the joint probability density function of two continuous random variables x and y is given by: f(x,y)

Answers

Answer 1

The value of k is 1/2π, and X and Y are independent because their joint density function factors into a product of their Probability density functions.

To find the value of k, we use the fact that the total area under the joint probability density function equals 1. That is:

integral from 0 to infinity of integral from 0 to infinity of kxye^(-x^2-y^2) dx dy = 1

Using polar coordinates (x = r cos(theta), y = r sin(theta)), the double integral can be written as:

integral from 0 to 2pi of integral from 0 to infinity of k r^3 e^(-r^2) dr d(theta) = 1

The integral over theta is just 2pi, so we can simplify to:

2pi k integral from 0 to infinity of r^3 e^(-r^2) dr = 1

Solving this integral, we get:

2pi k (-1/2) e^(-r^2)| from 0 to infinity = 1

Since e^(-r^2) goes to 0 as r goes to infinity, we have:

pi k = 1/2

Therefore, k = 1/(2pi).

To prove that X and Y are independent, we need to show that the joint probability density function can be factored into the product of the marginal probability density functions:

f(x,y) = f_X(x) * f_Y(y)

The marginal probability density function of X is given by:

f_X(x) = integral from 0 to infinity of kxye^(-x^2-y^2) dy

= kxe^(-x^2) * integral from 0 to infinity of ye^(-y^2) dy

The integral from 0 to infinity of ye^(-y^2) dy is a known integral equal to 1/2, so we have:

f_X(x) = kxe^(-x^2) / 2

Similarly, the marginal probability density function of Y is given by:

f_Y(y) = kye^(-y^2) / 2

Therefore, we have:

f_X(x) * f_Y(y) = k^2 xy e^(-(x^2+y^2))

Comparing this to the joint probability density function given in the problem, we can see that:

f(x,y) = f_X(x) * f_Y(y)

Thus, X and Y are independent.

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____The given question is incomplete, the complete question is given below:

The joint probability density function of two-dimensional continuous random variable (X,Y) is given by > 0, y >0; f(x,y) = 0, otherwise, kxye-(x2+y2). Find the value of k and prove also that X and Y are independent.


Related Questions

Explain the Pythagorean identity in terms of the unit circle.

Answers

The three Pythagorean trigonometric identities, which I’m sure one can find in any Algebra-Trigonometry textbook, are as follows:

sin² θ + cos² θ = 1

tan² θ + 1 = sec² θ

1 + cot² θ = csc² θ

where angle θ is any angle in standard position in the xy-plane.

Consistent with the definition of an identity, the above identities are true for all values of the variable, in this case angle θ, for which the functions involved are defined.

The Pythagorean Identities are so named because they are ultimately derived from a utilization of the Pythagorean Theorem, i.e., c² = a² + b², where c is the length of the hypotenuse of a right triangle and a and b are the lengths of the other two sides.

This derivation can be easily seen when considering the special case of the unit circle (r = 1). For any angle θ in standard position in the xy-plane and whose terminal side intersects the unit circle at the point (x, y), that is a distance r = 1 from the origin, we can construct a right triangle with hypotenuse c = r, with height a = y and with base b = x so that:

c² = a² + b² becomes:

r² = y² + x² = 1²

y² + x² = 1

We also know from our study of the unit circle that x = r(cos θ) = (1)(cos θ) = cos θ and y = r(sin θ) = (1)(sin θ) = sin θ; therefore, substituting, we get:

(sin θ)² + (cos θ)² = 1

1.) sin² θ + cos² θ = 1 which is the first Pythagorean Identity.

Now, if we divide through equation 1.) by cos² θ, we get the second Pythagorean Identity as follows:

(sin² θ + cos² θ)/cos² θ = 1/cos² θ

(sin² θ/cos² θ) + (cos² θ/cos² θ) = 1/cos² θ

(sin θ/cos θ)² + 1 = (1/cos θ)²

(tan θ)² + 1 = (sec θ)²

2.) tan² θ + 1 = sec² θ

Now, if we divide through equation 1.) by sin² θ, we get the third Pythagorean Identity as follows:

(sin² θ + cos² θ)/sin² θ = 1/sin² θ

(sin² θ/sin² θ) + (cos² θ/sin² θ) = 1/sin² θ

1 + (cos θ/sin θ)² = (1/sin θ)²

1 + (cot θ)² = (csc θ)²

3.) 1 + cot² θ = csc² θ

Christina wants to make a painting similar to the mountain scenery in a calendar of dimension 28.5 cm × 18 cm
. Select all the possible dimensions of the canvas she could use for her paintings?
42.75 cm × 51 cm
14.25 cm × 36 cm
57 cm × 27 cm
71.25 cm × 45 cm
46 cm × 68 cm

Answers

The possible dimension of the canvas she could use for her paintings is 71.25 cm × 45 cm

How to solve for possible dimensions that can be used?

The given dimensions of the mountain scenery in the calendar are 28.5 cm × 18 cm.

To make a painting similar to this scenery, Christina needs to use a canvas with the same aspect ratio as the mountain scenery in the calendar. Aspect ratio is the proportional relationship between the width and height of an image or object.

To find the aspect ratio of the mountain scenery in the calendar, we divide the width by the height:

aspect ratio = width / height = 28.5 cm / 18 cm = 1.5833

Now we can calculate the aspect ratio for each of the given canvas options by dividing the width by the height:

42.75 cm × 51 cm has an aspect ratio of 42.75 cm / 51 cm = 0.8382

14.25 cm × 36 cm has an aspect ratio of 14.25 cm / 36 cm = 0.3958

57 cm × 27 cm has an aspect ratio of 57 cm / 27 cm = 2.1111

71.25 cm × 45 cm has an aspect ratio of 71.25 cm / 45 cm = 1.5833

46 cm × 68 cm has an aspect ratio of 46 cm / 68 cm = 0.6765

Only the option 71.25 cm × 45 cm has the same aspect ratio (1.5833) as the mountain scenery in the calendar, so this is the correct answer.

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7. Jada walks up to a tank of water that can hold up to 10 gallons. When it is active, a
drain empties water from the tank at a constant rate. When Jada first sees the tank, it
contains 7 gallons of water. Three minutes later, the tank contains 5 gallons of water.
a. At what rate is the amount of water in the tank changing? Use a signed number, and
include the unit of measurement in your answer.
b. How many more minutes will it take for the tank to drain completely? Explain or
show your reasoning.
c. How many minutes before Jada arrived was the water tank completely full? Explain
or show your reasoning.

Answers

In the word problem,

a)Gallons per minute change is -2/3 gallons per minute

b) 7.5 minutes to drain the other 5 gallons

c)The tank was full 4 1/2 minutes before Jada arrived.

What is word problem?

Word problems are often described verbally as instances where a problem exists and one or more questions are posed, the solutions to which can be found by applying mathematical operations to the numerical information provided in the problem statement. Determining whether two provided statements are equal with respect to a collection of rewritings is known as a word problem in computational mathematics.

a) Gallons per minute change is -2/3 gallons per minute  since it decreases by 2 gallons in 3 minutes.

b)  we have 5 gallons and lose 2/3 gallons per minute

=> (2/3 gal/minute)(x minutes) = 5 gallons

=> x = 5(3/2) = 15/2 = 7.5 minutes to drain the other 5 gallons

c)  There was 10 gallons when the tank is full . That is 3 gallons more than the 7 gals there when Jada arrived.

=>  (2/3)(x) = 3

=> x = 3(3/2) = 9/2 = 4 1/2 minutes

The tank was full 4 1/2 minutes before Jada arrived.

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Can someone help quick i have 6 questions left

Answers

Step-by-step explanation:

remember the trigonometric triangle inside a circle.

sine is the up/down leg, cosine is the left/right leg.

the Hypotenuse (baseline) of the right-angled triangle is the angle-defining radius of the circle.

this is the basic definition inside the norm-circle with radius = 1.

for any other size sine and cosine need to be multiplied by the actual radius for the actual triangle side lengths.

I think you need the answer in simplified fraction and square root notification (you cut that off), but I give you also the decimal result, just in case.

so,

78 = cos(30)×y

78 = (sqrt(3)/2) × y

y = 78 / (sqrt(3)/2) = 2×78/sqrt(3) = 156/sqrt(3) =

= 90.06664199...

x = sin(30) × y = 0.5 × 156/sqrt(3) = 78/sqrt(3) =

= 45.033321...

Mary baked a batch of cookies. The cookies needed 2 1/4 pounds of flour. Jack made 3 1/2 times the amount of cookies that Mary made. How many pounds of flour did Jack need? 63 pounds 8 pounds 8 1/2 pounds 7 7/8 pounds

Answers

Jack needed 7 7/8 pounds of flour for his batch of cookies.

Define Pounds

"Pounds" is a unit of measurement for weight, commonly used in the United States and other countries that have adopted the imperial system of measurement. One pound is equal to 16 ounces, and there are 2.20462 pounds in one kilogram.

Mary used 2 1/4 pounds of flour for her batch of cookies.

finding out how much flour Jack needed, we first need to determine how many cookies Jack made.

Since Jack made 3 1/2 times the amount of cookies that Mary made, we can calculate:

Number of cookies Jack made = 3.5 x Number of cookies Mary made

Let's start by finding out how much flour Mary used per cookie:

2 1/4 pounds of flour / Number of cookies Mary made

So, if Mary made X cookies, Jack made 3.5X cookies.

Now we can set up an equation:

2 1/4 pounds of flour / X cookies = Amount of flour per cookie

Amount of flour per cookie x Number of cookies Jack made = Total amount of flour Jack needed

Amount of flour per cookie = 2 1/4 pounds of flour / X cookies

Number of cookies Jack made = 3.5X

Total amount of flour Jack needed = Amount of flour per cookie x Number of cookies Jack made

Total amount of flour Jack needed = (2 1/4 pounds of flour / X cookies) x (3.5X cookies)

Total amount of flour Jack needed = 7 7/8 pounds of flour

Therefore, Jack needed 7 7/8 pounds of flour for his batch of cookies.

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Find the largest interval which includes x = 0 for which the given initial value problem has a unique solution.(x−4)y′′+2y=x, y(0)=0, y′(0)=1.

Answers

The largest interval which includes x=0 for which the given initial value problem has a unique solution is (-∞,4).

The Differential Equation is : (x-4)y'' + 2y = x,

Now, we convert the differential-equation into the standard form,

We get,

⇒ y'' + 2y/(x-4) = x/(x-4)     ...equation(1)

We know that second order differential-equation is :

⇒ ay'' + by' + cy = R   ...equation(2),

Now, On comparing equation(2) with equation(1),

we get,

The values are : a = 1, b = 0 and c = 2/(x-4), R = x/(x-4),

The Values of "c" , "R" are continuous except at the value "4" because on substituting 4 in x of c and x of R the values of c and R approaches to ∞.

Therefore, The largest-interval which includes x=0 is (-∞,4).

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What are the zeros of the function f(x)= 240x -2x^2

Answers

Answer:

What are the zeros of the function f(x)= 240x -2x^2

Step-by-step explanation:

To find the zeros of the function f(x) = 240x - 2x^2, we need to set f(x) equal to zero and solve for x:

240x - 2x^2 = 0

We can factor out 2x from the left side:

2x(120 - x) = 0

So either 2x = 0 or 120 - x = 0. Solving for x in each case gives:

2x = 0 => x = 0

120 - x = 0 => x = 120

Therefore, the zeros of the function are x = 0 and x = 120.

If z=x+y ;where x varies p Square and y varies 1/p, find the equation connecting z and p . If z=5 when p=2 ,and z=6 when p=4 find z when p=6

Answers

The equation connecting z and p is:

z = x + (1/p)

To find z when p = 6, we can use the values of x and y when p = 2 and p = 4 to calculate the corresponding value of z when p = 6.

When p = 2, x = 4 and y = 1, so z = 5

When p = 4, x = 16 and y = 0.25, so z = 16.25

Therefore, when p = 6, x = 36 and y = 0.167, so z = 36.167

Find the expected value E(X) of the following data. Round your answer to one decimal place. 1 2 3 x 4 5 P(X = x) 0.1 0.2 0.3 0.2 0.2

Answers

The expected value of the given data is 3.2.

The expected value of a discrete random variable is the weighted average of its possible values, where the weights are their respective probabilities. Thus, the expected value of X is given by

E(X) =[tex]1*0.1 + 2*0.2 + 3*0.3 + 4*0.2 + 5*0.2\\ = 0.1 + 0.4 + 0.9 + 0.8 + 1\\ = 3.2[/tex]

Therefore, the expected value of X is 3.2, rounded to one decimal place.

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Find the value of cos H rounded to the nearest hundredth, if necessary.

Answers

Given:-

A right angled triangle right angled at I is given to us.The measure of the longest side (hypotenuse) is 41 and the other two sides are 40 and 9 .

To find:-

The value of cosH .

Answer:-

In the given right angled triangle, cosine is the ratio of base and hypotenuse. In this triangle with respect to angle H , IH is base, IJ is perpendicular and JH is hypotenuse. And their measures are ,

HJ = 41 IH = 40IJ = 9 .

And as mentioned earlier, we can find cosH as ,

[tex]\implies\cos\theta =\dfrac{b}{h} \\[/tex]

[tex]\implies \cos H =\dfrac{IH}{HJ} \\[/tex]

On substituting the respective values, we have;

[tex]\implies\cos H = \dfrac{40}{41}\\[/tex]

[tex]\implies \cos H = 0.975 [/tex]

Value rounded to nearest hundred will be ,

[tex]\implies \cos H =\boxed{0.98}[/tex]

Hence the value of cos H 0.98 .

Answer:

cos H = 0.98 (rounded to the nearest hundredth)

Step-by-step explanation:

Answer:

[tex]\cos R=\dfrac{3}{5}[/tex]

Step-by-step explanation:

To find the cosine of an angle in a right triangle, we can use the cosine trigonometric ratio.

The cosine trigonometric ratio is the ratio of the side adjacent to the angle to the hypotenuse.

[tex]\boxed{\cos \theta=\sf \dfrac{A}{H}}[/tex]

From inspection of the given right triangle HIJ, the side adjacent to angle H is IH, and the hypotenuse is HJ. Therefore:

θ = HA = IH = 40H = HJ = 41

Substitute these values into the formula:

[tex]\implies \cos H=\dfrac{40}{41}[/tex]

As 41 is a prime number, the fraction cannot be reduced any further.

The value of cos H rounded to the nearest hundredth is:

[tex]\implies \cos H=0.975609756...[/tex]

[tex]\implies \cos H=0.98[/tex]

given a function that is continuous on [a,b] and differentiable on (a,b), find a c in (a,b) that satisfies the mean value theorem.

Answers

According to the Mean Value Theorem, if f(x) is a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one c in (a, b) such that:

f(b) - f(a) = f'(c) x (b - a)

To find a c that satisfies the theorem, we need to solve for c:

c = (f(b) - f(a)) / (b - a) / f'(c)

However, this equation cannot be solved directly because c appears on both sides. Instead, we can use an iterative numerical method to approximate c.

One such method is the bisection method, which involves repeatedly dividing the interval (a, b) in half and checking which subinterval contains a root. Here's how it works:

1. Set c = (a + b) / 2, the midpoint of the interval.

2. Evaluate f'(c) using the derivative of f.

3. Calculate f(b) - f(a) and f'(c) x (b - a).

4. If f'(c) x (b - a) = f(b) - f(a), then c is the solution.

5. If f'(c) x (b - a) > f(b) - f(a), then the solution is in the left subinterval (a, c), so set b = c and go to step 1.

6. If f'(c) x (b - a) < f(b) - f(a), then the solution is in the right subinterval (c, b), so set a = c and go to step 1.

Repeat steps 1-6 until the solution is found to a desired degree of accuracy.

Note that this method assumes that f'(x) is continuous on [a, b], which is not always the case. In such cases, other numerical methods may be required.

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solve the quadratic equation 9×^2-15×-6=0​

Answers

Answer:

To solve the quadratic equation 9×^2-15×-6=0, we can use the quadratic formula, which is given by:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.

In this case, a = 9, b = -15, and c = -6, so we can substitute these values into the quadratic formula:

x = (-(-15) ± sqrt((-15)^2 - 4(9)(-6))) / 2(9)

Simplifying this expression gives:

x = (15 ± sqrt(225 + 216)) / 18

x = (15 ± sqrt(441)) / 18

x = (15 ± 21) / 18

So the two solutions to the quadratic equation are:

x = (15 + 21) / 18 = 2

x = (15 - 21) / 18 = -1/3

Therefore, the solutions to the quadratic equation 9×^2-15×-6=0 are x = 2 and x = -1/3.

consider a beam with the cross section shown, made of a material with an allowable stress of 119 mpa.

Answers

The largest couple M that can be applied  to the cross-section of beam  is 108.23 kN.m.

To determine the largest couple M that can be applied to the cross-section shown, we need to calculate the maximum shear stress in the cross-section and ensure that it is less than the allowable stress of the material.

The cross-section can be divided into two rectangles, and the centroid of the combined shape can be found to be at a distance of 2.5 mm from the top edge and 7.5 mm from the left edge. Using the formula for maximum shear stress, τ = VQ/It, we can find the maximum shear stress as:

V = M*d/A, where d is the distance from the neutral axis to the outermost fiber, and A is the area of the cross-section.

d = 5 mm + 2.5 mm = 7.5 mm

A = (10 mm * 5 mm) + (5 mm * 5 mm) = 75 mm^2

Therefore, V = (M*7.5)/75 = M/10

The second moment of area (I) of the combined shape can be found by summing the second moments of area of the two rectangles about their centroids:

I = (1/12 * 10 mm * (5 mm)^3) + 10 mm * (2.5 mm - 7.5 mm)^2 + (1/12 * 5 mm * (5 mm)^3) + 5 mm * (7.5 mm - 2.5 mm)^2

I = 8541.67 mm^4

The elastic modulus of the material is assumed to be constant and is equal to 200 GPa.

Using these values, we can find the maximum shear stress as:

τ = (M7.5)/(108541.67) * (10/5) = M/916.28 MPa

The maximum allowable stress is given as 118 MPa, so we set τ = 118 MPa and solve for M:

M/916.28 = 118 MPa

M = 108227.68 N.mm = 108.23 kN.m (rounded to two decimal places)

Therefore, the largest couple M that can be applied is 108.23 kN.m.

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_____The given question is incomplete, the complete question is given below:

Consider a beam with the cross section shown, made of a material with an allowable stress of 118 MPa. 10 mm 10 mm 801111n- 5mm 5 mm References eBook & Resources Section Break Difficulty: Easy 2· value: 10.00 points Determine the largest couple M that can be applied to the cross section shown. (Round the final answer to two decimal places.) The largest couple M that can be applied is kN m.

Find the exact value of the expression: tan270 degrees

Answers

Answer:

What is the Value of Tan 270 Degrees? The value of tan 270 degrees is undefined. Tan 270 degrees can also be expressed using the equivalent of the given angle (270 degrees) in radians (4.71238 . . .) ⇒ 270 degrees = 270° × (π/180°) rad = 3π/2 or 4.7123 . . .

please marke a a brainalist pls

Translate the phrase into an algebraic expression.
The product of 9 and x

Answers

The algebraic expression for the product of 9 and x is 9x. This can be expressed in steps as follows:

Step 1: Identify the values that are being multiplied together. In this case it is 9 and x.

Step 2: Write the two values side by side and place a multiplication sign between them.

Step 3: The algebraic expression for the product of 9 and x is then written as 9x.

find the value of x in the following figure​

Answers

Answer:

20

Step-by-step explanation:

Question Id: 467224
1
of a pound. Sylvia's dad bought 8 bags of potatoes at the store for a big family Thanksgiving meal. How many pounds c
A bag of potatoes weighs
potatoes did Sylvia's dad buy?
4x A
X
X
B
2
pounds
4 pounds
C 4 pounds
4 of 10
√x D 3 pounds

Answers

We can calculate Sylvia's dad purchased 8/3 pounds of potatoes for the Thanksgiving supper by using division.

Describe division?

Together with addition, subtraction, and multiplication, division is one of the four fundamental operations in mathematics. A bigger group is divided into smaller groups using the division process so that each group has an equal number of items. It is a mathematical technique for distributing and grouping things equally.

The division process is repeated subtraction. It is the polar opposite of multiplying. It is defined as the procedure for developing fair organisations. In order for the larger number taken to equal the multiplication of the smaller numbers, we divide larger numbers into smaller ones while dividing them.

Now in this question,

1 bag of potatoes is given as = 1/3 pounds

Then, 8 bags of potatoes= 8 × 1/3

=8/3 pounds

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A block of mass 2kg is attached to the spring of spring constant 50Nm −1. The block is pulled to a distance of 5 cm from its equilibrium position at x=0 on a horizontal frictionless surface from rest at t = 0. The displacement of the block at any time t is thenA. x= 0.05sin5tmB. x= 0.05cos5tmC. x= 0.5sin5tmD. x= 5sin5tm

Answers

The displacement of the block at any time t is then x= 0.05cos5tm. (option b).

Now, when the block is released, it starts oscillating back and forth about its equilibrium position due to the force exerted by the spring. This motion is described by the equation of motion for a simple harmonic oscillator:

x = Acos(ωt + φ)

The angular frequency ω of the oscillation is given by:

ω = √(k/m)

where k is the spring constant and m is the mass of the block.

Substituting the given values of k and m, we get:

ω = √(50/2) = 5 rad/s

The phase angle φ depends on the initial conditions of the system, i.e., the initial displacement and velocity of the block. Since the block is initially at rest, its initial velocity is zero and the phase angle is zero as well.

Therefore, the equation of motion for the displacement of the block is:

x = 0.05cos(5t)

Hence, option B, x = 0.05cos(5t), is the correct answer.

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A cylindrical aluminum can is being constructed to have a height h of 7 inches. If the can is to have a volume of 56 cubic inches, approximate its radius r. (Hint: V = 2²h)

The radius of the can is about _ inches.
(Type an integer or decimal rounded to two decimal places as needed)

Answers

When rounded to two decimal places, the radius of the can is approximately 1.6 inches.

What exactly is a cylinder?

Surface fοrmed by a straight line mοving parallel tο a fixed straight line and intersecting a fixed planar clοsed curve. a sοlid οr surface defined by a cylinder and twο parallel planes that cut all οf its elements. See Vοlume Fοrmulas Table, particularly fοr the right circular cylinder.

The volume of a cylinder can be calculated using the following formula:

V = πr²h

where

V denotes volume,

r denotes radius, and

h denotes height.

The cylindrical aluminium can has a height of 7 inches and a volume of 56 cubic inches. We can calculate the radius using the volume of a cylinder formula:

V = πr²h

56 = πr²(7)

56 = (22/7)r²(7)

56 = 22r²

56/22 = r²

2.54 = r²

r = [tex]\sqrt{2.54}[/tex]

r ≈ 1.6

As a result, when rounded to two decimal places, the radius of the can is approximately 1.6 inches.

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using the definition of compactness (i.e. you should not use the heine-borel theorem), show that the finite union of compact sets is compact.

Answers

To show that the finite union of compact sets is compact, we need to show that any open cover of the union has a finite subcover.  

Let A and B be two compact sets. Suppose that U is an open cover of A ∪ B. also U is also an open cover of A and an open cover ofB.   Since A is compact, there exists a finite subcover of U that coversA. Let this subcover be{ U1, U2,., Un}.   also, since B is compact, there exists a finite subcover of U that coversB. Let this subcover be{ V1, V2,., Vm}.  

Also the union of these two finite subcovers is a finite subcover of U that covers A ∪B. Specifically, the subcover is{ U1, U2,., Un, V1, V2,., Vm}.   thus, any open cover of the finite union of compact sets A ∪ B has a finite subcover, and  therefore A ∪ B is compact. By induction, we can extend this result to any finite union of compact sets.

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describe all the x -values at a distance of 13 or less from the number 8 . enter your answer in interval notation.

Answers

The set of all x-values that are at a distance of 13 or less from the number 8 in the interval notation is given by  [ -5, 21 ].

The distance between x and 8 is |x - 8|.

Find all the values of x such that |x - 8| ≤ 13.

This inequality can be rewritten as follow,

|x - 8| ≤ 13

⇒ -13 ≤ x - 8 ≤ 13

Now,

Adding 8 to all sides of the inequality we get,

⇒  -13 +  8 ≤ x - 8 + 8 ≤ 13 + 8

⇒ -5 ≤ x ≤ 21

Therefore, all the x-values which are at a distance of 13 or less from the number 8 represented in the interval notation as [ -5, 21 ].

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Instructions: Use the following data set to find the sample statistics for the
following data set. (Round answers to the nearest hundredth, if necessary.
{51, 48, 42, 43, 48, 48, 46, 15, 29, 45,
47, 55, 46, 35, 47, 48, 54, 26, 53, 42}
1.
2.
<<>>
11
4. Mode =
(>>
3. Median =
=
11

Answers

By using sample standard deviation - (1) N or n = 20

(2) X or μ = 43.4

(3) σ = 9.78 and s = 10.04

Why does sample standard deviation exist?

When we talk about a sample standard deviation, we're not talking about population standard deviation. The average amount by which a group of values deviates from their mean is shown by the statistical measure of variability known as the standard deviation.

We are given with the following data set below;

{51, 48, 42, 43, 48, 48, 46, 15, 29, 45, 47, 55, 46, 35, 47, 48, 54, 26, 53, 42}

(1) As we can see in the above data that there are 20 data values in our data set which means that the value of N or n (numner of observations) is 20.

(2) The formula for calculating Mean of the data, i.e. X or μ is given by;

       Mean  =  51 +48+ 42+ 43+ 48 + 48 + 46+ 15+ 29+ 45+ 47+ 55+ 46+ 35+ 47+ 48+ 54+ 26+ 53+ 42/20

                = 868/20 = 43.4

So, the value of  X or μ   is 43.4.

(3) The formula for calculating population standard deviation () is given by;

  Standard deviation, σ  = √(51 - 43.4)² + (48 - 43.4)² + .........+(42 - 43.4)²/20

        =  9.78

Similarly, the formula for calculating sample standard deviation (s) is given by;

       Sample Standard deviation, s =  

                       √(51 - 43.4)² + (48 - 43.4)² + .........+(42 - 43.4)²/20 - 1

                      =  10.04

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3 Open Ended Two fractions have a common denominator
of 8. What could the two fractions be?
3. what cou

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two fractions with a common denominator of 8 can be expressed in the form of a/b and c/8, where a and c are integers. As long as a and c are not both multiples of 8  then these fractions would have a common denominator of 8.

What is common denominator ?

A number that can be divided exactly by all of the denominators in a group of fractions is referred to as a common denominator. 2. A noun that counts. A trait or attitude that all members of a group share is known as a common denominator.

According to the given information:

Since the two fractions have a common denominator of 8, they can be written in the form of a/b and c/8, where a and c are integers.

There are many possible combinations of integers that could satisfy this condition. Here are some examples:

1/8 and 3/8

2/8 (which simplifies to 1/4) and 6/8 (which simplifies to 3/4)

4/8 (which simplifies to 1/2) and 7/8

5/8 and 2/8 (which simplifies to 1/4)

3/8 and 4/8 (which simplifies to 1/2)

In general, any two fractions with a common denominator of 8 can be expressed in the form of a/b and c/8, where a and c are integers. As long as a and c are not both multiples of 8  then these fractions would have a common denominator of 8.

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without calculation, find one eigenvalue and two linearly independent eigenvectors of a d [2 4 5 5 5 5 5 5 5 5 5 3 5]. justify your answer.

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To find the eigenvalues and eigenvectors of a matrix, we need to solve the equation Ax = λx, where A is the matrix, λ is the eigenvalue, and x is the eigenvector. In this case, we have a 3x3 matrix, so we expect to find three eigenvalues and three eigenvectors.

Without performing the calculations, it is difficult to determine the exact eigenvalues and eigenvectors of matrix D. However, based on the structure of the matrix, we can make some observations. We notice that the matrix has a repeating pattern of the number 5, which suggests that 5 is a dominant eigenvalue. Additionally, the matrix is symmetric, so we know that the eigenvectors will be orthogonal.

One possible approach to finding the eigenvectors is to use the Gram-Schmidt process, which is a method for orthogonalizing a set of vectors. We can start with a vector that is parallel to one of the columns of the matrix, and then subtract the projection of that vector onto the next column to obtain a vector that is orthogonal to the first. We can continue this process for each column to obtain a set of orthogonal vectors, which will be our eigenvectors.

Based on these observations, we can hypothesize that 5 is an eigenvalue of D and that we can find two linearly independent eigenvectors by using the Gram-Schmidt process on the columns of the matrix. However, we would need to perform the calculations to confirm this hypothesis and determine the exact values of the eigenvectors.

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Modulus of Rigidity or Shear Modulus (G) The modulus of rigidity or shear modulus is a measure of the rigidity of the material when in "shear' - when it is twisting. It is a ratio of the shear stress and the shear strain of the material: Shear Stress F/ A1 (6.1) Shear Strain Ax/h This formula only works when the material is stressed in its elastic region. דן Polar Moment of Inertia (J) This is an equation that shows the ability of a circular cross-section beam or specimen to resis torsion (twisting). A higher polar moment of inertia shows that the beam or specimen can resist i higher torsion or twisting force. The diameter of the beam determines polar moment of inertia A larger diameter gives a larger polar moment of inertia. #D* J = 32 (6.2) The general equation for the torque in a circular cross-section beam or specimen is: TG (6.3) Where is in radian. Torque The twisting force (torque) at the end of a specimen is the moment of force on the torque arm: T = F x Torque Arm Length (m) (6.4) Shear Stress

Answers

Modulus of rigidity (shear modulus) measures a material's rigidity in shear stress/strain. Polar moment of inertia measures the ability of a circular beam to resist torsion measured with J = 32 / (pi x D^4) , and torque is the twisting force on a specimen measured as  T = G x J x θ.

The modulus of rigidity or shear modulus, represented by G, is a measure of a material's rigidity when subjected to shear stress. Shear stress is the force applied perpendicular to the cross-sectional area of a material, while shear strain is the resulting deformation or twisting of the material.

The equation G = shear stress / shear strain is only valid in the elastic region of a material, where it can return to its original shape after the force is removed.

The polar moment of inertia, J, is a measure of a circular cross-section beam or specimen's resistance to torsion or twisting. A larger diameter of the beam results in a larger polar moment of inertia.

The equation J = 32 / (pi x D^4) is used to calculate the polar moment of inertia, where D is the diameter of the beam.

The torque in a circular cross-section beam or specimen is given by the equation T = G x J x θ, where G is the shear modulus, J is the polar moment of inertia, and theta is the angle of twist in radians.

The torque arm length and the applied force F are used to calculate the twisting force or torque in the specimen.

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hellpppppppppppp. need it for math and am just confused

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Step-by-step explanation:

this is solution which is given above in photo

write a quadratic equation in the form x2+bx+c=0 that has the following roots: -8±4i

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A quadratic equation in the form [tex]x^2[/tex]+bx+c=0 that has the roots -8±4i is [tex]x^2[/tex] + 16x + 48 = 0

If the roots of a quadratic equation are -8+4i and -8-4i, then the factors of the quadratic equation are (x-(-8+4i))(x-(-8-4i))=0.

Simplifying this expression, we get:

(x+8-4i)(x+8+4i) = 0

Expanding this expression, we get:

[tex]x^2[/tex] + (8-4i+8+4i)x + (8-4i)(8+4i) = 0

Simplifying this expression, we get:

[tex]x^2[/tex] + 16x + ([tex]8^2[/tex] - [tex](4i)^2[/tex]) = 0

[tex]x^2[/tex] + 16x + 48 = 0

Therefore, the quadratic equation in the form [tex]x^2[/tex]+bx+c=0 that has the roots -8±4i is:

[tex]x^2[/tex] + 16x + 48 = 0

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the temperature on monday was ₋7∝.
the temperature on tuesday was 5∝ lower than on monday.
the temperature on wednesday was 8∝ higher than on tuesday.
find the temperature on wednesday.

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Answer:

пошел в

Step-by-step explanation:

evaluate the following limits?
a=?
b=?

Answers

The value of the limits are:

lim x → ∞√(9+3x²)/(2+7x) = 1lim x→-∞ √(9+3x^2)/(2+7x)  = -∞.

What is the value of the limits?

The highest degree terms in the numerator and denominator are both 3x^2.

So, as x approaches infinity, the expression behaves like √(3x^2)/√(3x^2) = 1.

Therefore, the limit evaluates to:

lim x → ∞√(9+3x²)/(2+7x) = lim x → ∞(√(3x²)/√(3x²))

lim x → ∞√(9+3x²)/(2+7x) = 1.

(b) The highest degree term in the numerator is 3x^2, while the highest degree term in the denominator is 7x.

Therefore, as x approaches negative infinity, the expression behaves like:

√(3x^2)/√(7x) = √(3/7)(x^2/x) = √(3/7)x.

Since the coefficient of x is positive, the expression approaches negative infinity as x approaches negative infinity.

Therefore, the limit evaluates to:

lim x→-∞ √(9+3x^2)/(2+7x) = lim x→-∞ √(3/7)x

lim x→-∞ √(9+3x^2)/(2+7x)  = -∞.

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Complete question:

Evaluate the following limits. If needed, enter 'INF' for ∞ and '-INF for -∞.

(a)

lim x → ∞√9+3x²/2+7x

(b)

lim x→-∞  √9+3x2/2+7x

The Stamp-M-Out Company manufactures rubber stamps. An inspector finds that there are 10 defective stamps in a sample of 700. a) What is the probability that a randomly selected stamp will be defective?

b) According to Stamp-M-Out Company quality control standards no more than 3.5% of stamps produced may be defective. Does Stamp-M-Out Company need to adjust its manufacturing process to meet this standard?

Answers

A defective stamp is likely to be chosen at random 1.4% of the time, or about 0.014 times. The observed percentage probability  faulty stamps is below the 3.5% maximum permitted rate,

How can I figure out probability?

Name an event from one outcome. Step 2: Compile a list of all potential outcomes, including any positive ones. Step 3: Subtract the number of favorable outcomes from the total number of possibilities that are feasible.

P(faulty) = quantity of defective stamps divided by total quantity of stamps

P(defective) = 10/700.

0.014 P(defective)

Consequently, there is a 1.4% chance (or about 0.014) that a randomly chosen stamp will be flawed.

B-The observed percentage of flawed stamps is:

10 / 700 0.5 – 0.014

Divide this rate by 100 to get the percentage:

0.014 x 100 approximately 1.4%

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