Find the length of the indicated line segment

To find P_w(W), we need to determine the probability that W takes on each possible value. Since W is defined as the minimum of X and Y, we can see that W can take on any value between 1 and 100.

To find P_w(W), we need to sum the joint probabilities for all pairs (X, Y) that give us a minimum of W. For example, if we want to find P_w(1), we need to add up all the joint probabilities where either X=1 or Y=1 (since the minimum of X and Y must be 1).

P_w(1) = P(X=1, Y=1) = 1/10000

For P_w(2), we need to add up all the joint probabilities where either X=1 or Y=1 (since the minimum of X and Y must be 2), and so on:

P_w(2) = P(X=1, Y=2) + P(X=2, Y=1) = 2/10000

P_w(3) = P(X=1, Y=3) + P(X=2, Y=3) + P(X=3, Y=1) = 3/10000

Continuing this pattern, we can see that

P_w(w) = w/10000

for w=1, 2, ..., 100.

Therefore, the probability distribution of W is given by

P_w(W) = {1/10000 for W=1, 2, ..., 100; 0 otherwise.}

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Answer:

d. Missing x:2 Missing y:5

Step-by-step explanation:

To determine the missing data values, we need to first determine the equation of the linear function that represents the given data. We can use the two given data points (x=0, y=3) and (x=-1, y=2) to find the slope of the function:

slope = (y2 - y1) / (x2 - x1) = (2 - 3) / (-1 - 0) = -1

Next, we can use the point-slope form of a linear equation to find the y-intercept of the function:

y - y1 = m(x - x1)

y - 3 = -1(x - 0)

y - 3 = -x

y = -x + 3

Using this equation, we can determine the missing data values:

When x=4, y = -4 + 3 = -1.

When x=2, y = -2 + 3 = 1.

Therefore, the correct option is:

d. Missing x:2 Missing y:5

The minimum marginal cost occurs at q = 30.

The lowest average cost is 7.

The marginal cost at q = 30 is 16.

what is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas.

To find the total cost of producing q goods, we need to multiply the average cost by the number of goods produced:

C(q) = a(q) * q

Substituting a(q) = 0.01q² - 0.6q + 13, we get:

C(q) = (0.01q² - 0.6q + 13) * q

= 0.01q³ - 0.6q² + 13q

To find the minimum marginal cost, we need to take the derivative of the average cost function:

a'(q) = 0.02q - 0.6

Setting a'(q) = 0 to find the critical point, we get:

0.02q - 0.6 = 0

q = 30

Therefore, the minimum marginal cost occurs at q = 30.

To find the production level at which the average cost is a minimum, we need to find the minimum point of the average cost function. We can do this by taking the derivative of the average cost function and setting it equal to zero:

a'(q) = 0.02q - 0.6 = 0

q = 30

Therefore, the production level at which the average cost is a minimum is q = 30.

To find the lowest average cost, we can substitute q = 30 into the average cost function:

a(30) = 0.01(30)² - 0.6(30) + 13

= 7

Therefore, the lowest average cost is 7.

To compute the marginal cost at q = 30, we need to take the derivative of the total cost function:

C(q) = 0.01q³ - 0.6q² + 13q

C'(q) = 0.03q² - 1.2q + 13

Substituting q = 30, we get:

C'(30) = 0.03(30)² - 1.2(30) + 13

= 16

Therefore, the marginal cost at q = 30 is 16.

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Answer:

de 200 Pessoa que forum pesquisadas

The volume of the cone with a height of 9 inches and a diameter of 14 inches is 147π cubic inches. So, the correct answer is D).

To calculate the volume of a cone, we use the formula

V = (1/3)πr²h

where "r" is the radius of the base and "h" is the height of the cone.

In this problem, we are given the diameter of the base, which is 14 inches. To find the radius, we divide the diameter by 2

r = 14/2 = 7 inches

We are also given the height, which is 9 inches.

Now we can substitute these values into the formula

V = (1/3)π(7²)(9)

V = (1/3)π(49)(9)

V = (1/3)(441π)

V = 147π

So the volume of the cone is 147π cubic inches.

So the answer is option (D) 147π.

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The solution for the given initial-value problem using Laplace transform is :

y(t) = sin(t) + [1 -cos(t-π)]U(t-2π) - [1 - cos(t-2π)]U(t-2π)

To solve this initial value problem using Laplace transform, we first need to take the Laplace transform of both sides of the equation:

L[y''](s) + L[y](s) = L[f(t)](s)

Using the properties of Laplace transform, we can simplify this expression to:

s^2Y(s) + Y(s) = 1/s - e^(-πs)/s + e^(-2πs)/s

We can now solve for Y(s):

Y(s) = 1/(s^2 + 1) - e^(-πs)/(s^2 + 1) + e^(-2πs)/(s^2 + 1)

Using partial fraction decomposition, we can write this as:

Y(s) = (1/s) - (sin(t)/2) + [1/2 - cos(t-π)]e^(-πs) - [1/2 - cos(t-2π)]e^(-2πs)

Taking the inverse Laplace transform of Y(s), we get:

y(t) = sin(t) + [1 -cos(t-π)]U(t-2π) - [1 - cos(t-2π)]U(t-2π)

This is the same answer as given in the book.

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The eigenvalues and eigenspaces of A are: λ1 = 1 - sqrt(7), eigenspace span{(6 - sqrt(7))/5, 1} and λ2 = 1 + sqrt(7), eigenspace span{(6 + sqrt(7))/5, 1}

To find the eigenvalues and eigenspaces of the matrix A, we need to solve the characteristic equation det(A - λI) = 0, where I is the 2x2 identity matrix.

det(A - λI) = det(leftbracket2.gif 1 5 rightbracket2.gif6 0 - λleftbracket1.gif 0 0 1 rightbracket)

= (2 - λ)(-λ) - (1)(6)

= λ² - 2λ - 6

Using the quadratic formula, we get:

λ = (2 ± sqrt(2² - 4(1)(-6))) / 2

λ = 1 ± sqrt(7)

Therefore, the eigenvalues are λ1 = 1 - sqrt(7) and λ2 = 1 + sqrt(7).

Next, we find the eigenvectors for each eigenvalue by solving the system of equations (A - λI)x = 0.

For λ1 = 1 - sqrt(7), we have:

(A - λ1I)x = leftbracket2.gif 1 5 rightbracket2.gif6 0 - (1 - sqrt(7))leftbracket1.gif 0 0 1 rightbracketx = leftbracket0.gif 0 5 6 - sqrt(7) rightbracketx = 0

Reducing the augmented matrix to row echelon form, we get:

leftbracket0.gif 0 5 6 - sqrt(7) rightbracket --> leftbracket0.gif 0 1 6/(5 + sqrt(7)) rightbracket --> leftbracket0.gif 0 0 0 0 rightbracket

So, the eigenvector corresponding to λ1 is any non-zero solution to the equation 5x2 + (6 - sqrt(7))x1 = 0. We can choose x2 = 1, which gives x1 = (-6 + sqrt(7))/5. Therefore, the eigenspace corresponding to λ1 is span{(6 - sqrt(7))/5, 1}.

For λ2 = 1 + sqrt(7), we have:

(A - λ2I)x = leftbracket2.gif 1 5 rightbracket6 0 - (1 + sqrt(7))leftbracket1.gif 0 0 1 rightbracketx = leftbracket0.gif 0 5 6 + sqrt(7) rightbracketx = 0

Reducing the augmented matrix to row echelon form, we get:

leftbracket0.gif 0 5 6 + sqrt(7) rightbracket --> leftbracket0.gif 0 1 (6 + sqrt(7))/5 rightbracket --> leftbracket0.gif 0 0 0 0 rightbracket

So, the eigenvector corresponding to λ2 is any non-zero solution to the equation 5x2 + (6 + sqrt(7))x1 = 0. We can choose x2 = 1, which gives x1 = (-6 - sqrt(7))/5. Therefore, the eigenspace corresponding to λ2 is span{(6 + sqrt(7))/5, 1}.

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Thus, the value of the give composite function is found as: f(-9) = 38.

Typically, a composite function is a function that is embedded within another function. The process of creating a function involves replacing one function for another. For instance, the composite function of f (x) with g is called f [g (x)] (x). You can read the composite function f [g (x)] as "f of g of x." In contrast to the function f (x), the function g (x) is referred to as an inner function.

Given that:

f(x) = x² + 6x + 11

g(x) = -5x + 1

To find: f(g(2)) , Input x = 2 at  in the function g(x).

g(2) = -5(2) + 1

g(2) = -10 + 1

g(2) = -9

Now,

f(g(2)) = f(-9) = (-9)² + 6(-9) + 11

f(-9) = 81 - 54 + 11

f(-9) = 38

Thus, the value of the give composite function is found as: f(-9) = 38.

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The set of all symmetric 3x3 matrices satisfies all three conditions for a subspace, it is indeed a subspace of M33

To determine whether the set of all symmetric 3x3 matrices is a subspace of M33, we need to check if it satisfies the three conditions for a subspace:

Closure under addition: If A and B are both symmetric 3x3 matrices, then A+B will also be a symmetric 3x3 matrix since [tex](A+B)^T = A^T + B^T = A + B[/tex]. Therefore, the set is closed under addition.

Closure under scalar multiplication: If A is a symmetric 3x3 matrix and c is a scalar, then cA will also be a symmetric 3x3 matrix since [tex](cA)^T = cA^T = cA[/tex]. Therefore, the set is closed under scalar multiplication.

Contains the zero vector: The zero vector in M33 is the matrix of all zeroes. This matrix is also a symmetric 3x3 matrix since all its entries are equal. Therefore, the set contains the zero vector.

Since the set of all symmetric 3x3 matrices satisfies all three conditions for a subspace, it is indeed a subspace of M33.

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Answer:

Step-by-step explanation:To find the inverse of the function f(x), we can follow these steps:

Replace f(x) with y:

y = 3x - 4

Swap x and y:

x = 3y - 4

Solve for y:

x + 4 = 3y

y = (x + 4)/3

Replace y with f^-1(x):

f^-1(x) = (x + 4)/3

Therefore, the inverse of the function f(x) is f^-1(x) = (x + 4)/3.

Note that to find f^-1(x), we swapped x and y in step 2, and solved for y in step 3. The resulting expression for y gives us the inverse function f^-1(x).

We need at least 27 terms of the series to ensure that the remainder is less than 0.0005.

We need to find the number of terms required to satisfy the following inequality:

| R | < 0.0005

where R is the remainder after truncating the series to n terms.

The nth term of the series is given by:

[tex]an = 5n^5[/tex]

The sum of the first n terms can be expressed as:

[tex]Sn = 5(1^5 + 2^5 + ... + n^5)[/tex]

Using the formula for the sum of the first n natural numbers, we can simplify this to:

[tex]Sn = 5(n(n+1)/2)^2(n^2 + n + 1)[/tex]

We can now express the remainder R as:

[tex]R = 5((n+1)^5 + (n+2)^5 + ...)[/tex]

Using the inequality (n+1[tex])^5[/tex] > [tex]n^5[/tex], we can simplify this to:

R < [tex]5((n+1)^5 + (n+1)^5 + ...)[/tex] = [tex]5/(1-(n+1)^(-5))[/tex]

We want R to be less than 0.0005, so we can set up the inequality:

[tex]5/(1-(n+1)^{(-5))[/tex] < 0.0005

Solving for n, we get:

n ≥ 26.86

Since n must be an integer, the smallest value of n that satisfies this inequality is:

n = 27

Therefore, we need at least 27 terms of the series to ensure that the remainder is less than 0.0005.

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Answer:

B,C

Step-by-step explanation:

B and C work.

A and D do not work.

Derivative of r(t) =(e^(-t), 8t - t^3, ln(t)) is (-e^(-t), 8 - 3t^2, 1/t).

Explanation: -

The derivative of the given vector function r(t) = (e^(-t), 8t - t^3, ln(t)) first find the derivative for each component separately and the following formulas.

d/dt (e^(t)) = e^(t)

d/dt (x^(n)) = n x^(n-1)

d/dt (ln(t)) = 1/t

1. For the first component by the use of chain rule, e^(-t), take the derivative with respect to t:
d/dt (e^(-t)) = -e^(-t)

2. For the second component, 8t - t^3, take the derivative with respect to t:
d/dt (8t - t^3) = 8 - 3t^2

3. For the third component, ln(t), take the derivative with respect to t:
d/dt (ln(t)) = 1/t

Now, combine the derivatives of each component to form the derivative vector r'(t):
r'(t) = (-e^(-t), 8 - 3t^2, 1/t)

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The answer to the question for the following equation lim n → [infinity] 1 8 n 5n = lim n → [infinity] eln(y) is that lim n → ∞ (1/(8n^5)) = 0

Given the problem, we need to find the limit as n approaches infinity for the equation: lim n → ∞ (1/(8n^5)).

We'll also need to express this limit in terms of e^(ln(y)).

Let's follow these steps:

1. Write down the given equation: lim n → ∞ (1/(8n^5))

2. Apply the properties of limits: lim n → ∞ (1/n^5) * (1/8)

3. Since 1/8 is a constant, we can rewrite it as lim n → ∞ (1/n^5) * (1/8)

4. Now, find the limit as n approaches infinity for 1/n^5: As n increases, the value of 1/n^5 approaches 0, so lim n → ∞ (1/n^5) = 0.

5. Multiply the limit by the constant: 0 * (1/8) = 0

6. Now, express this limit in terms of e^(ln(y)): Since 0 is our limit, we can write it as e^(ln(0)). However, the natural logarithm of 0 is undefined, so we cannot express the limit in this form.

So, the answer to the question is that lim n → ∞ (1/(8n^5)) = 0, but it cannot be expressed in terms of e^(ln(y)).

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Answer:

--------------------------

Given x-values in the table.

Use the equation of the function to find the corresponding y-values:

When x = - 27:

When x = 0:

When x = 27:

So the missing numbers are: - 3, 0 and 3.

The matching choice is B.

The sampling distribution of the sample mean of the observations on the amount of nitrogen removed by the four buffer strips with widths of 6 feet is the theoretical probability distribution of all possible sample means that could be obtained by randomly selecting samples of size 6 from the population of nitrogen removal observations.

Assuming the sample means are normally distributed, the mean of the sampling distribution of the sample means would be equal to the population mean of nitrogen removal by the buffer strips, while the standard deviation would be equal to the population standard deviation divided by the square root of the sample size.

The Central Limit Theorem states that, as the sample size increases, the sampling distribution of the sample means becomes increasingly normal, regardless of the distribution of the original population. This means that, if we take enough samples of size 6, the distribution of their means will approach a normal distribution.

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For the given graphs of a linear equation and a quadratic equation. There are 2 number of solutions to the system.

The coordinates of a ordered pair(s) which satisfy all of the system's equations make up the solution set. In other words, the equations will be true for certain x and y numbers. As a result, when a system of equations is graphed, all of the places at which the graphs cross are the solution.

Depending on how many solutions a system of linear equations has, it can be classified. Systems of equations fall into one of two categories:

For the question:

As, there are two intersecting points for the graphs of a linear equation and a quadratic equation. Thus, there are 2 number of solutions to the system.

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The sum of the geometric series 4 + 3 + 9/4 + 27/16 +...  is 16.

To find the sum of the given geometric series, we need to determine the common ratio (r) and the first term (a).

We can see that each term of the series is obtained by multiplying the previous term by 3/4. Therefore, the common ratio is 3/4.

The first term (a) is 4.

Using the formula for the sum of a finite geometric series, we can find the sum of the first n terms of the series

Sn = a(1 - r^n) / (1 - r)

Substituting the values of a and r, we get

Sn = 4(1 - (3/4)^n) / (1 - 3/4)

Simplifying the expression

Sn = 16(1 - (3/4)^n)

Since this is an infinite geometric series (the ratio r is less than 1), the sum of the series can be found by taking the limit as n approaches infinity

S = [tex]\lim_{n \to \infty}[/tex] 16(1 - (3/4)^n)

S = 16(1 - 0) = 16

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The given question is incomplete, the complete question is:

Find the sum of the geometric series  4 + 3 + 9/4 + 27/16 +...

The function y=f(x)=2x−ln(2x) has a global minimum at x=1 and a global maximum at x=2.5 within the interval 1<x<2.5, and there are no local non-global maximum points within the interval.

To find the points where y = f(x) = 2x - ln(2x) has a global maximum, global minimum, and local, non-global maximum on the interval 1 < x < 2.5, we need to find the critical points and analyze the behavior of the function.

1. Find the first derivative: f'(x) = 2 - (1/x)
2. Set f'(x) to zero and solve for x: 2 - (1/x) = 0 => x = 1/2 (but it's outside the interval, so discard it)

So, the critical point of f(x) is at x= 1/2. However, we need to check if this critical point is within the given interval 1<x<2.5. Since 1/2​ is not within that interval, we can conclude that f(x) does not have any critical points within the given interval.

Since there's no critical point within the interval, we need to check the endpoints of the interval:

1. f(1) = 2(1) - ln(2(1)) = 2 - ln(2)
2. f(2.5) = 2(2.5) - ln(2(2.5)) = 5 - ln(5)

Since f(1) < f(2.5), we can conclude that:
Global minimum: At x = 1, f(x) ≈ 2 - ln(2) ≈ 0.31
Global maximum: At x = 2.5, f(x) ≈ 5 - ln(5) ≈ 3.39

So, we can see that f( 1 ) is the global minimum point and f( 2.5 ) is the global maximum point within the given interval.

Local, non-global maximum: Not present within the interval 1 < x < 2.5

In summary, the function y=f(x)=2x−ln(2x) has a global minimum at x=1 and a global maximum at x=2.5 within the interval  1<x<2.5, and there are no local non-global maximum points within the interval.

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Answer: a

Step-by-step explanation:

Answer:

2

Step-by-step explanation:

Since line m is parallel to line n, any line that is perpendicular to line n will also be perpendicular to line m. Therefore, we just need to determine the number of lines perpendicular to line n that pass through point p.

If we draw a diagram, we can see that there are two such lines: one that is perpendicular to line n and passes through the endpoint of line segment p on line m, and another that is perpendicular to line n and passes through the other endpoint of line segment p on line m. These two lines are the only ones that are perpendicular to line n and pass through point p, so the answer is 2.

The equation of the circle with center at (-2, 3) and passing through the point (1, 8) is (x + 2)² + (y - 3)² = 34.

The standard form equation of a circle with center (h, k) and radius r is expressed as:

(x - h)² + (y - k)² = r²

Given that: the center of the circle is (-2, 3) and the circle passes through the point (1, 8).

First, we find the radius of the circle, we can use the distance formula between the center and the point on the circle:

r = √[(x2 - x1)² + (y2 - y1)²]

r = √[(1 - (-2))² + (8 - 3)²]

r = √[3² + 5²]

r = √34

So, the equation of the circle is:

(x - (-2))² + (y - 3)² = (√34)²

Simplifying and expanding the equation, we get:

(x + 2)² + (y - 3)² = 34

Therefore, the equation of the circle is (x + 2)² + (y - 3)² = 34.

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The statement "The shape of a sampling distribution of sample means that follows the requirements of the central limit theorem will be approximately bell-shaped" is true.

The central limit theorem states that as the sample size increases, the distribution of sample means approaches a normal distribution. This normal distribution is approximately bell-shaped. Therefore, the shape of a sampling distribution of sample means that follows the requirements of the central limit theorem will be approximately bell-shaped.

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Answer:

The bicycle will be worth less than 1000 after 4 years.

Step-by-step explanation:

The derivative value of dy for the given values of x and dx. y = x 1 x − 1 , x = 2, dx = 0.05 is -0.05.

The given function is y = x/(x-1). We need to find dy when x = 2 and dx = 0.05.

First, we find the derivative of the function with respect to x using the quotient rule:

y' = [(x-1)(1) - x(1)] / (x-1)²

= -1 / (x-1)²

Next, we substitute x = 2 into the derivative expression to get the slope of the tangent line at x = 2:

y' = -1 / (2-1)² = -1

This means that for every 1 unit increase in x, y decreases by 1 unit. So when dx = 0.05, the change in y is:

dy = y' × dx = (-1) × 0.05 = -0.05

Therefore, when x = 2 and dx = 0.05, the value of dy is -0.05. The main mathematics topic used here is calculus, specifically the quotient rule and finding the derivative.

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The derivative value of dy for the given values of x and dx. y = x 1 x − 1 , x = 2, dx = 0.05 is -0.05.

The given function is y = x/(x-1). We need to find dy when x = 2 and dx = 0.05.

First, we find the derivative of the function with respect to x using the quotient rule:

y' = [(x-1)(1) - x(1)] / (x-1)²

= -1 / (x-1)²

Next, we substitute x = 2 into the derivative expression to get the slope of the tangent line at x = 2:

y' = -1 / (2-1)² = -1

This means that for every 1 unit increase in x, y decreases by 1 unit. So when dx = 0.05, the change in y is:

dy = y' × dx = (-1) × 0.05 = -0.05

Therefore, when x = 2 and dx = 0.05, the value of dy is -0.05. The main mathematics topic used here is calculus, specifically the quotient rule and finding the derivative.

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The required answer is the inverse of J(X0) does not exist.

The Jacobian matrix represents the differential of f at every point where f is differentiable. In detail, if h is a displacement vector represented by a column matrix, the matrix product J(x) ⋅ h is another displacement vector, that is the best linear approximation of the change of f in a neighborhood of x, if f(x) is differentiable at x.

To find the Jacobian matrix J for this initial guess xo of (0,-1), we first need to find the partial derivatives of each equation with respect to x and y:

∂f1/∂x = 0     ∂f1/∂y = 0
∂f2/∂x = 0     ∂f2/∂y = 1

Therefore, the Jacobian matrix J is:

J = [∂f1/∂x ∂f1/∂y; ∂f2/∂x ∂f2/∂y] = [0 0; 0 1]

Next, to find J(20), we simply substitute x=20 and y=20 into the Jacobian matrix:

J(20) = [0 0; 0 1]

Finally, we can use Newton's method to find the next iteration X1:

X1 = X0 - J(X0)^(-1) * F(X0)

where X0 is the initial guess, J(X0) is the Jacobian matrix at X0, and F(X0) is the function evaluated at X0.

Plugging in the values we have:

X0 = (0,-1)
J(X0) = [0 0; 0 1]
F(X0) = [2 + 6; 5.23 + (-1) - 5] = [8; 0.23]

Now, we need to find the inverse of J(X0):

J(X0)^(-1) = [1/0 0; 0 1/1] = [undefined 0; 0 1]

Since the inverse of J(X0) does not exist, we cannot proceed with one iteration of Newton's method.
The given nonlinear system of equations is not written correctly. Please provide the correct system of equations, including the variables, so I can help you find the Jacobian matrix and apply Newton's method.

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0.47474747 can be expressed as the ratio of integers 47/33.

We can write it as 47/100.

To express 0.47474747 as a ratio of integers, we can write it as 47/99. This is because the repeating decimal can be represented as an infinite geometric series:

0.47474747 = 0.47 + 0.0047 + 0.000047 + ...

The sum of this infinite series can be found using the formula S = a/(1-r), where a is the first term (0.0047) and r is the common ratio (0.01).

S = 0.0047/(1-0.01) = 0.0047/0.99 = 47/9900

Simplifying this fraction by dividing both numerator and denominator by 100 gives 47/990, which can be further simplified by dividing both numerator and denominator by 3 to get 47/33.

Therefore, 0.47474747 can be expressed as the ratio of integers 47/33.

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The explicit formula for the sequence [tex]{a_n} is a_n = 2n + 4[/tex].

The pattern suggests that the sequence is increasing by 2 for each term. So we can write the formula for the nth term as:

[tex]a_n = a_1 + (n-1)d[/tex]

where a_1 is the first term, d is the common difference (which is 2 in this case), and n is the term number.

Substituting the given values, we get:

[tex]a_n = 6 + (n-1)2[/tex]

Simplifying, we get:

[tex]a_n = 2n + 4[/tex]

Therefore, the explicit formula for the sequence. [tex]{a_n} is a_n = 2n + 4[/tex]

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The values of a and b given vectors are a = 4 and b = 8.

for a set of vectors to be orthogonal, the dot product of any two distinct vectors in the set should be zero.

Let's check if this condition is satisfied for the given vectors:

u_1 • u_2 = (2)(4) + (1)(-5) + (-1)(3) = 8 - 5 - 3 = 0

u_1 • u_3 = (2)(2) + (1)(a) + (-1)(b) = 4 + a - b

u_2 • u_3 = (4)(2) + (-5)(a) + (3)(b) = 8 - 5a + 3b

We need to find values of a and b such that u_1, u_2, and u_3 form an orthogonal set. So we need u_1 • u_3 = 0 and u_2 • u_3 = 0.

u_1 • u_3 = 4 + a - b = 0, so a - b = -4 ...(1)

u_2 • u_3 = 8 - 5a + 3b = 0, so 5a - 3b = 8 ...(2)

We now have two equations in two variables (a and b). Solving these equations simultaneously, we get:

a = 4, b = 8

Substituting these values back into the dot products, we can check that u_1, u_2, and u_3 form an orthogonal set:

u_1 • u_2 = 0

u_1 • u_3 = 4 + 4 - 8 = 0

u_2 • u_3 = 8 - 20 + 24 = 0

Therefore, the values of a and b that make u_1, u_2, and u_3 an orthogonal set are a = 4 and b = 8.

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The statement "f(w ∩ x) = f(w) ∩ f(x)" is false in general for a function f: a → b and subsets w, x ⊆ a.

To see why, consider the following counterexample:

Let f: {1,2} → {1} be the constant function defined by f(1) = f(2) = 1.

Let w = {1} and x = {2}. Then w ∩ x = ∅, the empty set. Therefore, f(w ∩ x) = f(∅) = ∅, the empty set.

However, f(w) = {1} and f(x) = {1}, so f(w) ∩ f(x) = {1} ∩ {1} = {1}.

Since ∅ ≠ {1}, we can see that the equation f(w ∩ x) = f(w) ∩ f(x) does not hold in this case. Therefore, the statement is false in general.

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The statement "f(w ∩ x) = f(w) ∩ f(x)" is false in general for a function f: a → b and subsets w, x ⊆ a.

To see why, consider the following counterexample:

Let f: {1,2} → {1} be the constant function defined by f(1) = f(2) = 1.

Let w = {1} and x = {2}. Then w ∩ x = ∅, the empty set. Therefore, f(w ∩ x) = f(∅) = ∅, the empty set.

However, f(w) = {1} and f(x) = {1}, so f(w) ∩ f(x) = {1} ∩ {1} = {1}.

Since ∅ ≠ {1}, we can see that the equation f(w ∩ x) = f(w) ∩ f(x) does not hold in this case. Therefore, the statement is false in general.

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