribution.

Substituting the values into the formula, we get:

variance = (0.18 - 0.1)^2 / 12 = 0.00556

Calculate the standard deviation:

The standard deviation is the square root of the variance:

standard deviation = sqrt(variance) = sqrt(0.00556) = 0.0746

Therefore, the standard deviation of the coating thickness for this process is approximately 0.0746.

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The differential equation for the total amount raised F(t) t hours after the fundraising begins, with an initial condition of F(0) = 10, is dF/dt = k× (6000 - F)×F.

The fundraising principle can be expressed mathematically as

dF/dt = k× (6000 - F)×F,

where k is the proportionality constant, (6000 - F) is the amount still needed to reach the target, and F is the amount raised so far.

The differential equation above is a first-order nonlinear differential equation, and it describes the rate of change of F with respect to time t.

To find the initial condition, we can use the fact that the student contributes $10 before the fundraising begins. Thus, when t=0, F(0) = 10.

Therefore, the initial condition for the differential equation is F(0) = 10.

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Answer:

the bank will require you to make a final payment of $22,004.52 at the end of the loan in five years.

Step-by-step explanation:

To calculate the final payment that the bank requires you to make, we need to find the present value of the five annual payments of $4000 each, and then compound that present value to the end of the loan in five years at the interest rate of 5.63%.

Let's begin by calculating the present value of the five annual payments. We can use the formula for the present value of an annuity:

PV = C * [(1 - (1 + r)^-n) / r]

where:

PV = present value

C = annual payment amount

r = interest rate per period (annual rate divided by number of periods per year)

n = number of periods

Plugging in the given values, we get:

PV = $4000 * [(1 - (1 + 0.0563/1)^-5) / (0.0563/1)]

= $4000 * [(1 - (1.0563)^-5) / 0.0563]

= $4000 * 4.169942

= $16,679.77

So the present value of the five annual payments is $16,679.77.

Next, we need to compound this present value to the end of the loan in five years. We can use the formula for future value:

FV = PV * (1 + r)^n

where:

FV = future value

PV = present value

r = interest rate per period

n = number of periods

Plugging in the given values, we get:

FV = $16,679.77 * (1 + 0.0563/1)^5

= $16,679.77 * 1.319695

= $22,004.52

Therefore, the bank will require you to make a final payment of $22,004.52 at the end of the loan in five years.

Answer:

Step-by-step explanation:

Sorry It looks so blurry, find the radius then square it, times it by pie/3.14

The car will have covered 118/3 miles during the first 2 hours of driving.

The velocity of the car is given by v(t) = 20 + 19t - 6t². To find the distance covered by the car during the first 2 hours of driving, we need to integrate the velocity function from 0 to 2.

This gives us the total displacement of the car during the first 2 hours, which we can then take the absolute value of to get the distance.

s(2) - s(0) = ∫₀² v(t) dt

           = ∫₀² (20 + 19t - 6t²) dt

           = [20t + (19/2)t² - 2t³] from 0 to 2

           = [40 + 19(2) - 2(2³/3)] - [0 + 0 - 0]

           = 40 + 38/3

           = 118/3 miles

Therefore, the car will have covered 118/3 miles during the first 2 hours of driving.

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Answer:

The length of the piece of wood can be found using the Pythagorean theorem. The ramp is the hypotenuse of a right triangle with one leg being the height of the cinder block (30 cm) and the other leg being the distance from the block to where the ramp begins (72 cm). So, the length of the piece of wood is [tex]√(30² + 72²) = √(900 + 5184) = √(6084) = 78 cm.[/tex]

Step-by-step explanation:

Thus, the amount after the 7 years compounded quarterly is found as $4326.12.

A quarterly compounded rate means that the principal amount typically compounded four times over the course of a full year. According to the compound interest procedure, if the duration of compounding is longer inside a year, the investors would receive higher future values for their investment.

Given that:

For for the quarterly compounding:

A = P[tex](1 + r/n)^{nt}[/tex]

A = 1773.00[tex](1 + .1295/4)^{4*7}[/tex]

A = 1773.00*2.44

A = 4326.12

Thus, the amount after the 7 years compounded quarterly is found as $4326.12.

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The function f(x)=(x²-4x)⁴ has intercepts at (0,0) and (4,0). Its derivative has critical numbers at x=0 and x=4. The function is decreasing on (-∞,0) and (0,4) and (4,∞) and increasing on (-∞,0) and (4,∞). There are two extrema at (0,0) and (4,0), both of which are RMIN.

To find the intercepts, we set f(x) = 0 and solve for x

f(x) = (x² - 4x)⁴ = 0

Factor out x² - 4x

x² - 4x = 0

x(x - 4) = 0

So the intercepts are (0,0) and (4,0).

To find the derivative of f(x), we apply the chain rule and the power rule

f'(x) = 4(x² - 4x)³(2x - 4)

Setting f'(x) = 0 and solving for x, we get the critical numbers

f'(x) = 4(x² - 4x)³(2x - 4) = 0

x² - 4x = 0

x(x - 4) = 0

So the critical numbers are x = 0 and x = 4.

To find where the function is decreasing, we look at the intervals between the critical numbers and at the intervals outside the critical numbers

For x < 0, f'(x) > 0, so f(x) is decreasing.

For 0 < x < 4, f'(x) < 0, so f(x) is decreasing.

For x > 4, f'(x) > 0, so f(x) is decreasing.

Therefore, the function is decreasing on (-∞,0) and (0,4) and (4,∞).

To find where the function is increasing, we look at the intervals outside the critical numbers

For x < 0, f'(x) > 0, so f(x) is increasing.

For x > 4, f'(x) > 0, so f(x) is increasing.

Therefore, the function is increasing on (-∞,0) and (4,∞).

To find the extrema, we look at the critical numbers and the endpoints of the intervals

At x = 0, f(x) = 0.

At x = 4, f(x) = 0.

So we have two extrema, both of which are RMIN (0,0) and (4,0).

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The volume of the prism is 825 in3.

The correct answer is option D: 825 in3.

The volume of a rectangular prism is the amount of space occupied by the prism in three-dimensional space. It is calculated by multiplying the length, width, and height of the prism.

The volume of a rectangular prism is given by the formula V = l x w x h, where l, w, and h are the length, width, and height of the prism, respectively.

In this case, the length is 15 inches, the width is 11 inches, and the height is 5 inches.

Therefore, the volume of the rectangular prism is:

V = l x w x h

V = 15 in x 11 in x 5 in

V = 825 in3

So the volume of the prism is 825 in3.

Therefore, the correct answer is option D: 825 in3.

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The volume of the prism is 825 in3.

The correct answer is option D: 825 in3.

What is rectangular prism?

The volume of a rectangular prism is the amount of space occupied by the prism in three-dimensional space. It is calculated by multiplying the length, width, and height of the prism.

The volume of a rectangular prism is given by the formula V = l x w x h, where l, w, and h are the length, width, and height of the prism, respectively.

In this case, the length is 15 inches, the width is 11 inches, and the height is 5 inches.

Therefore, the volume of the rectangular prism is:

V = l x w x h

V = 15 in x 11 in x 5 in

V = 825 in3

So the volume of the prism is 825 in3.

Therefore, the correct answer is option D: 825 in3.

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The dimensions of the rectangular field of maximum area are 35 m by 35 m, which corresponds to option C

To find the dimensions of the rectangular field of maximum area using 140 m of fencing material, you can follow these steps:
1. Let the length of the rectangle be L meters, and the width be W meters.
2. The perimeter of the rectangle is given by 2L + 2W = 140 m.
3. Rearrange the formula to solve for L: L = (140 - 2W) / 2.
4. The area of the rectangle is given by A = L * W.
5. Substitute the expression for L from step 3 into the area formula: A = ((140 - 2W) / 2) * W.
6. Simplify the equation: A = (140W - 2W^2) / 2.
7. To find the maximum area, take the first derivative of A with respect to W and set it equal to 0: dA/dW = 140/2 - 2W = 0.
8. Solve for W: W = 35 m.
9. Substitute W back into the formula for L: L = (140 - 2(35)) / 2 = 35 m.

The dimensions of the rectangular field of the maximum area that can be made from 140 m of fencing material are 35 m by 35 m
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This 9-year-old girl's performance places her at the 70th percentile.

To determine the girl's percentile based on her sit-up performance, you need to consider the percentage of girls her age who can perform fewer or equal sit-ups in one minute.

Since 30% of girls her age can perform more sit-ups,

it means that 70% of girls her age can perform fewer or equal sit-ups in one minute.

Therefore, this 9-year-old girl's performance places her at the 70th percentile.

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The 9-year-old girl's performance in sit-ups is in the 30th percentile.

Based on the information given, you found that a 9-year-old girl can perform more sit-ups in one minute than 30% of the girls her age.

1. Understand the meaning of percentile:

A percentile indicates the relative standing of a data point within a data set, showing the percentage of scores that are equal to or below the data point.

2. Interpret the given information:

In this case, 30% of girls her age can perform fewer sit-ups than she can in one minute.

3. Calculate the percentile:

Since 30% of the girls perform fewer sit-ups than her, this girl's performance is at the 30th percentile. This means that she performs better than or equal to 30% of the girls her age.

In conclusion, this 9-year-old girl's performance in sit-ups places her at the 30th percentile.

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The value of k that makes the given function a valid probability density function is k = 6.

To be a valid probability density function, the given function must satisfy the following two conditions:

The function must be non-negative for all possible values of X.

The integral of the function over all possible values of X must equal 1.

Using these conditions, we can determine the value of k as follows:

For the function to be non-negative, kx(1-x) must be non-negative for all possible values of X. This requires that k must be non-negative as well.

To find the value of k such that the integral of the function over all possible values of X is equal to 1, we integrate the given function from 0 to 1 and set the result equal to 1:

∫[tex]0^1 kx(1-x) dx = 1[/tex]

Solving the integral gives:

k/6 = 1

k = 6

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The distribution of X(n) is (n/4ⁿ) * x^ⁿ⁻¹ with mean 4n/(n+1) and variance 16/3n². The distribution of X(1) is (n/4ⁿ) * (4-x)ⁿ⁻¹ with mean 4(1-1/n) and variance 16/3n². The function of the geometric mean GM(x) = [tex](4/n)^{1/n}[/tex] and GM(x) = exp(1/n * Sum(ln(Xi))).

Since the samples are from the uniform distribution on (0,4), the distribution of X(n) and X(1) can be derived as follows

P(X(n) ≤ x) = P(all samples ≤ x) = (x/4)^n

P(X(1) ≥ x) = P(all samples ≥ x) = (4-x)^n/4^n

Using these probabilities, the cumulative distribution functions (CDFs) for X(n) and X(1) can be obtained

F(X(n)) = P(X(n) ≤ x) = (x/4)ⁿ for 0 ≤ x ≤ 4

F(X(1)) = 1 - P(X(1) > x) = 1 - (4-x)ⁿ/4ⁿ for 0 ≤ x ≤ 4

The probability density functions (PDFs) can be obtained by differentiating the CDFs

f(X(n)) = (n/4ⁿ) * x^ⁿ⁻¹ for 0 ≤ x ≤ 4

f(X(1)) = (n/4ⁿ) * (4-x)ⁿ⁻¹ for 0 ≤ x ≤ 4

The mean and variance of X(n) and X(1) can be calculated as follows

Mean(X(n)) = 4n/(n+1)

Var(X(n)) = (16n-48)/(n+1)²

Mean(X(1)) = 4(1-1/n)

Var(X(1)) = 16/(3n²)

Using Y = -ln(X()), we have

[tex]P(Y \leq y) = P(X() \geq e^{-y} = 4 - e^{-y}^{n/4^{n}})[/tex]

The CDF of Y can be obtained by substituting X() with [tex]e^{-Y}[/tex]

[tex]P(Y \leq y) = 4 - e^{-y}^{n/4^{n}})[/tex]

The PDF of Y can be obtained by differentiating the CDF

[tex]f(Y) = (n/4^n) * e^{-ny} * (4-e^{-y}^{n-1}[/tex]

The geometric mean can be written as

GM(x) = exp(1/n * sum(ln(x(i))))

Using the definition of Y and the PDF of Y, the geometric mean can be written as

GM(x) = exp(-1/n * sum(ln(X(i)))) = exp(-1/n * sum(-ln(Y(i)))) = exp(1/n * sum(ln(Y(i))))

GM(x) = exp(1/n * integral(ln(y) * f(y) dy, 0, infinity))

Substituting the PDF of Y in the above integral

GM(x) = exp(1/n * integral(ln(y) * (n/4ⁿ) * [tex]e^{-ny}[/tex] * (4-[tex]e^{-y}[/tex])ⁿ⁻¹ dy, 0, infinity))

Using integration by parts, the above integral can be simplified as

GM(x) = [tex](4/n)^{1/n}[/tex]

The result in above part shows that the geometric mean of the samples follows a distribution that does not depend on the values of the samples. Specifically, it is equal to[tex](4/n)^{1/n}[/tex] which approaches 1 as n gets larger. This suggests that the geometric mean is a consistent estimator of the true mean of the distribution.

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The value of particular solution is,

⇒ y (p0 = (4/5)x^(5/2) - (4/15)x^(7/2).

Now, we need to find the Wronskian of the given solutions;

⇒ y₁ = e²ˣ and y₂ = x e⁻²ˣ.

Hence, We get;

⇒ W(y₁, y₂) = |e²ˣ   xe⁻²ˣ|

                 = -2e⁰

                  = -2

Next, we can find the particular solution using the formula:

⇒ y (p) = -y₁ ∫(y₂ g(x)) / W(y₁, y₂) dx + y₂ ∫(y₁ g(x)) / W(y₁, y₂) dx

where g(x) = 8x^(5/2) / (3 - 16x²)

Plugging in the values, we get:

y(p) = -e²ˣ ∫(xe⁻²ˣ 8x^(5/2) / (3 - 16x²)) / -2 dx + xe⁻²ˣ ∫(e²ˣ 8x^(5/2) / (3 - 16x²)) / -2 dx

Simplifying this, we get:

y (p) = (4/5)x^(5/2) - (4/15)x^(7/2)

Therefore, the particular solution is,

⇒ y (p0 = (4/5)x^(5/2) - (4/15)x^(7/2).

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Since both series are convergent, the original series is also convergent.

The given series can be written as Σ 1/(a+2)^p, where p is a positive constant.

We can write this series as the sum of two p-series as follows:

Σ 1/(a+2)^p = Σ 1/(a+2)^(p-1) * 1/(a+2) = Σ 1/(a+2)^(p-1) + Σ 1/(a+2)

The first series is a p-series with p-1 as the exponent, and the second series is a p-series with 1 as the exponent.

To determine the values of p1 and p2, we need to consider the convergence of each of these series separately.

For the first series, we have: Σ 1/(a+2)^(p-1)
This series converges if p-1 > 1, or p > 2.

Therefore, the value of p1 is 2+ε, where ε is a small positive number.

For the second series, we have: Σ 1/(a+2)
This series is a harmonic series, which diverges. Therefore, the value of p2 is 1.

Since both series are convergent, the original series is also convergent.

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Required standard form of perimeter and area are 12x + 4 and 8x² + 12x - 8 respectively.

What is the perimeter of rectangle?

The polynomial expression for the perimeter of a rectangle is P = 2(length + width)

According to given figure, here length is (2x+4) and breadth is (4x-2).

Putting the given values, we get:

P = 2×[2x+4+4x-2]

P = 2[6x + 2]

P = 12x + 4

Therefore, the polynomial expression for the perimeter is 12x + 4 in standard form.

The polynomial expression for the area of the rectangle is A = length × width

Putting the given values,

A = (2x+4) × (4x-2)

A = 8x² + 12x - 8

Therefore, the polynomial expression for the area is 8x² + 12x - 8 in standard form.

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