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The statement "Var(aX+b) = \(a^{2}\)Var(X)" has been proved by using the definition of variance.

The variance of a random variable X measures the spread or variability of its values. Using the definition of variance, we can prove that the variance of aX+b, where a and b are constants, is equal to \(a^{2}\) times the variance of X."

To prove the statement, let's start by calculating the variance of aX+b. The variance of a random variable Y is defined as Var(Y) \(= E[(Y - E(Y))^{2}]\), where E(Y) is the expected value of Y.

Using this definition, we have Var(aX+b) \(= E[(aX+b - E(aX+b))^2]\).

Now, simplifying this expression. We know that E(aX+b) = aE(X) + b, as the expected value of a constant times a random variable is equal to the constant times the expected value of the random variable.

Expanding the squared term, we get Var(aX+b) \(= E[(aX+b - aE(X) - b)^{2}]\).

Simplifying further, we have Var(aX+b) = \(E[(a(X - E(X)))^2]\).

Notice that (X - E(X)) is equivalent to X - E(X), as a constant (a) can be factored out.

Now, applying the property of linearity of expectation, we have Var(aX+b) \(= E[a^{2}(X - E(X))^2]\).

Again, using the definition of variance, this can be written as

Var(aX+b) \(= a^{2}E[(X - E(X))^2] = a^{2}Var(X)\)

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Answer:

\(-6 < k < 2\)

Step-by-step explanation:

Given

\(y = x^2 - 2x\)

\(y =kx -4\)

Required

Possible values of k

The general quadratic equation is:

\(ax^2 + bx + c = 0\)

Subtract \(y = x^2 - 2x\) and \(y =kx -4\)

\(y - y = x^2 - 2x - kx +4\)

\(0 = x^2 - 2x - kx +4\)

Factorize:

\(0 = x^2 +x(-2 - k) +4\)

Rewrite as:

\(x^2 +x(-2 - k) +4=0\)

Compare the above equation to: \(ax^2 + bx + c = 0\)

\(a = 1\)

\(b= -2-k\)

\(c =4\)

For the equation to have two distinct solution, the following must be true:

\(b^2 - 4ac > 0\)

So, we have:

\((-2-k)^2 -4*1*4>0\)

\((-2-k)^2 -16>0\)

Expand

\(4 +4k+k^2-16>0\)

Rewrite as:

\(k^2 + 4k - 16 + 4 >0\)

\(k^2 + 4k - 12 >0\)

Expand

\(k^2 + 6k-2k - 12 >0\)

Factorize

\(k(k + 6)-2(k + 6) >0\)

Factor out k + 6

\((k -2)(k + 6) >0\)

Split:

\(k -2 > 0\) or \(k + 6> 0\)

So:

\(k > 2\) or k \(> -6\)

To make the above inequality true, we set:

\(k < 2\) or \(k >-6\)

So, the set of values of k is:

\(-6 < k < 2\)

Answer:

24

Step-by-step explanation:

Clara builds 3 parts of the fence per hour--

if she works for 8 hours-- we know she built 3 fences each hour-- you would multiply,

8(hours) x 3 (fences) = 24  

we know this is right because 3 parts are built for every hour

(3+3+3+3+3+3+3+3 = 24)

she worked for 8 hours and built 3 parts of the fence every hour

in total, she built 24 parts of the fence.

Answer:

Marcos height is 63 inches. Marcos height is 3 inches less than 1.5 times Jennifer’s height.

What is Jennifer height in inches

A: 40in

B : 44in

C: 64.5in

D 58.5in

The answer is 44in(B)

Step-by-step explanation:

Hope this helps :D

Answer:

see explanation

Step-by-step explanation:

Calculate AC and AB using the distance formula

d = \(\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }\)

with (x₁, y₁ ) = A(1,8) and (x₂, y₂ ) = C(4,2)

AC = \(\sqrt{(4-1)^2+(2-8)^2}\)

      = \(\sqrt{3^2+(-6)^2}\)

       = \(\sqrt{9+36}\)

       = \(\sqrt{45}\) = \(\sqrt{9(5)}\) = 3\(\sqrt{5}\)

Repeat with

(x₁, y₁ ) = A(1, 8) and (x₂, y₂ ) = B(2, 6)

AB = \(\sqrt{(2-1)^2+(6-8)^2}\)

      = \(\sqrt{1^2+(-2)^2}\)

       = \(\sqrt{1+4}\)

       = \(\sqrt{5}\)

So AB = \(\sqrt{5}\) and AC = 3\(\sqrt{5}\)

Thus

AC = 3AB

Answer:

Proven below

Step-by-step explanation:

Distance Between Points in the Plane

Given two points A(x,y) and B(w,z), the distance between them is:

\(d=\sqrt{(w-x)^2+(z-y)^2}\)

Let's calculate the distance AC, where A(1,8) and C(4,2):

\(d_{AC}=\sqrt{(4-1)^2+(2-8)^2}\)

\(d_{AC}=\sqrt{3^2+(-6)^2}\)

\(d_{AC}=\sqrt{9+36}=\sqrt{45}\)

Since 45=9*5:

\(d_{AC}=\sqrt{9*5}=3\sqrt{5}\)

Calculate the distance AB, where A(1,8) and B(2,6)

\(d_{AB}=\sqrt{(2-1)^2+(6-8)^2}\)

\(d_{AB}=\sqrt{1^2+(-2)^2}\)

\(d_{AB}=\sqrt{1+4}=\sqrt{5}\)

It follows that:

\(d_{AC}=3d_{AB}\)

Answer:its 5

Step-by-step explanation: cuz

Main Answer:The solution to the initial-value problem is:

y(x) = (\(x^{2}\) + x + 7) / |x|  

Supporting Question and Answer:

What method can be used to solve the initial-value problem ?

The method of integrating factors can be used to solve the  initial-value problem.

Body of the Solution:To solve the given initial-value problem, we can use the method of integrating factors. The equation

x dy/ dx + y = 2x + 1 can be written as follow :

dy/dx + (1/x) × y = 2 + (1/x)

Comparing this equation with the standard form dy/dx + P(x) × y = Q(x), we have:

P(x) = 1/x and

Q(x) = 2 + (1/x)

The integrating factor (IF) can be found by taking the exponential of the integral of P(x):

IF = exp ∫(1/x) dx

= exp(ln|x|)

= |x|

Multiplying the entire equation by the integrating factor, we get:

|x| dy/dx + y = 2|x| + 1

Now, we can rewrite the left side of the equation as the derivative of the product of the integrating factor and y:

d(|x| y)/dx = 2|x| + 1

Integrating both sides with respect to x:

∫d(|x|y)/dx dx = ∫(2|x| + 1) dx

Integrating, we have:

|x| y = 2∫|x| dx + ∫dx

Since the absolute value function has different definitions depending on the sign of x, we need to consider two cases

For x > 0:

∫|x| dx = ∫x dx

= (1/2)\(x^{2}\)

For x < 0:

∫|x| dx = ∫(-x) dx

= (-1/2)\(x^{2}\)

So, combining the two cases, we have:

|xy = 2 (1/2)\(x^{2}\) + x + C   [ C is the intigrating constant ]

Simplifying the equation:

|x|y =\(x^{2}\) + x + C

Now, substituting the initial condition y(1) = 9, we have:

|1|9 = 1^2 + 1 + C

9 = 1 + 1 + C

9 = 2 + C

C = 9 - 2

C = 7

Plugging the value of C back into the equation:

|x|y = \(x^{2}\) + x + 7

To find y(x), we divide both sides by |x|:

y = (\(x^{2}\) + x + 7) / |x|

Final Answer:Therefore, the solution to the initial-value problem is:

y(x) = (\(x^{2}\) + x + 7) / |x|  

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The solution to the initial-value problem is: y(x) = ( + x + 7) / |x|  

The method of integrating factors can be used to solve the  initial-value problem.

To solve the given initial-value problem, we can use the method of integrating factors. The equation

x dy/ dx + y = 2x + 1 can be written as follow :

dy/dx + (1/x) × y = 2 + (1/x)

Comparing this equation with the standard form dy/dx + P(x) × y = Q(x), we have:

P(x) = 1/x and

Q(x) = 2 + (1/x)

The integrating factor (IF) can be found by taking the exponential of the integral of P(x):

IF = exp ∫(1/x) dx

= exp(ln|x|)

= |x|

Multiplying the entire equation by the integrating factor, we get:

|x| dy/dx + y = 2|x| + 1

Now, we can rewrite the left side of the equation as the derivative of the product of the integrating factor and y:

d(|x| y)/dx = 2|x| + 1

Integrating both sides with respect to x:

∫d(|x|y)/dx dx = ∫(2|x| + 1) dx

Integrating, we have:

|x| y = 2∫|x| dx + ∫dx

Since the absolute value function has different definitions depending on the sign of x, we need to consider two cases

For x > 0:

∫|x| dx = ∫x dx

= (1/2)

For x < 0:

∫|x| dx = ∫(-x) dx

= (-1/2)

So, combining the two cases, we have:

|xy = 2 (1/2) + x + C   [ C is the intigrating constant ]

Simplifying the equation:

|x|y = + x + C

Now, substituting the initial condition y(1) = 9, we have:

|1|9 = 1^2 + 1 + C

9 = 1 + 1 + C

9 = 2 + C

C = 9 - 2

C = 7

Plugging the value of C back into the equation:

|x|y =  + x + 7

To find y(x), we divide both sides by |x|:

y = ( + x + 7) / |x|

Therefore, the solution to the initial-value problem is:

y(x) = ( + x + 7) / |x|  

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Hi there!  

»»————-　★　————-««

I believe your answer is:

\(c = \pm\sqrt{43}\)

»»————-　★　————-««  

Here’s why:

I am assuming that the two is an exponent.  

⸻⸻⸻⸻

\(\boxed{\text{Solving for 'c'...}}\\\\c^2 = 43\\----------------\\\rightarrow \sqrt{c^2} = \sqrt{43}\\\\\rightarrow \boxed{c = \pm\sqrt{43}}}\)

⸻⸻⸻⸻

»»————-　★　————-««  

Hope this helps you. I apologize if it’s incorrect.

Answer: 19.6 feet

Step-by-step explanation:

Using the Pythagorean theorem,

\(x^2 +(x+6)^2 =48^2\\\\x^2 +x^2 +12x+36=2304\\\\2x^2 +12x-2268=0\\\\x^2 +6x-1134=0\\\\x=\frac{-6 \pm \sqrt{6^2 -4(1)(-1134)}}{2(1)}\\\\x \approx 30.8 \text{ } (x > 0)\\\\\implies x+(x+6) \approx 67.6\\\\\therefore (x+(x+6))-48 \approx 19.6\)

The equation of the tangent plane to the surface \(xy^2 + yz^2 + zx^2\) = 1 at the point (1, 0, 1) is 2x - y + z = 3.

To find the equation of the tangent plane, we need to determine the normal vector to the surface at the given point (1, 0, 1). The normal vector is perpendicular to the tangent plane.

First, we calculate the partial derivatives of the surface equation with respect to x, y, and z:

∂(\(xy^2 + yz^2 + zx^2\))/∂x = \(y^2 + 2zx\),

∂(\(xy^2 + yz^2 + zx^2\))/∂y =\(2xy + z^2\),

∂(\(xy^2 + yz^2 + zx^2\))/∂z = \(x^{2} +2yz\).

Evaluating these partial derivatives at the point (1, 0, 1), we get:

∂(\(xy^2 + yz^2 + zx^2\))/∂x =\(0^2\) + 2(1)(1) = 2,

∂(\(xy^2 + yz^2 + zx^2\))/∂y = 2(1)(0) + 1^2 = 1,

∂(\(xy^2 + yz^2 + zx^2\))/∂z = \(1^{2}\) + 2(0)(1) = 1.

So, the normal vector to the surface at (1, 0, 1) is (2, 1, 1). Using the point-normal form of a plane equation, we can write the equation of the tangent plane as:

2(x - 1) + 1(y - 0) + 1(z - 1) = 0,

which simplifies to:

2x - y + z = 3.

Therefore, the equation of the tangent plane to the surface \(xy^2 + yz^2 + zx^2\)= 1 at the point (1, 0, 1) is 2x - y + z = 3.

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When t = 4, the expression 3⋅2ᵗ evaluates to 48.

Any mathematical statement with variables, numbers, and an arithmetic operation between them is called an expression or an algebraic expression. For instance, the expression 4m + 5 has the terms 4m and 5 as well as the variable m of the given expression, all of which are separated by the arithmetic sign +.

An expression consists of one or more numbers or variables along with one more operation.

When t=4, the expression 3⋅2ᵗ evaluates to:

3⋅2⁴ = 3⋅16 = 48

Hence, when t = 4, the expression 3⋅2ᵗ evaluates to 48.

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Answer:

y = - 2 | x - 2 | + 4

Step-by-step explanation:

the general equation of an absolute value function is

y = a|x - h| + k

where (h, k ) are the coordinates of the vertex and a is the stretch factor

here (h, k ) = (2, 4 ) , then

y = - 2|x - 2| + 4

Answer:

See below ~

Step-by-step explanation:

Formula for absolute value function :

y = a |x - h| + k

==============================================================

Given :

⇒ a = -2

⇒ Vertex = (h, k) = (2, 4) [based on graph]

=============================================================

Solving by substitution :

⇒ y = -2|x - 2| + 4

-1/7, since parallel lines have equal slopes.

We have shown that asi = λisi for i = 1, ..., n. Also, if x = α1s1...αnsn, then asx = λ(asx)

To solve this Matrix Equations. Now, let's consider x = α1s1...αnsn, where αi represents scalar constants. that asi = λisi, we'll start with the given equation:

a = sλs^(-1)

Multiplying both sides of the equation by s on the right:

as = sλs^(-1) s

Since s^(-1) * s is the identity matrix, we have:

as = sλ

Now, let's multiply both sides of the equation by si:

asi = sλsi

Since λ is a diagonal matrix, it commutes with si:

λsi = siλ

Substituting this back into the equation, we get:

asi = s(siλ)

Now, recall that siλ represents a diagonal matrix with elements si * λii, where λii is the ith diagonal element of λ.

Therefore, we can rewrite the equation as:

asi = λisi

So, we have shown that asi = λisi for i = 1, ..., n.

Now, let's consider x = α1s1...αnsn, where αi represents scalar constants.

To find asx, we substitute x into the expression for a:

asx = a(α1s1...αnsn)

Since matrix multiplication is associative, we can rearrange the order of multiplication:

asx = (aα1)(s1α2s2...αnsn)

From part (a), we know that aα1 = λ1s1α1, so we can substitute that in:

asx = (λ1s1α1)(s1α2s2...αnsn)

Again, using the associativity of matrix multiplication, we rearrange the order:

asx = (λ1s1)(s1α1α2s2...αnsn)

From part (a), we know that asi = λisi, so we can substitute that in:

asx = (λ1s1)(siα1α2s2...αnsn)

Using the associativity again, we rearrange:

asx = λ1(s1si)(α1α2s2...αnsn)

Since s1si is a diagonal matrix, it commutes with the remaining terms:

asx = λ1(siα1α2s2...αnsn)(s1si)

This simplifies to:

asx = λ1(sis1)(α1α2s2...αnsn)

Again, using part (a), we know that asi = λisi, so we substitute that in:

asx = λ1(λisi)(α1α2s2...αnsn)

Since λ1 is a scalar constant, it commutes with the remaining terms:

asx = (λ1λisi)(α1α2s2...αnsn)

Simplifying further:

asx = λ(asx)

We can see that asx is equal to λ times itself, so we have:

asx = λ(asx)

Therefore, we have shown that if x = α1s1...αnsn, then asx = λ(asx).

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Answer:

1888 × \(10^{8}\)

or

188.8 × \(10^{9}\)

or

18.88 × \(10^{10}\)

Step-by-step explanation:

Lets look at this in a more simple way:

\((5.9\) × \(10^{6} )\) × \((3.2\) × \(10^{4} )\)

First, let's move \(10^{6}\) and switch it's places with 3.2:

(5.9 × 3.2) × (\(10^{6}\) × \(10^{4}\))

Then, let's simplify:

18.88 × \(10^{10}\)

Finally, Celebrate! WooHoo! You just learned how to do this!

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There are basically 6 ratios used for finding the elements in Trigonometry. They are called trigonometric functions. The six trigonometric functions are sine, cosine, secant, cosecant, tangent and cotangent.

By using a right-angled triangle as a reference, the trigonometric functions and identities are derived:

sin Î= Opposite Side/Hypotenuse

cos Î= Adjacent Side/Hypotenuse

tan Î= Opposite Side/Adjacent Side

sec Î= Hypotenuse/Adjacent Side

cosec Î= Hypotenuse/Opposite Side

cot Î= Adjacent Side/Opposite Side

All these are taken from a right-angled triangle.

When the height and base side of the right triangle are known, we can find out the sine, cosine, tangent, secant, cosecant, and cotangent values using trigonometric formulas.

The reciprocal trigonometric identities are also derived by using the trigonometric functions.

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Answer:

1.25X+3

Step-by-step explanation:

(-4,8) AND (4,-2)

Rise

-------

Run

8+2 =  10

-4-4=  -8

1.25

from the y intercept 0, the line is a 3, (0,3)

Answer:

(a−9b)(a2+b)

Step-by-step explanation:

A vector space is an abstract mathematical concept that is used to describe a collection of elements, known as vectors.

Vector is a mathematical object that has both magnitude and direction. It can be represented graphically by a line segment with a starting point and an endpoint. Vectors are used in physics, engineering, and mathematics to represent physical quantities such as force, velocity, and acceleration. They can also be used to represent geometric figures such as lines, points, and planes. Vectors are important in many applications, such as navigation, graphics, and robotics.

It is an important concept in linear algebra, and is used to describe the behavior of linear equations and systems.

A vector space is made up of a set of vectors, which can be seen as elements of the space. Each vector is a combination of scalar numbers, called components, and the space is made up of all possible combinations of these components. The components can be real numbers, complex numbers, or even functions.

The vector space is defined by certain operations, called vector addition and scalar multiplication. Vector addition is the process of adding two vectors together, and scalar multiplication is a process of multiplying a vector by a scalar number. Using these operations, the vector space can be used to solve linear equations and systems of equations.

The vector space is also used to describe the geometry of the space. It is used to describe dimensions, angles, and distances between vectors. It is also used to describe shapes and objects within the space, and to describe how they interact with each other.

Finally, the vector space is used to represent linear transformations, which are important in physics and engineering. A linear transformation is a process of mapping a vector to another vector, and is used to describe the behavior of physical systems. This can be used to solve complicated equations and systems of equations.

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Answer:

-8

Step-by-step explanation:

-5+2-3-2

Combine

-3 -3-2

-6-2

-8

Answer:

-8

Step-by-step explanation:

-5+2 = -3

-3-3= -6

-6 -2 = -8

Answer:

66.9 degrees faranhiet

Step-by-step explanation:

That's what it looks like

Answer:  650 miles

Step-by-step explanation:

If it takes 5 hours to go from Paris to Geneva and a train is travelling at 130 miles per hour then multiply the total number of miles per hour by 5 to find how many miles you will have to travel from Paris to Geneva.

130 * 5 =  650

Answer:

The answer should be 8.

Step-by-step explanation:

Answer:

4

Step-by-step explanation:

Rate of change forumla

\(m = \frac{y2 - y1}{x2 - x1} \)

We have our X points, -7 and -2. Now to find the Y value at those places.

For -7, Y is 5

For -2, Y is 25

Now we plug this into the equation

\( = \frac{25 - 5}{ - 2 - ( - 7)} = \frac{20}{5} = 4\)

no noeeeeeeeeeeeeeeeeed

The number that goes beneath the radical symbol is 40.

To find the distance between the two points (-4,3) and (2,5), you can use the distance formula:

distance = √[(x2 - x1)^2 + (y2 - y1)^2]

Plugging in the values, you get:

distance = √[(2 - (-4))^2 + (5 - 3)^2]

Simplifying the expression, you get:

distance = √[6^2 + 2^2]

Which simplifies to:

distance = √[36 + 4]

Simplifying further, you get:

distance = √40

So the distance between the two points is √40. The number that goes beneath the radical symbol is 40.

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Answer:

40

Step-by-step explanation:

calculate the distance d using the distance formula

d = \(\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }\)

with (x₁, y₁ ) = (- 4, 3 ) and (x₂, y₂ ) = (2, 5 )

d = \(\sqrt{(2-(-4))^2+(5-3)^2}\)

  = \(\sqrt{(2+4)^2+2^2}\)

  = \(\sqrt{6^2+4}\)

  = \(\sqrt{36+4}\)

  = \(\sqrt{40}\)

Answer: 46

Step-by-step explanation:

When we are trying to find 8% of 575, so we just multiply 0.08 * 575 and we get an answer of 46.

She needs to complete 20 laps for a total distance of 0.5 kilometer.

1 kilometers = 1000 meters

0.5 kilometers = 500 meters

Length of swimming pool = 25 meters

That means, she covers 25 meters in 1 lap.

Laps to cover 500 meters = 500/25

                                            = 20

Hence, she needs to do 20 laps.

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Given a Math exam with 7 short answer questions and Peter intending to answer three of them, there are a total of 35 different combinations of short answer questions he can choose.

To determine the number of combinations, we can use the concept of combinations in combinatorics. The formula for combinations is given by:

C(n, r) = n! / (r! * (n - r)!)

Where:

C(n, r) represents the number of combinations of choosing r items from a set of n items.

n! denotes the factorial of n, which is the product of all positive integers up to n.

In this case, Peter wants to answer three out of the seven questions, so we can calculate it as:

C(7, 3) = 7! / (3! * (7 - 3)!) = 7! / (3! * 4!)

Simplifying further:

7! = 7 * 6 * 5 * 4 * 3 * 2 * 1

3! = 3 * 2 * 1

4! = 4 * 3 * 2 * 1

C(7, 3) = (7 * 6 * 5 * 4 * 3 * 2 * 1) / [(3 * 2 * 1) * (4 * 3 * 2 * 1)]

After canceling out common terms, we get:

C(7, 3) = 7 * 6 * 5 / (3 * 2 * 1) = 35

Therefore, there are 35 different combinations of short answer questions Peter can choose to answer.

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