Problem 6-33 Consider a system having four components with reliabilities through time t of: (1) 0.80 (2) 0.66(3) 0.78 (4) 0.89

Approximate f(3.99) by adding the differential to f(4): f(3.99) ≈ 2 + (-0.0025) = 1.9975. Using a calculator, the square root of 3.99 is approximately 1.9975.

To approximate the value of an expression using differentials, we need a function and a point close to the given value. It seems that some information is missing from your question, so I will provide an example using a different expression.
Suppose we want to approximate the square root of 3.99 using differentials. We can use the function f(x) = √x and the point x = 4 (which is close to 3.99).
First, find the derivative of f(x): f'(x) = 1 / (2√x)
Now, calculate the differential: dy = f'(x) * dx
Since x = 4 and dx = 3.99 - 4 = -0.01, we get: dy = f'(4) * (-0.01) = 1 / (2√4) * (-0.01) = -0.0025
Now, find the value of f(x) at x = 4: f(4) = √4 = 2
Finally, approximate f(3.99) by adding the differential to f(4): f(3.99) ≈ 2 + (-0.0025) = 1.9975
Using a calculator, the square root of 3.99 is approximately 1.9975. The answers match up to four decimal places.

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We have proven that [tex](I - A)^2 = I - A[/tex], which means I - A is also idempotent and a square matrix.

To show that I - A is idempotent, we need to show that[tex](I - A)^2 = I - A[/tex].

Expanding:

[tex](I - A)^2 = (I - A)(I - A) = I^2 - IA - AI + A^2 = I - 2A + A^2[/tex]

Since A is idempotent, we know that A^2 = A. Substituting that into above equation, we get:
[tex](I - A)^2 = I - 2A + A = I - A[/tex]

Therefore, we have shown that[tex](I - A)^2 = I - A[/tex], which means that I - A is also idempotent.
Hi! I'd be happy to help you with your question involving idempotent matrices. To show that I - A is also idempotent, we need to prove that [tex](I - A)^2 = I - A[/tex], where I is the identity matrix. Here are the step-by-step calculations:

1. Calculate [tex](I - A)^2[/tex]:

[tex](I - A)^2 = (I - A)(I - A)[/tex]

2. Expand the product using matrix multiplication:

(I - A)(I - A) = I(I) - I(A) - A(I) + A(A)

3. Apply the properties of the identity matrix and the definition of idempotent matrix:

I(I) = I, I(A) = A, A(I) = A, and A(A) =[tex]A^2[/tex] = A

So, the expression becomes:

I - A - A + A

4. Simplify the expression:

I - A - A + A = I - A

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Connect the endpoints of each new line segment to create the scaled polygon a.

To draw a scaled copy of polygon a using a scale factor of 2, follow these steps:

Choose a point on the paper that will be the centre of your scaling transformation.

Draw a line from the centre point to each vertex of the original polygon a.

Measure the length of each line segment.

Multiply each length measurement by a scale factor of 2.

From the centre point, draw a new line for each scaled segment with the new, scaled length.

Connect the endpoints of each new line segment to create the scaled polygon a.

Remember to label your scaled polygon a to indicate that it is a scaled copy and to note the scale factor used.

Complete Question:

Here are 3 polygons.

(Below mentioned diagram)

a) Draw a scaled copy of polygon a suing a scale factors of 2.

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The probability that all five balls drawn out of the box at random are white is approximately 0.001082.

To find the probability that all five balls drawn out of the box at random are white, follow these steps:

1. Calculate the total number of balls in the box: 5 white balls + 6 black balls = 11 balls
2. Determine the probability of drawing the first white ball: 5 white balls / 11 total balls = 5/11
3. After drawing the first white ball, there are now 4 white balls and 10 total balls remaining. Determine the probability of drawing the second white ball: 4 white balls / 10 total balls = 4/10
4. After drawing the second white ball, there are now 3 white balls and 9 total balls remaining. Determine the probability of drawing the third white ball: 3 white balls / 9 total balls = 1/3
5. After drawing the third white ball, there are now 2 white balls and 8 total balls remaining. Determine the probability of drawing the fourth white ball: 2 white balls / 8 total balls = 1/4
6. After drawing the fourth white ball, there is now 1 white ball and 7 total balls remaining. Determine the probability of drawing the fifth white ball: 1 white ball / 7 total balls = 1/7

To find the probability of all five events occurring, multiply the probabilities together: (5/11) * (4/10) * (1/3) * (1/4) * (1/7) = 0.00108225108

So, the probability that all five balls drawn out of the box at random are white is approximately 0.001082, or 0.1082%.

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The composite function g(f(x)) = 3x² + 9x - 25. Given that f(x) = x²  + 3x - 8 and g(x) = 3x - 1, we need to find the composite function g(f(x)). This means we'll substitute the entire f(x) function into the g(x) function.

Step 1: Identify f(x) and g(x)
f(x) = x²  + 3x - 8
g(x) = 3x - 1
Step 2: Substitute f(x) into g(x) for the variable x
g(f(x)) = 3(f(x)) - 1
Step 3: Replace f(x) with its expression, which is x^2 + 3x - 8
g(f(x)) = 3(x²  + 3x - 8) - 1
Step 4: Distribute the 3 to each term inside the parentheses
g(f(x)) = 3x²  + 9x - 24 - 1
Step 5: Combine like terms (in this case, just the constants)
g(f(x)) = 3x²  + 9x - 25
So, the composite function g(f(x)) = 3x²  + 9x - 25. If anyone has difficulty with these problems, we recommend reviewing Sections 1.1-1.3 for a better understanding of function compositions and related topics.

To find the function g o f, we need to substitute the function f(x) into the function g(x) wherever we see x. So, g o f(x) = g(f(x)).
First, we find f(x):
f(x) = x²  + 3x - 8
Now we substitute f(x) into g(x):
g(f(x)) = g(x²  + 3x - 8)
= 3(x²  + 3x - 8) - 1
= 3x²  + 9x - 25
Therefore, g o f(x) = 3x²  + 9x - 25.
Given that f(x) = x²  + 3x - 8 and g(x) = 3x - 1, we need to find the composite function g(f(x)). This means we'll substitute the entire f(x) function into the g(x) function.

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Answer:   $7.15

Step-by-step explanation: So first we divide 440 cents from 8.That equals $0.55 that per 1/8 pound of coffee beans. So then 5+8 = 13 and the you times it by 55.You get 715 cents and then you cover it into dollars and get 7 dollars and 15 cents.

The probability of x being greater than 5 is approximately 0.3679.

To compute p(x > 5) for x with an exp(0.2) distribution, we can use the probability density function (PDF) of the exponential distribution:

f(x) = 0.2e^(-0.2x)

The probability of x being greater than 5 is given by the integral of the PDF from 5 to infinity:

p(x > 5) = integral from 5 to infinity of f(x) dx

= integral from 5 to infinity of 0.2e^(-0.2x) dx

= [-e^(-0.2x)] from 5 to infinity

= e⁻¹ˣ

= 0.3679

Therefore, the probability of x being greater than 5 is approximately 0.3679.

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The perimeter, in units, of the triangle is x² - 9

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

The three sides of a triangle have lengths of

x units, (x-4) units, and (x² - 2x - 5) units

The perimeter, in units, of the triangle is the sum of the side lenths

So, we have

Perimeter = x + x - 4 + x² - 2x - 5

Evaluate the like terms

So, we have

Perimeter = x² - 9

Hence, the perimeter is x² - 9

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By using bayes’ theorem;

P(h|a) = 0.0625.

We can use Bayes' theorem to calculate P(h|a) as follows:

P(h|a) = P(a|h) * P(h) / P(a)

where P(a) is the total probability of event a, given by:

P(a) = P(a|h) * P(h) + P(a|m) * P(m) + P(a|l) * P(l)

We are given that P(a|h) = 0.2, P(a|m) = 0.3, and P(a|l) = 0.2. We are also given that the events h, m, and l are collectively exhaustive, which means that their probabilities add up to 1. Therefore, we have:

P(m) + P(l) = 0.4 + P(l) = 1 - P(h) = 0.9

Solving for P(l), we get:

P(l) = 0.5

Now we can use Bayes' theorem to calculate P(h|a) as follows:

P(h|a) = P(a|h) * P(h) / P(a)

= 0.2 * 0.1 / (0.2 * 0.1 + 0.3 * 0.4 + 0.2 * 0.5)

= 0.02 / 0.32

= 0.0625

Therefore, P(h|a) = 0.0625.

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Yes, the double torus is a surface.

A surface is a two-dimensional manifold that is locally Euclidean, meaning that every point on the surface has a neighborhood that is homeomorphic (topologically equivalent) to an open disk in the Euclidean plane.

The double torus, like other tori, meets this definition because each point on the double torus has a neighborhood that can be mapped to an open disk in the Euclidean plane, preserving the local topological structure.

A surface is a two-dimensional object that can be embedded in three-dimensional space, and the double torus fits this definition. It can be visualized as a doughnut shape with two holes, and can be constructed by taking two copies of a standard torus and gluing them together along their inner holes.

The resulting object is a closed, orientable surface that can be smoothly deformed without tearing or intersecting itself. Therefore, the double torus is indeed a surface.

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The explicit formula for the sequence {an} is:

an = 14 + 5n + 100(n-1) for even n
an = -(14 + 5n + 100(n-1)) for odd n

To find the explicit formula for the sequence {an} with the given terms 14, -19, 114, -119, 124, ...,
Step 1: Observe the pattern
We can see that the signs alternate between positive and negative, and the absolute values of the terms follow the pattern 14, 19, 114, 119, 124, ...
Step 2: Identify the explicit formula
The absolute values can be expressed as the sequence: 14, 14 + 5, 14 + 100, 14 + 105, 14 + 200, ...
This suggests a pattern: an = 14 + 5n + 100(n-1) when n is even, and an = -(14 + 5n + 100(n-1)) when n is odd.

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The constant (equilibrium) solution to the differential equation x′(t) = x(t) is x(t) = Ce^(t), where C is a constant. This equilibrium is stable if C < 0, semi-stable if C = 0, and unstable if C > 0.

To find the equilibrium solutions, we set x′(t) = x(t). This gives us the equation:

x′(t) - x(t) = 0

This is a first-order linear homogeneous differential equation. The general solution is x(t) = Ce^(t), where C is a constant determined by the initial condition. To determine stability, we analyze how x(t) behaves as t goes to infinity:

1. If C < 0, x(t) approaches 0 as t goes to infinity, which means the equilibrium is stable.
2. If C = 0, x(t) remains constant at 0, which indicates a semi-stable equilibrium.
3. If C > 0, x(t) grows unbounded as t goes to infinity, indicating an unstable equilibrium.

So, the constant (equilibrium) solution x(t) = Ce^(t) can be stable, semi-stable, or unstable depending on the value of C.

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Two events E and F are independent if the occurrence of event E in a probability experiment does not affect the probability of event F.

Probability is the measure of the likelihood or chance of an event occurring, expressed as a number between 0 and 1, with 0 indicating impossibility and 1 indicating certainty.

According to the given information:

In probability theory, two events E and F are independent if the occurrence of one event does not affect the probability of the other event occurring. In other words, the probability of event F occurring is the same whether or not event E occurs. This is different from mutually exclusive events, which cannot occur simultaneously, and dependent events, where the occurrence of one event affects the probability of the other event occurring. Disjoint events are similar to mutually exclusive events, and conditional events involve the probability of an event given that another event has already occurred. Understanding the concepts of independent and dependent events is crucial in probability and statistics, as it can help in calculating joint probabilities and conditional probabilities.

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The standard deviation of the wait time is a measure of how spread out the wait times are from the average wait time. It tells us how much variability or dispersion there is in the wait times.

To calculate the standard deviation of the wait time, we need to first find the average wait time and then calculate the difference between each individual wait time and the average wait time. We then square each of these differences, add them up, divide by the number of wait times, and finally take the square root of that result. This gives us the standard deviation. The answer to your specific question will depend on the data provided and the calculations performed.

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The standard deviation of the wait time is a measure of how spread out the wait times are from the average wait time. It tells us how much variability or dispersion there is in the wait times.

To calculate the standard deviation of the wait time, we need to first find the average wait time and then calculate the difference between each individual wait time and the average wait time. We then square each of these differences, add them up, divide by the number of wait times, and finally take the square root of that result. This gives us the standard deviation. The answer to your specific question will depend on the data provided and the calculations performed.

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To find the tangent line approximation for 1 ‾‾‾‾‾√ near =2, we first need to find the derivative of the function f(x) = 1 ‾‾‾‾‾√.

Using the power rule of differentiation, we get: f'(x) = 1/2(x)^(-1/2) Now, we can substitute the value a = 2 and f'(a) = f'(2) = 1/2(2)^(-1/2) = 1/2√2 into the formula: f(x) = f^1(a)(x-a) + f(a) to get the equation of the tangent line at x = 2: y = 1/2√2(x-2) + 1 Therefore, the tangent line approximation for 1 ‾‾‾‾‾√ near =2 is y = 1/2√2(x-2) + 1,

where the slope of the line is given by f'(2) and the point (2,1) lies on the line.  Use the formula for the tangent line approximation: f(x) ≈ f^1(a)(x-a) + f(a). For x near 2, f(x) ≈ (1/2√2)(x-2) + √2. This is the tangent line approximation for f(x) = √x near x = 2.

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The area of the regular hexagon is approximately 23.4 square units.

We have,

To find the area of a regular hexagon with a given radius, we can use the formula:

Area = (3√3 / 2) x r²

Where r is the radius of the hexagon.

Substituting r = 3 into the formula, we get:

Area = (3√3 / 2) x 3^2

    = (3√3 / 2) x 9

    = 23.38 (rounded to the nearest tenth)

Therefore,

The area of the regular hexagon is approximately 23.4 square units.

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The particular solution for the differential equation is:

yp = 1/2cos(x) + 1/4sin(x)

To find the form of the particular solution for the differential equation y''-2y'y=1*sin(x), we can use the method of undetermined coefficients.

Assuming a particular solution of the form:

yp = Acos(x) + Bsin(x)

We can find the first and second derivatives of yp:

yp' = -Asin(x) + Bcos(x)

yp'' = -Acos(x) - Bsin(x)

Substituting these into the differential equation, we get:

[tex](-Acos(x) - Bsin(x)) - 2(-Asin(x) + Bcos(x))(-Asin(x) + Bcos(x)) = sin(x)[/tex]

Expanding the terms, we get:

[tex](-Acos(x) - Bsin(x)) + 2(Asin^2(x) - 2ABsin(x)cos(x) + Bcos^2(x)) = sin(x)[/tex]

Simplifying and equating coefficients of sin(x) and cos(x), we get the following system of equations:

-A + 2B = 0

2A*B = 1

Solving for A and B, we get:

A = 1/2

B = 1/4

Therefore, the particular solution is:

yp = 1/2cos(x) + 1/4sin(x)

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The quotient (h(x+3))/(h(x)) is 125. This tells us that the value of h changes by a factor of 125 when the input is increased by 3.

To find the quotient (h(x+3))/(h(x)), we will first evaluate the function h(x) for the given inputs and then divide the two results.

The function h is given by h(x) = 5^(x).

1. Evaluate h(x+3): h(x+3) = 5^(x+3)
2. Evaluate h(x): h(x) = 5^x
3. Find the quotient: (h(x+3))/(h(x)) = (5^(x+3))/(5^x)

Using the properties of exponents, we can simplify the expression further:
(5^(x+3))/(5^x) = 5^(x+3-x) = 5^3 = 125

The quotient (h(x+3))/(h(x)) is 125. This tells us that the value of h changes by a factor of 125 when the input is increased by 3.

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To evaluate the definite integral ∫ 7 3 ( 5 x − 20 ) d x ∫37(5x-20)dx in terms of areas, we can interpret it as the area bounded by the x-axis, the line y=5x-20, and the vertical lines x=3 and x=7.

Using the power rule of integration, we can first simplify the integrand:

∫ 7 3 ( 5 x − 20 ) d x = ∫ 7 3 5 x d x − ∫ 7 3 20 d x
= [ 5 2 x 2 ] 7 3 − [ 20 x ] 7 3
= ( 5 2 ( 7 2 − 3 2 ) ) − ( 20 ( 7 − 3 ) )
= 70

Therefore, the definite integral evaluates to 70, which represents the area of the region bounded by the x-axis, the line y=5x-20, and the vertical lines x=3 and x=7.

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The value of x that gives a value of y = 0 from the graph is

Graphs that represent the absolute value of a real number, define the absolute value graph. The non-negative value of the number represents its absolute value regardless of its sign.

Denoted as f(x) = |x|, the graph of the absolute value function resembles a V-shape with its central point set at the origin (0,0).

Using the attached graph it can be seen that the value of x that gives a value of y = 0 are

(-3, 0) and (-1, 0)

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The slopes and the y-intercept of a linear function that is represented by the table is: D. the slope is 2/5 and the y-intercept is -1/3.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):

y - y₁ = m(x - x₁)

Where:

First of all, we would determine the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (-2/15 + 1/30)/(-1/2 + 3/4)

Slope (m) = -0.1/0.25

Slope (m) = -0.4 or 2/5.

At data point (-3/4, -1/30) and a slope of 2/5, a linear function in slope-intercept form for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - (-1/30) = 2/5(x + 3/4)

y = 2x/5 - 1/3

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Required initial value is y(t) = (16/17)cos(4t) + (28/17)sin(4t) - (4/17).

What is initial value?

In mathematics, the initial value is the value of a function or a variable at a particular starting point or initial time. It is typically used in the context of differential equations or initial value problems, where the goal is to find a solution to an equation that satisfies aset of initial conditions.

The given differential equation is a second-order linear homogeneous ordinary differential equation with constant coefficients, which has the characteristic equation r² + 16 = 0. The roots of this equation are r = ±4i, which are complex conjugates of each other. Therefore, the general solution of the differential equation is given by [tex]y(t) = c_1 cos(4t) + c_2 sin(4t)[/tex] where [tex]c_1 \: and \: c_2[/tex] are arbitrary constants that can be determined using the initial conditions.

To find [tex]c_1 \: and \: c_2[/tex], we need to use the initial conditions y(4) = 4 and y'(4) = 7. Substituting t = 4, y = 4, and y' = 7 into the general solution,

[tex]4 = c_1 cos(16) + c_2 sin(16) \\ 7 = -4c_1 sin(16) + 4c_2 cos(16)[/tex]

Solving these two equations for [tex]c_1 \: and \: c_2[/tex], we obtain:

[tex]c_1 = (4cos(16) - 7sin(16))/(-4sin(16)) \\ c_2 = (4 - c_1 cos(16))/sin(16)[/tex]

Therefore, the solution to the initial-value problem is

y(t) = [(4cos(16) - 7sin(16))/(-4sin(16))]cos(4t) + [(4 - (4cos(16) - 7sin(16))/(-4sin(16)))sin(4t)]

Simplifying this expression using trigonometric identities, we get:

y(t) = (16/17)cos(4t) + (28/17)sin(4t) - (4/17)

Thus, the solution to the initial-value problem is y(t) = (16/17)cos(4t) + (28/17)sin(4t) - (4/17).

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Correct question is "solve the initial-value problem.y''+16y = 0, y(4) = 4, = 0, y'(4) = 7"

The value of the integral is 4π.

What is integral?

An integral is a mathematical concept that represents the area under a curve or the volume enclosed by a surface.

To evaluate the integral of the function [tex]f(x,y,z) = 1/\sqrt{(x^2+y^2+z^2)[/tex] over the region W, which is the bottom half of a sphere of radius 3, we can use spherical coordinates. In spherical coordinates, the position of a point in 3D space is given by the radius ρ, the polar angle θ, and the azimuthal angle ϕ.

The sphere of radius 3 centered at the origin has equation ρ=3, and the bottom half of the sphere is given by θ ranging from 0 to π, and ϕ ranging from 0 to 2π. Therefore, the integral can be expressed as:

[tex]\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{3} \frac{1}{\rho^2} \rho^2 \sin(\phi) , d\rho , d\phi , d\theta[/tex]

where sin(φ) is the Jacobian of the spherical coordinate transformation.

Evaluating the integral, we get:

[tex]\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{3} \frac{1}{\rho^2} \rho^2 \sin(\phi) , d\rho , d\phi , d\theta[/tex]

[tex]\int_{0}^{2\pi}\int_{0}^{\pi} [-\cos(\phi)]\Bigg|_{0}^{3} , d\phi , d\theta[/tex]

[tex]= \int\limits^2_0\pi2d[/tex]θ

= 4π

Therefore, the value of the integral is 4π.

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The value of the integral is 4π.

What is integral?

An integral is a mathematical concept that represents the area under a curve or the volume enclosed by a surface.

To evaluate the integral of the function [tex]f(x,y,z) = 1/\sqrt{(x^2+y^2+z^2)[/tex] over the region W, which is the bottom half of a sphere of radius 3, we can use spherical coordinates. In spherical coordinates, the position of a point in 3D space is given by the radius ρ, the polar angle θ, and the azimuthal angle ϕ.

The sphere of radius 3 centered at the origin has equation ρ=3, and the bottom half of the sphere is given by θ ranging from 0 to π, and ϕ ranging from 0 to 2π. Therefore, the integral can be expressed as:

[tex]\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{3} \frac{1}{\rho^2} \rho^2 \sin(\phi) , d\rho , d\phi , d\theta[/tex]

where sin(φ) is the Jacobian of the spherical coordinate transformation.

Evaluating the integral, we get:

[tex]\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{3} \frac{1}{\rho^2} \rho^2 \sin(\phi) , d\rho , d\phi , d\theta[/tex]

[tex]\int_{0}^{2\pi}\int_{0}^{\pi} [-\cos(\phi)]\Bigg|_{0}^{3} , d\phi , d\theta[/tex]

[tex]= \int\limits^2_0\pi2d[/tex]θ

= 4π

Therefore, the value of the integral is 4π.

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a) The random variable is the life of a dishwasher, denoted as X, which represents the number of years a dishwasher will last.

b) To find the probability that a dishwasher will last less than 66 years, we need to calculate the z-score for 66 years using the given mean and standard deviation values. Using the z-score formula, we find that the z-score for 66 years is -429.6. We can then use a standard normal distribution table or calculator to find the probability, which is very close to zero.

c) To find the probability that a dishwasher will last between 88 and 1010 years, we need to calculate the z-scores for both 88 and 1010 using the given mean and standard deviation values. The z-scores for 88 and 1010 are -1019.2 and -177.6, respectively. We can then use a standard normal distribution table or calculator to find the probabilities, which are also very close to zero. The probability that a dishwasher will last between 88 and 1010 years is the difference between these probabilities, which is also very close to zero.

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The estimate for 3√65 is 49/12.

To use a linear approximation (or differentials) to estimate the given number 3√65, follow these steps:

1. Choose a number close to 65 that has an easy-to-calculate cube root, such as 64 (since the cube root of 64 is 4).

2. Define the function f(x) = 3√x.

3. Calculate the derivative f'(x) = (1/3)x^(-2/3).

4. Evaluate f'(x) at the chosen number (x=64): f'(64) = (1/3)(64)^(-2/3) = 1/12.

5. Apply the linear approximation formula: Δy ≈ f'(x)Δx, where Δy is the change in f(x) and Δx is the change in x.

6. Find the change in x (Δx): Δx = 65 - 64 = 1.

7. Calculate the change in y (Δy): Δy ≈ f'(64)Δx = (1/12)(1) = 1/12.

8. Add the change in y (Δy) to the initial function value f(64): 3√65 ≈ 3√64 + Δy = 4 + 1/12 = 49/12.

So, using linear approximation, the estimate for 3√65 is 49/12.

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Answer:

61

Step-by-step explanation:

(1) To find the resonant frequencies of the model T(s) = 7/s(s2 +6s+58), we first need to factor the denominator:

s(s2 +6s+58) = s(s+3-√31i)(s+3+√31i)

The resonant frequencies occur at the poles of the transfer function, which are the roots of the denominator. Therefore, the resonant frequencies are:

ω1 = 0 (from the pole at s = 0)

ω2 = √31 (from the poles at s = -3±√31i)

(2) To find the resonant frequencies of the model T(s) = 7/ (3s2 +18s+174)(2s2 +85+58), we first need to factor the denominator:

(3s2 +18s+174)(2s2 +85+58) = 6(s+3+i√11)(s+3-i√11)(s+(-7+i√85)/2)(s+(-7-i√85)/2)

The resonant frequencies occur at the poles of the transfer function, which are the roots of the denominator. Therefore, the resonant frequencies are:

ω1 = √11 (from the poles at s = -3±i√11)

ω2 = √85/2 (from the poles at s = (-7±i√85)/2)

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if the mean and median of a population are the same, then its distribution is symmetric.

If the mean and median of a population are the same, then the distribution is symmetric. In a symmetric distribution, the mean and median are equal and the distribution is mirror image about its central point. This is because in a symmetric distribution, the same number of observations falls on either side of the central point, making the mean and median equal.

A normal distribution is an example of a symmetric distribution, but not all symmetric distributions are normal. Skewed distributions have unequal mean and median, and uniform distributions have a constant probability density function, which would result in a different mean and median.

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