an air filter is rated to catch 90% of airborne particles. if the average particle diameter is 0.5 microns and the population standard deviation is 0.2 microns, what is the largest diameter particle (in microns) that will pass through the filter? assume that the diameter of particles in the air is normally distributed.

Steady-state probability for each system state π_1 = 3π_0 ≈ 0.495.

What is probability?

Probability is a measure of the likelihood of an event occurring.

To analyze the system, we can use the Markov chain approach. We can define the state of the system as the number of busy servers, ranging from 0 to 3. Let's denote the state of the system at time t as X(t). The transition rates between states depend on the arrival and service rates, as follows:

For X(t) = 0, the arrival rate is λ = 8 per hour, and the departure rate is μ = 1/20 per minute per server. Therefore, the transition rate from state 0 to state 1 is λ, and the transition rate from state 1 to state 0 is 3μ.

For X(t) = 1, the arrival rate is λ = 8 per hour, and the departure rate is μ = 1/20 per minute per server. Therefore, the transition rate from state 1 to state 2 is λ, and the transition rates from state 2 to state 1 and from state 1 to state 0 are both 2μ.

For X(t) = 2, the arrival rate is λ = 8 per hour, and the departure rate is μ = 1/20 per minute per server. Therefore, the transition rate from state 2 to state 3 is λ, and the transition rates from state 3 to state 2 and from state 2 to state 1 are both μ.

For X(t) = 3, the arrival rate is λ = 0 (since there is no waiting space), and the departure rate is μ = 1/15 per minute per server. Therefore, the transition rate from state 3 to state 2 is 3μ.

To find the steady-state probabilities for each system state, we can use the balance equations:

π_i * q_i,j = π_j * q_j,i

where π_i is the steady-state probability of being in state i, and q_i,j is the transition rate from state i to state j.

We can set up a system of four equations (one for each state) and solve for the unknown probabilities. The equations are:

λπ_0 = 3μπ_1

λπ_1 = 2μπ_2 + 2μπ_0

λπ_2 = μπ_3 + 2μπ_1

3μπ_3 = μπ_2

We also have the normalization condition:

π_0 + π_1 + π_2 + π_3 = 1

Solving the system of equations, we get:

π_0 = (1 - ρ) * (1 - ρ²) * (1 + 3ρ + 9ρ²) / (1 + 3ρ + 9ρ² + 9ρ³)

π_1 = 3π_0

π_2 = 3ρπ_0

π_3 = ρ³π_0

where ρ = λ/3μ is the traffic intensity.

Substituting the given values, we get:

ρ = (8/3) / (3 * (1/20)) = 32/3

π_0 = (1 - 32/3) * (1 - (32/3)^2) * (1 + 3*(32/3) + 9*(32/3)²) / (1 + 3*(32/3) + 9*(32/3)² + 9*(32/3)³) ≈ 0.165

π_1 = 3π_0 ≈ 0.495

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Answer:

[tex] \frac{250(n + 4)}{n} \geqslant 320[/tex]

[tex] \frac{250n + 1000}{n} \geqslant 320[/tex]

[tex]250 + \frac{1000}{n} \geqslant 320[/tex]

[tex] \frac{1000}{n} \geqslant 70[/tex]

[tex] \frac{n}{1000} \leqslant \frac{1}{70} [/tex]

[tex]n \leqslant \frac{1000}{70} [/tex]

[tex]n \leqslant 14[/tex]

No, the velocity vector v(t) of a curve r(t) is not always perpendicular to the acceleration vector a(t). To understand this concept, we first need to understand what these vectors are.

The velocity vector v(t) of a curve r(t) represents the rate of change of position of an object with respect to time. In other words, it tells us how fast an object is moving and in what direction. It is a tangent vector to the curve r(t) at any given point on the curve.

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The required canoeist's paddling rate in calm water is 3 mph.

Let's denote the canoeist's paddling rate in calm water as "x" mph.

When the canoeist paddles upstream against the current, their effective speed is the difference between their paddling rate and the current's rate, or (x - 3) mph. The distance traveled upstream is also 8 miles.

Using the formula:

distance = rate x time

We can set up two equations based on the distance traveled and the effective speed for each leg of the trip:

Downstream leg: 8 = (x + 3) * t1

Upstream leg: 8 = (x - 3) * t2

Since the total time for the round trip is 2 hours, we know that:

t1 + t2 = 2

Now we can solve for x by substituting t1 = 8/(x+3) and t2 = 8/(x-3) into the equation above:

8/(x+3) + 8/(x-3) = 2

Multiplying both sides by (x+3)(x-3) gives:

8(x-3) + 8(x+3) = 2(x+3)(x-3)

Simplifying this expression gives:

16x = 48

x = 3

Therefore, the canoeist's paddling rate in calm water is 3 mph.

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Answer:

= 512 ft²

Step-by-step explanation:

All the rectangular figure:

Area₁ = (32ft + 8ft) (8ft + 8ft)

Area₁ = (40ft)(16ft)

Area₁ = 640ft²

Area of NOT shaded region;

There are 2 similar squares:

Area₂ = 2(8ft*8ft)

Area₂ = 2(64ft²)

Area₂ = 128ft²

Then:

The shaded area is:

Area₁ - Area₂ = Shaded Area

Shaded area = 640 ft² - 128ft²

Shaded area = 512 ft²

The parameterization of the line from (-1, 0) to (3, -2) is P(t) = (-1 + 4t, -2t), where t varies from 0 to 1.

Direction vector = (3, -2) - (-1, 0) = (3 + 1, -2 - 0) = (4, -2)

In order to get a parameterization of the line, we can use the equation:

P(t) = P0 + t * v

where P(t) is a point on the line, P0 is a known point on the line (in this case, (-1, 0)), v is the direction vector of the line, and t is a parameter that varies along the line.

Substituting the values we have, we get:

P(t) = (-1, 0) + t * (4, -2)

Simplifying, we get:

P(t) = (-1 + 4t, -2t)

So the parameterization of the line from (-1, 0) to (3, -2) is P(t) = (-1 + 4t, -2t), where t varies from 0 to 1.

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Answer:

b.11

Step-by-step explanation:

 b is the answer

Answer:

-72-27m or switch them for -27m-27

Step-by-step explanation:

We are given the problem:

9(-8-3m)

and are asked to combine like terms.

Combining like terms means to combine terms/values that share similar properties.  In this case, it would be single numbers or if we had more than 1 value with "m" as the variable, that.

We would have to use the distributive property in this case to combine like terms, so distribute the 9 to all terms in the parenthesis.

-72-27m

We usually put the variable and coefficient first, so it would be -27m-72.

Hope this helps! :)

The equilibrium level of GDP can be found using the equation Y = C + I + G, where Y is GDP, C is consumption, I is investment, and G is government spending.

Given the values of C, I, and G, we can substitute them into the equation and solve for Y.
In this case, we have Y = 300 + 100 + 200 = 600. Therefore, the equilibrium level of GDP is 600.

The equilibrium level of GDP is the level at which aggregate demand (the sum of consumption, investment, and government spending) equals aggregate supply (the total output produced by the economy). In this problem, we are given the values of consumption, investment, and government spending, and asked to find the equilibrium level of GDP. Using the equation Y = C + I + G, we can substitute in the given values and solve for Y. This gives us the equilibrium level of GDP, which represents the point at which the economy is in equilibrium and there is no tendency for output or prices to change.

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The PDF of Y, when X is uniformly distributed between 0 and 1, is given by fY(y) = 1/y for y > 0.

Let's denote the PDF of X as fX(x), which represents the probability density function of X. We want to find the PDF of Y, denoted as fY(y), which represents the probability density function of Y = eˣ.

To derive the PDF of Y, we need to understand the transformation that occurs when we apply the exponential function to X. The transformation Y = eˣ is a monotonic transformation, which means that it preserves the order of the values of X. In other words, if X1 < X2, then eˣ1 < eˣ2.

To find the PDF of Y, we will use the cumulative distribution function (CDF) approach. The CDF of Y, denoted as FY(y), gives us the probability that Y takes on a value less than or equal to y. Mathematically, FY(y) = P(Y ≤ y).

We can express the CDF of Y in terms of the CDF of X. Since Y = eˣ, we have FY(y) = P(eˣ ≤ y). Now, we can solve this inequality for X by taking the natural logarithm (ln) of both sides: ln(Y) ≤ X.

Next, we can write the inequality in terms of the CDF of X. Since X is uniformly distributed between 0 and 1, its CDF, denoted as FX(x), is given by FX(x) = x for 0 ≤ x ≤ 1.

Substituting ln(Y) ≤ X, we get ln(Y) ≤ FX(x). To find the probability that this inequality holds, we integrate the CDF of X from 0 to the value of X that satisfies ln(Y) ≤ X. This gives us:

FY(y) = P(Y ≤ y) = P(ln(Y) ≤ X) = P(ln(Y) ≤ FX(x)) = ∫[0,X] fX(x) dx,

where fX(x) is the PDF of X.

Since X is uniformly distributed between 0 and 1, its PDF is a constant function over the interval [0,1]. Therefore, fX(x) = 1 for 0 ≤ x ≤ 1, and fX(x) = 0 otherwise.

We can now compute FY(y) by evaluating the integral for different values of y. However, to obtain the PDF of Y, we need to differentiate FY(y) with respect to y. By applying the Fundamental Theorem of Calculus, we get:

fY(y) = d/dy [FY(y)] = d/dy ∫[0,X] fX(x) dx.

Since fX(x) = 1 for 0 ≤ x ≤ 1, we can take the derivative of the integral with respect to y, resulting in:

fY(y) = d/dy [FY(y)] = d/dy ∫[0,X] 1 dx = d/dy (X) = d/dy (ln(y)) = 1/y,

where y > 0.

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The number of 48 squares cover the surface of a rectangle.

Area of a Square = Side × Side. Therefore, the area of square = [tex]Side^2[/tex] square units. and the perimeter of a square = 4 × side units.

We have the information :

The side of a square is 2cm

and, Length of the rectangle is = 24 cm

Breadth of the rectangle is = 8cm

We have to find the how many square cover the surface of a rectangle.

The area of the square is

2 × 2

=4

The area of the rectangle is

L × W

=24 ×8

=192

The number of squares is

192 ÷ 4

=48

Hence, The number of 48 squares cover the surface of a rectangle.

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The approximate probability that the mean number of siblings in the sample of size 50 is at most 2 is 0.1562 or 15.62%.

What is probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.

To answer this question, we need to use the central limit theorem, which states that the sample mean of a large enough sample from any population with a finite mean and variance will follow a normal distribution with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

In this case, we have a sample size of 50, which is considered large enough for the central limit theorem to apply. Therefore, the mean of the sample means will be equal to the population mean, which is 2.2, and the standard deviation of the sample means will be equal to the population standard deviation divided by the square root of the sample size, which is 1.4/sqrt(50) = 0.198.

To find the probability that the mean number of siblings in the sample of size 50 is at most 2, we need to calculate the z-score and use the standard normal distribution table or calculator. The z-score can be calculated as:

z = (2 - 2.2) / 0.198 = -1.01

Using the standard normal distribution table or calculator, we can find that the probability of getting a z-score of -1.01 or less is approximately 0.1562.

Therefore, the approximate probability that the mean number of siblings in the sample of size 50 is at most 2 is 0.1562 or 15.62%.

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The equivalent value of the exponential expressions are solved

Given data ,

Let the exponential equation be represented as A

Now , the value of A is

A = (1.4*10 by the power of one)(8*10by the power of 4)

On simplifying , we get

A = 1.4 x 10¹ x 8 x 10⁴

A = 11.2 x 10⁵

A = 1.12 x 10⁶

Now , the exponential equation is B

where B = ( 1.1 * 10 by the power of negative 5) ( 3 * 10² negative power)

B = 1.1 x 10⁻⁵ x 3 x 10⁻²

B = 3.3 x 10⁻⁷

Hence , the exponents are solved

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Er lekt elk uur 0,3 liter water uit de emmer. Per dag zijn er 24 uur, dus er lekt per dag 0,3 x 24 = 7,2 liter water uit de emmer.

Als er aanvankelijk 10,8 liter water in de emmer zit en er elke dag 7,2 liter uit lekt, dan zal de emmer leeg zijn na 10,8 / 7,2 = 1,5 dagen.

Dus de emmer zal na 1,5 dagen leeg zijn.

Answer: The answer is D.

Step-by-step explanation: i did the test

Your question is about a sample survey on illegal drug use and the potential effects on the results due to people's fear  In this case, the correct answer is: c. this is a non-sampling error that causes bias.

c. this is a non-sampling error that causes bias. When respondents are not truthful in their answers due to fear of repercussions, this is a form of response bias, which is a type of non-sampling error. This can lead to biased results since the true prevalence of illegal drug use in the population is not accurately represented.
Your question is about a sample survey on illegal drug use and the potential effects on the results due to people's fear of their information being shared with the police or employers. In this case, the correct answer is: c. this is a non-sampling error that causes bias.

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A smaller cylinder rod in the example of question 5 would provide a greater retraction force.

To understand why this is the case, we need to look at the formula for force: force = pressure x area. In the case of a hydraulic cylinder, the force is generated by pressure acting on the surface area of a piston. A smaller cylinder rod would have a smaller surface area than a larger one, which means that the same amount of pressure would generate a greater force.

For example, let's say that we have two hydraulic cylinders with the same pressure and the same fluid flow rate, but one has a smaller cylinder rod than the other. The smaller cylinder rod would have a smaller surface area, so the force generated by the pressure would be greater. This greater force would result in greater retraction of the rod, making it move faster and with greater force.

In conclusion, a smaller cylinder rod would provide a greater retraction force due to the smaller surface area on which pressure acts, resulting in a greater force for the same amount of pressure.

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The density of the material it is made of is equal to 0.9952 g/cm³.

In Mathematics and Geometry, the volume of a cylinder can be calculated by using this formula:

Volume of a cylinder, V = πr²h

Where:

By substituting the parameters, we have the following:

Volume, V = 3.14 × 4² × 10

Volume, V = 502.4 cm³.

Density = mass/volume

Density = 500/502.4

Density = 0.9952 g/cm³.

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Answer:

150 + 12 = 162 cubic cm

Step-by-step explanation:

There's no question but I am going to guess you need the total volume of this figure.

So bottom first - - -

=10x3x5 = 150 cubic cm

Now the top - - -

= 2 x 3 x 2 = 12 cubic cm

Add them up:

150 + 12 = 162 cubic cm

After considering the given data we come to the conclusion that the conversion of the given quadratic equation into vertex form is f(x) = (x + 3)² - 6.

In order to form the quadratic function in vertex form, we have to apply the formula f(x) = a(x - h)² + k.
Then the vertex form of the given quadratic function is
Y = x² + 6x + 3,
The completion of the square are as follows
Y = x² + 6x + 3
Y = (x² + 6x + 9) - 9 + 3
Y = (x + 3)² - 6
Hence, the vertex form of the given quadratic function is f(x) = (x + 3)² - 6. The vertex of this parabola is (-3,-6).
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The correct optionr is A. Al(x) is the area of the cross section through the solid at the point x

Al(x) is not the volume of the cross section, the function that describes the cross section, or the function that describes the solid. It is simply the area of the cross section at a specific point.

If a cut is made through a solid object perpendicular to the x-axis at a particular point x, Al(x) represents the area of the cross section through the solid at that point.

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The gradient of the linear function graphed in this problem is given as follows:

4.

The slope-intercept equation for a linear function is presented as follows:

y = mx + b

m represents the gradient of the function, which is by how much the dependent variable y increases or decreases when the independent variable x is added by one.

In this problem, we have that when x increases by one, y increases by 4, hence the gradient of the linear function is given as follows:

4.

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Answer:

?

Step-by-step explanation:

For a right-tailed test of a hypothesis for a population means with n = 14 and a test statistic t = 1.863, the p-value is between 0.10 and 0.05.

Hi! You have a question regarding a right-tailed hypothesis test for a population mean with a sample size of n = 14 and a test statistic t = 1.863. You want to determine the p-value range for this test.

To find the p-value range, follow these steps:
1. Identify the degrees of freedom (df) for the t-distribution: df = n - 1 = 14 - 1 = 13.
2. Use a t-distribution table or a calculator to find the p-value range corresponding to the test statistic (t = 1.863) and degrees of freedom (df = 13).

Using a t-distribution table or calculator, you'll find that the p-value for this test is between 0.10 and 0.05.

So, for a right-tailed test of a hypothesis for a population means with n = 14 and a test statistic t = 1.863, the p-value is between 0.10 and 0.05.

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Answer:

a)  $12/child    

b)  $260    

c)  c = 12n + 260    

d)  n can be any whole number up to the maximum number of children allowed.

Step-by-step explanation:

A)

440 - 380 = 60

15 - 10 = 5

60 / 5 = 12

There's a $12 additional charge for each child.

B)

10 * 12 = 120 (total additional charge)

Subtract the total additional charge from the total price to find the initial charge.

380 - 120 = 260

Initial fee is $260

C)

C is going to equal cost/total cost and n will equal the number of children attending.

c = 12n + 260

This is because you pay an extra $12 for every child attending and the initial price is $260. You want to add these together to find the total price.

D)

n can equal any whole number as long as it doesn't exceed a set limit (if there is one set). Of course there can't be half a child so it will always be numbers like 10, 15, 20, 25, 32, etc.

The way that the sample size would have to change to decrease the length of the confidence interval is to increase from 400 to 1600.

The confidence interval's length is directly proportional to the sample size. The specific relationship between these two factors follows an inverse proportion that correlates to the square root of the sample size.

One could represent this correlation through a proportionality statement: the larger the sample size, the smaller the confidence interval's length becomes.

Given that L1 = 0. 06 and n1 = 400, we want to find n2 such that L 2 = 0. 03:

L1 / L2 = √(n 2 / n 1)

The values would then be:

L1 / L2 = √ ( n2 / n1 )

0.06 / 0.03 = √ ( n2 / 400)

2 = √ (n2 / 400)

2² = ( √ (n2 / 400))²

4 = n2 / 400

n2 = 4 x 400

n2 = 1600

In conclusion, the sample size needs to increase to 1, 600.

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The number of miles that Hummer H1 can go would be = 230 miles.

To calculate the number of miles the vehicle can travel when a certain amount of gallons are given would be done following the steps below.

The number of miles for 1 gallon of gas = 10 miles

The number of miles for 23 gallons of gas = X miles

That is :

1 gallon = 10 miles

23 gallons = X miles

make X miles the subject of formula;

X miles = 23×10/1

= 230 miles.

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Answer:

by 29.25 %

Step-by-step explanation:

that is the ans

Answer:

6.67%

Step-by-step explanation:

there are 125 + 100 = 225 students.

boys:

125 decreased by 8%

decrease in 8% means 100% - 8% = 92%

125 X 0.92 = 115 boys.

girls:

100 decreased by 5%

decrease in 5% means 100% - 5% = 95%

100 X 0.95 = 95 girls

now the total number of girls and boys on the honor roll = 95 + 115 = 210 students.

that is a reduction of 225 - 210 = 15.

we want the reduction in percentage

15/225 = 0.0667

= 6.67%

The given expression can be simplified using trigonometric identities. By using the identity cos(a-b) = cos(a)cos(b) + sin(a)sin(b), we get:

cos(pi/4-x) = cos(pi/4)cos(x) + sin(pi/4)sin(x)

Substituting the given values of cos(x) and sin(x), we get:

cos(pi/4-x) = (1/sqrt(2))(-1/2) + (1/sqrt(2))(sqrt(3)/2)

Simplifying this expression, we get:

cos(pi/4-x) = -sqrt(2)/4 + sqrt(6)/4

The given expression involves the cosine of the difference between two angles. By using the identity cos(a-b) = cos(a)cos(b) + sin(a)sin(b), we can simplify the expression in terms of the cosine and sine of the individual angles. We are given the value of cos(x) and we can use the identity sin^2(x) + cos^2(x) = 1 to find the value of sin(x). Once we have the values of sin(x) and cos(x), we can substitute them in the above identity to get the simplified expression.

In this particular problem, we are given the value of cos(x) and the fact that x is in the second quadrant, which implies that sin(x) is positive. Using these values, we can simplify the expression to get the final answer. It is important to note that the answer is requested in exact form as a fraction, and not as a decimal approximation.

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