Determine if (-6, 9) is a solution of the system, 6x+y=-27 5x-y=-38

The area of the triangle is 14.7 units squared.

The area of a triangle can be found as follows:

area of a triangle = 1 / 2 ab sin C

Therefore, the angle C is the included angle.

Therefore,

area of the triangle  XYZ =  1 / 2 × (7) × (4.3) sin 79

area of the triangle  XYZ = 30.1 / 2 sin 79°

area of the triangle  XYZ = 15.05 sin 79

area of the triangle  XYZ = 15.05 × 0.98162718344

area of the triangle  XYZ = 14.7244077517

area of the triangle  XYZ = 14.7 units²

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To find the value of m so that 5ba³√a / b³2a² = 5aᵐ / 2b², we can first simplify the left-hand side of the equation. We can do this by using the following rules:

a³√a = a²

b³2a² = b²

This gives us the following equation:

5ba² / b² = 5aᵐ / 2b²

We can then solve for m by multiplying both sides of the equation by 2b² and dividing both sides by 5a². This gives us the following equation:

m = 2

The first step is to simplify the left-hand side of the equation. We can do this by using the following rules:

a³√a = a²

b³2a² = b²

This gives us the following equation:

5ba² / b² = 5aᵐ / 2b²

We can then solve for m by multiplying both sides of the equation by 2b² and dividing both sides by 5a². This gives us the following equation:

m = 2

The final step is to express the answer in decimal form. Since 2 is an integer, the answer is simply 2.0.

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To convert the given expression, √(2 - x² - y²) * x * y * dz * dy * dx, to spherical coordinates, we need to express x, y, and z in terms of spherical coordinates (ρ, θ, φ).

To convert the given expression to spherical coordinates, we need to express x, y, and z in terms of spherical coordinates (ρ, θ, φ).

1. Expressing x, y, and z in terms of spherical coordinates:

In spherical coordinates, we have:

x = ρsin(φ)cos(θ)

y = ρsin(φ)sin(θ)

z = ρcos(φ)

2. Converting the given expression:

The expression to be converted is:

√(2 - x² - y²) * x * y * dz * dy * dx

Substituting the values of x, y, and z in terms of spherical coordinates, we get:

√(2 - (ρsin(φ)cos(θ))² - (ρsin(φ)sin(θ))²) * (ρsin(φ)cos(θ)) * (ρsin(φ)sin(θ)) * ρ²sin(φ) dρ * dθ * dφ

Simplifying the expression:

ρ⁴sin⁴(φ) * √(2 - ρ²sin²(φ)(cos²(θ) + sin²(θ))) dρ * dθ * dφ

So, the expression in spherical coordinates is:

ρ⁴sin⁴(φ) * √(2 - ρ²sin²(φ)) dρ * dθ * dφ

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Hence, the given polynomial is factorized as (x+9)(x-5).

The polynomial x(x + 9)-5(x +9) can be factored completely as:(x+9)(x-5).

The given polynomial is x(x+9)-5(x+9)

Expanding the brackets we get, x²+9x-5x-45x²+4x-45

Gathering like terms, we get: x²+4x-45

Now we need to factorize this quadratic expression.

We can split the middle term as +9x-5x=4x

Thus, we can write the quadratic expression as:x²+9x-5x-45

Taking common factor from the first two terms and the last two terms separately, we get:

x(x+9)-5(x+9)

Now we can see that there is a common factor of (x+9).

So, we can write the given expression as:(x+9)(x-5)

Hence, the given polynomial is factorized as (x+9)(x-5).

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a. As the angle measure D varies from 0 degrees to 90 degrees, cos(D) varies from 1 to 0, and sin(D) varies from 0 to 1. In other words, when D is 0 degrees, cos(D) is 1 and sin(D) is 0, while when D is 90 degrees, cos(D) is 0 and sin(D) is 1.

b. As the angle measure D varies from 180 degrees to 270 degrees, cos(D) varies from -1 to 0, and sin(D) varies from -1 to 0. In this range, cos(D) is negative and decreases from -1 to 0, while sin(D) is also negative and decreases from -1 to 0.

c. The domain of cos(D) is all real numbers, as cos(D) is defined for any angle measure D. The domain of sin(D) is also all real numbers, as sin(D) is defined for any angle measure D.

d. The range of cos(D) is [-1, 1], meaning that cos(D) can take any value between -1 and 1, inclusive. The range of sin(D) is also [-1, 1], meaning that sin(D) can take any value between -1 and 1, inclusive. Both cos(D) and sin(D) oscillate between these extreme values as the angle measure D varies.

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the probability of selecting a female student without aid is approximately 0.0602.

To find the probability of selecting a female student without aid, we need to calculate the probability of selecting a female student and then multiply it by the probability of not receiving aid among female students.

Let's start with the probability of selecting a female student:

P(female) = Number of female students / Total number of students

= 1,822,972 / (8,003,975 + 1,822,972)

= 0.185924059 (approximately)

Next, we calculate the probability of not receiving aid among female students:

P(without aid | female) = 1 - P(receiving aid | female)

= 1 - (67.6% / 100%)

= 1 - 0.676

= 0.324

Finally, we multiply the two probabilities to find the probability of selecting a female student without aid:

P(female without aid) = P(female) * P(without aid | female)

= 0.185924059 * 0.324

= 0.060202 (approximately)

Therefore, the probability of selecting a female student without aid is approximately 0.0602.

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Given question is incomplete, the complete question is below

In a recent year, 8,003,975 male students and 1,822,972 female students were enrolled as undergraduates. Receiving aid were 63.6% of the male students and 67.6% of the female students. Of those receiving aid, 43.8% of the males got federal aid and 50.8% of the females got federal aid. Choose 1 student at random. (Hint: Make a tree diagram.) Find the probability of selecting a student from the following. Carry your intermediate computations to at least 4 decimal places.

A female student without ad flemale without aid

The decimal number 0.0625 can be expressed as the fraction 1/16 in its simplest form.

To convert the decimal number 0.0625 into a fraction, we can follow these steps:

Step 1: Determine the number of decimal places in the given decimal. In this case, there are four decimal places.

Step 2: Write the given decimal as the numerator of the fraction, and the denominator as 1 followed by the same number of zeros as the decimal places. In this case, the numerator is 0625 and the denominator is 10000.

Step 3: Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 0625 and 10000 is 625. Dividing both the numerator and denominator by 625, we get the fraction 1/16.

Therefore, the decimal number 0.0625 can be expressed as the fraction 1/16 in its simplest form.

This conversion is possible because we can observe a pattern in the given decimal numbers 0.3 and 0.11. We can see that 0.3 is equivalent to 3/10, and 0.11 is equivalent to 11/100. The pattern is that the decimal number is written as the numerator, and the denominator is obtained by placing a 1 followed by the same number of zeros as the decimal places. Following this pattern, we can convert 0.0625 into the fraction 1/16.

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To settle both debts in one month, a single payment of $4,921.99 is required.

To calculate the single payment required, we need to consider the present values of the two debts with respect to the focal date (one month from now). The present value of each debt can be determined using the formula for present value of a single sum with simple interest: PV = FV / (1 + r * t), where PV is the present value, FV is the future value (debt payment), r is the interest rate, and t is the time in years.

Step 1: Calculate the present value of the first debt payment of $2,900 due in five months: PV1 = $2,900 / (1 + 0.044 * (5/12)).

Step 2: Calculate the present value of the second debt payment of $2,100 due in eight months: PV2 = $2,100 / (1 + 0.044 * (8/12)).

Step 3: Add the present values of the two debts to get the total single payment required: Total Payment = PV1 + PV2 = $4,921.99.

Therefore, a single payment of approximately $4,921.99 is required to settle both debts in one month.

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The equation of the tangent line to the curve y = -7ln(2³ - 26) at the point (3, 0) is y = 0.

The derivative of the function y = -7ln(2³ - 26).

Using the chain rule, the derivative of ln(u) is (1/u) * du/dx, so:

dy/dx = -7 * (1 / (2³ - 26)) * d(2³ - 26)/dx

Now, differentiate 2³ - 26:

d(2³ - 26)/dx = d(8 - 26)/dx = d(-18)/dx = 0

Therefore, the derivative dy/dx simplifies to:

dy/dx = -7 * (1 / (2³ - 26)) * 0 = 0

The slope of the tangent line at the point (3, 0).

Since the derivative dy/dx is zero, it means the tangent line is horizontal, and its slope is zero.

The equation of the tangent line using the point-slope form.

The point-slope form of a linear equation is: y - y₁ = m(x - x₁), where (x₁, y₁) is the given point and m is the slope.

Using the point (3, 0) and slope 0, we have:

y - 0 = 0(x - 3)

y = 0

Therefore, the equation of the tangent line to the curve y = -7ln(2³ - 26) at the point (3, 0) is y = 0.

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Answer:

a)

Given the runner is jogging at a constant speed of 3.1 mph, we can construct a function representing distance by multiplying 3.1mph by t, the number of hours (I assume).

Answer: d(t) = 3.1t

or d(t) = 3.1 * t

3.1 is being multiplied by t because 3.1 mph is the speed, and t is time.

Distance = rate (which is speed) * time (t)

b)

To find the inverse, time in terms of distance, we must manipulate the equation.

d(t) will be expressed as d.

d = 3.1t

Manipulate this by dividing by 3.1 to solve for time:

[tex]\frac{d}{3.1} = t[/tex]

Given a distance, we can now solve directly for time.

Answer: t(d) = [tex]\frac{d}{3.1}[/tex]

or t(d) = d / 3.1

To find the value of F0.05 with numerator degrees of freedom (df1) = 7 and denominator degrees of freedom (df2) = 17, we can use the F-distribution table.

The F-distribution table provides critical values for different levels of significance (α) and degrees of freedom (df1 and df2).

Since α = 0.05 and the alternative hypothesis (Ha) is "greater than" (>), we are interested in finding the critical value that corresponds to an upper tail area of 0.05.

In the F-distribution table, the column headings represent the numerator degrees of freedom (df1), and the row headings represent the denominator degrees of freedom (df2).

Looking up the row for df2 = 17 and scanning across until we find the column for df1 = 7, we can locate the corresponding critical value.

The critical value F0.05 with df1 = 7 and df2 = 17 is approximately 2.462.

Therefore, F0.05 = 2.462.

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The linear speed of the car in miles per hour is 71.39 mph.

The radius of the wheel on a car is 20 inches. If the wheel is revolving at 346 revolutions per minute, what is the linear speed of the car in miles per hour?Firstly, we can compute the distance travelled in one minute of the wheel's motion as:Distance = circumference of the wheel = 2πr.

Where r is the radius of the wheelWe know that the radius of the wheel, r = 20 inchesTherefore, distance travelled in one minute = 2π × 20= 40π inchesIf the wheel is revolving at 346 revolutions per minute, then distance travelled by the wheel in one minute = 40π × 346 = 13840π inches. One mile is equal to 63360 inches (by definition).Hence distance travelled by the wheel in one hour = 13840π × 60= 830400π inches per hourWe now convert from inches to miles:Distance travelled in one hour = 830400π ÷ 63360 miles/hour≈ 131.24 mph

Hence, the linear speed of the car in miles per hour is 71.39 mph (rounded to two decimal places).

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Let Q be the universal set and A₁, A₂, ..., Aₙ be subsets of Q. The indicator function IA(W) is defined as 1 if w ∈ A and 0 if w ∉ A. We want to prove the property: I(A₁ ∩ A₂ ∩ ... ∩ Aₙ) = min{IA₁, IA₂, ..., IAₙ}.

To prove the property of the indicator function, we need to show that I(A₁ ∩ A₂ ∩ ... ∩ Aₙ) = min{IA₁, IA₂, ..., IAₙ}.

Let's consider an arbitrary point w in the universal set Q. We can break down the proof into two cases:

1. If w ∈ A₁ ∩ A₂ ∩ ... ∩ Aₙ:

In this case, w belongs to the intersection of all the sets A₁, A₂, ..., Aₙ. Therefore, IA₁(w) = IA₂(w) = ... = IAₙ(w) = 1. Hence, the minimum value among IA₁, IA₂, ..., IAₙ is 1. Therefore, min{IA₁, IA₂, ..., IAₙ}(w) = 1. On the other hand, I(A₁ ∩ A₂ ∩ ... ∩ Aₙ)(w) is also equal to 1 since w belongs to the intersection. Thus, min{IA₁, IA₂, ..., IAₙ}(w) = I(A₁ ∩ A₂ ∩ ... ∩ Aₙ)(w).

2. If w ∉ A₁ ∩ A₂ ∩ ... ∩ Aₙ:

In this case, w does not belong to the intersection of the sets A₁, A₂, ..., Aₙ. Therefore, at least one of the indicator functions, say IAₖ(w), is 0. Thus, min{IA₁, IA₂, ..., IAₙ}(w) = 0. On the other hand, I(A₁ ∩ A₂ ∩ ... ∩ Aₙ)(w) is also equal to 0 since w does not belong to the intersection. Hence, min{IA₁, IA₂, ..., IAₙ}(w) = I(A₁ ∩ A₂ ∩ ... ∩ Aₙ)(w).

Since the property holds for all points w in the universal set Q, we can conclude that I(A₁ ∩ A₂ ∩ ... ∩ Aₙ) = min{IA₁, IA₂, ..., IAₙ}.

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The appropriate reciprocal identity to find the exact value of sin 0 for the given value of csc 8. Rationalize denominators when applicable the exact value of `sin 8` for the given value of `csc 8 = √44` is `√11 / 4`.

Given csc 8 = √44,

we need to find sin 8 using the appropriate reciprocal identity.

We can use the reciprocal identity of sine and cosecant, which is,

`sin θ = 1/csc θ`.

Simplify `csc 8 = √44`

First, simplify `csc 8 = 1/sin 8` to `sin 8 = 1/csc 8`.

Now, replace `csc 8` with `√44` to get `sin 8 = 1/√44`

Rationalize the denominator by multiplying both the numerator and denominator by `√44`.

sin 8 = `1/√44 × √44/√44`

= `√44/44` = `√4 × √11 / 4 × 11`

= `√11 / 4`

Therefore, the exact value of `sin 8` for the given value of `csc 8 = √44` is `√11 / 4`.

Hence, the answer is `sin 8= √11/4`.

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Using Z-test, The seniors in the GED program have a significantly higher average reading comprehension score compared to the graduating seniors in the local high school.

1. Research hypothesis: The average reading comprehension score of seniors in the GED program (μ_GED) is greater than the average reading comprehension score of graduating seniors in the local high school (μ_high school).

2. Null hypothesis: There is no difference in the average reading comprehension scores between seniors in the GED program and graduating seniors in the local high school (μ_GED = μ_high school).

To determine if the research hypothesis can be supported, we can perform a one-sample Z-test. With a sample mean of 79.53 and a population mean of 72.55, the test statistic (Z-score) can be calculated as follows:

[tex]Z = (sample mean - population mean) / (population standard deviation / \sqrt{sample size[/tex]

[tex]Z = (79.53 - 72.55) / (12.62 / \sqrt25)[/tex]

[tex]Z = 6.98 / (12.62 / 5)[/tex]

[tex]Z \approx 6.98 / 2.524[/tex]

[tex]Z \approx2.764[/tex]

At a 0.05 significance level, the critical Z-score is +1.96. Since the calculated Z-score (2.764) is greater than the critical value, we reject the null hypothesis. This means that the research hypothesis can be supported at the 0.05 significance level.

At a 0.01 significance level, the critical Z-score is +2.05. Again, the calculated Z-score (2.764) is greater than the critical value, so we reject the null hypothesis. The research hypothesis can be supported at the 0.01 significance level as well.

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(a) The level of significance, denoted by α, is given as 1%, which means the desired probability of making a Type I error (rejecting a true null hypothesis) is 1%.

(b) To find the value of the chi-square statistic, we need to compare the observed frequencies (the number of tools from the current excavation site) with the expected frequencies (the regional percent of stone tools multiplied by the total number of tools in the sample).

First, let's calculate the expected frequencies for each raw material:

Expected frequency of Basalt = 61.3% * 1486 = 910.918

Expected frequency of Obsidian = 10.6% * 1486 = 157.316

Expected frequency of Welded Tuff = 11.4% * 1486 = 169.404

Expected frequency of Pedernal chert = 13.1% * 1486 = 194.666

Expected frequency of Other = 3.6% * 1486 = 53.496

Next, we can calculate the chi-square statistic using the formula:

χ² = Σ [(Observed frequency - Expected frequency)² / Expected frequency]

χ² = [(905-910.918)² / 910.918] + [(150-157.316)² / 157.316] + [(162-169.404)² / 169.404] + [(207-194.666)² / 194.666] + [(62-53.496)² / 53.496]

χ = 6.352

The degrees of freedom for the chi-square test can be calculated as (number of categories - 1). In this case, we have 5 categories of raw materials, so the degrees of freedom would be 5 - 1 = 4.

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Step-by-step explanation:

Slope , m , betwen the two points

(y1-y2) / (x1-x2) =  (-2 -2) /(5-7) = -4/-2 = 2

SO  y = mx + b form would be

        y = 2x + b

              sub in one of the points to calculate 'b'

           -2 = 2(5) + b    shows b = -12

  so equation is   y =  2x -12  

The location of the point P is -3.2

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

The graph of the number line (See attachment)

On the number line , we can see that

using the above as a guide, we have the following:

P = -3 - 0.2

So, we have

P = -3.2

Hence, the location of the point P is -3.2

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The two unit vectors orthogonal to [-1] [1]

[2] and [0]

[-2] and [-1] are

First vector: [2, -1, 0]

Second vector: [1, 2, 0]

To find two unit vectors orthogonal to a given vector, we can use the cross product. Let's consider the given vector as [a, b, c]. We can then find the cross product of [a, b, c] with [0, 0, 1] to obtain a vector orthogonal to both. Finally, we normalize the obtained vector to make it a unit vector.

In this case, the given vector is [-1, 1, 2]. By taking the cross product of [-1, 1, 2] and [0, 0, 1], we get [2, -1, 0]. To obtain a second unit vector orthogonal to the given vector, we can swap the components and change the sign of one component. Thus, the second vector is [1, 2, 0].

The area of the parallelogram can be calculated using the formula A = |a x b|, where a and b are two adjacent sides of the parallelogram and |a x b| denotes the magnitude of their cross product.

Given the vertices (3, 1, 0), (7, 2, 0), (12, 5, 0), and (16, 6, 0), we can take two adjacent sides: (7, 2, 0) - (3, 1, 0) and (12, 5, 0) - (7, 2, 0).

Calculating the cross product of these two sides gives the normal vector [0, 0, 1], which has a magnitude of 1. Therefore, the area of the parallelogram is |[0, 0, 1]| = 1.

The area of the triangle can be calculated using the same formula, A = |a x b|, where a and b are two sides of the triangle.

Given the vertices (0, 0, 0), (1, -3, 5), and (1, -2, 4), we can take two sides: (1, -3, 5) - (0, 0, 0) and (1, -2, 4) - (0, 0, 0).

Calculating the cross product of these two sides gives the normal vector [-3, -1, -3], which has a magnitude of sqrt(19). Therefore, the area of the triangle is |[-3, -1, -3]| = sqrt(19).

To find the volume of the parallelepiped determined by the vectors a = [6, 1, 1], b = [1, 6, 1], and c = [1, 1, 10], we can use the scalar triple product.

The volume V can be calculated as V = |a · (b x c)|, where · denotes the dot product and x denotes the cross product.

Taking the cross product of b and c gives the vector [-59, 9, 5], and then taking the dot product of a with that vector gives -334. Therefore, the volume of the parallelepiped is |(-334)| = 334.

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We need to determine the number of months it takes for the total cost of Club A and Club B to be equal. Club A has a membership fee of $19 and a monthly fee of $27, while Club B has a membership fee of $29 and a monthly fee of $22.

Let's represent the total cost for Club A after "m" months as A(m) and the total cost for Club B after "m" months as B(m). We can set up the equation A(m) = B(m) to find the number of months when the total costs are equal.

For Club A, the total cost after "m" months is given by:

A(m) = 19 + 27m

For Club B, the total cost after "m" months is given by:

B(m) = 29 + 22m

Setting A(m) equal to B(m):

19 + 27m = 29 + 22m

To find the number of months when the costs are equal, we need to solve for "m" in the equation above.

First, let's subtract 22m from both sides:

19 + 5m = 29

Next, subtract 19 from both sides:

5m = 10

Finally, divide both sides by 5:

m = 2

Therefore, after 2 months, the total cost of Club A and Club B will be the same.

To find the total cost for each club after 2 months, we substitute m = 2 into the respective equations:

For Club A:

A(2) = 19 + 27(2)

= 19 + 54

= 73

For Club B:

B(2) = 29 + 22(2)

= 29 + 44

= 73

Hence, after 2 months, the total cost for both Club A and Club B will be $73.

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The z-score for P(? ≤ z ≤ ?) = 0.60 is approximately 0.25.

The z-score for P(z ≥ ?) = 0.30 is approximately -0.52.

How to find the Z score

P(Z ≤ z) = 0.60

We can use a standard normal distribution table or a calculator to find that the z-score corresponding to a cumulative probability of 0.60 is approximately 0.25.

Therefore, the z-score for P(? ≤ z ≤ ?) = 0.60 is approximately 0.25.

For the second question:

We want to find the z-score such that the area under the standard normal distribution curve to the right of z is 0.30. In other words:

P(Z ≥ z) = 0.30

Using a standard normal distribution table or calculator, we can find that the z-score corresponding to a cumulative probability of 0.30 is approximately -0.52 (since we want the area to the right of z, we take the negative of the z-score).

Therefore, the z-score for P(z ≥ ?) = 0.30 is approximately -0.52.

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The standard error of the mean decreases when the sample size increases or the standard deviation decreases.

Standard error of the mean (SEM) is a measure of how much the mean of a sample deviates from the true mean of the population. The SEM is calculated as the standard deviation of the sample divided by the square root of the sample size.

Hence, the SEM is affected by changes in the sample size and the standard deviation of the sample.

As per the given options, the standard error of the mean will decrease when the sample size increases or the standard deviation decreases.

This can be explained as follows:

When the sample size increases, the sample mean becomes more representative of the true population mean.

This reduces the variability of the sample mean, which in turn reduces the SEM.

The standard error of the mean (SEM) is a measure of how much the mean of a sample deviates from the true mean of the population. It is calculated as the standard deviation of the sample divided by the square root of the sample size.

The SEM is affected by changes in the sample size and the standard deviation of the sample.

Specifically, the SEM decreases when the sample size increases or the standard deviation decreases.When the sample size increases, the sample mean becomes more representative of the true population mean. s.

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C = 20000 + (20000/n)

Thus, we have expressed C in terms of n.

Let the constant part of the average cost be represented by k. Since the average cost varies inversely with the number of machines produced, we can express this relationship as k/n. Therefore, we have:

C = k + (k/n)

Given that the average cost is $25000 when 20 machines are produced, we can substitute these values into the equation:

25000 = k + (k/20)

Simplifying this equation, we get:

20k = 500000

k = 25000

Now, we can substitute the value of k into the equation to find C in terms of n:

C = 25000 + (25000/n)

Similarly, when 40 machines are produced and the average cost is $20000, we can substitute these values into the equation to find k:

20000 = k + (k/40)

40k = 800000

k = 20000

Substituting the value of k into the equation, we have:

C = 20000 + (20000/n)

Thus, we have expressed C in terms of n.

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We are given a line as [x, y, z] = [10, 5, 16] + t [3, 1, 5]. We have to show that this line is contained in each of the given planes.

a) The equation of plane a is given as x + 2y - z - 4 = 0. Let's check if the line is contained in the plane or not. If the point on the line belongs to the plane, then all the points on the line will belong to the plane. Let's find out the coordinates of a point on the line: Put t = 0 in [x, y, z] = [10, 5, 16] + t[3, 1, 5]We get a point (10, 5, 16) on the line. Now let's check if the point (10, 5, 16) lies on the plane a. x + 2y - z - 4 = 0 => 10 + 2(5) - 16 - 4 = 0 => 0 = 0Since (10, 5, 16) lies on the plane a, all points on the line will lie on the plane a. So the line [x, y, z] = [10, 5, 16] + t[3, 1, 5] is contained in the plane x + 2y - z - 4 = 0. b) The equation of plane b is given as 9x - 2y - 5z = 0Let's check if the line is contained in the plane or not. If the point on the line belongs to the plane, then all the points on the line will belong to the plane.

Let's find out the coordinates of a point on the line: Put t = 0 in [x, y, z] = [10, 5, 16] + t[3, 1, 5]. We get a point (10, 5, 16) on the line. Now let's check if the point (10, 5, 16) lies on the plane b.9x - 2y - 5z = 0 => 9(10) - 2(5) - 5(16) = 0 => 90 - 10 - 80 = 0 => 0 = 0Since (10, 5, 16) lies on the plane b, all points on the line will lie on the plane b. So the line [x, y, z] = [10, 5, 16] + t[3, 1, 5] is contained in the plane 9x - 2y - 5z = 0.

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(a) To find the general solution yh(t) of the corresponding homogeneous ODE ÿ(t) + 10y(t) + 25y(t) = 0, we can assume a solution of the form yh(t) = e^(rt), where r is a constant.

Substituting this into the ODE, we get:

(r^2 + 10r + 25)e^(rt) = 0

Since e^(rt) is never zero, the only way for the equation to hold is if the quadratic term (r^2 + 10r + 25) is equal to zero.

Solving r^2 + 10r + 25 = 0, we find that the roots are r = -5.

Therefore, the general solution yh(t) of the homogeneous ODE is:

yh(t) = Ae^(-5t) + Be^(-5t), where A and B are arbitrary constants.

(b) Suppose p(t) = 3sin(2t). To find an appropriate form for the particular solution y(t), we can try a solution of the form yp(t) = A sin(2t) + B cos(2t), where A and B are constants.

Taking the derivatives of yp(t), we have:

ÿp(t) = 2A cos(2t) - 2B sin(2t)

yp(t) = A sin(2t) + B cos(2t)

Substituting these into the ODE, we get:

(2A cos(2t) - 2B sin(2t)) + 10(A sin(2t) + B cos(2t)) + 25(A sin(2t) + B cos(2t)) = 3sin(2t)

Simplifying, we obtain:

(12A + 18B)sin(2t) + (12B - 18A)cos(2t) = 3sin(2t)

For this equation to hold for all values of t, the coefficients of sin(2t) and cos(2t) must be equal to the corresponding coefficients on the right side.

Therefore, we can conclude that an appropriate form for the particular solution y(t) is:

y(t) = (12A + 18B)sin(2t) + (12B - 18A)cos(2t), where A and B are arbitrary constants.

Among the given options, the correct answer is:

f. a sin(2t) + b cos(2t), where a = 18 and b = -18, corresponding to A and B in the general solution.

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Examples of something the individual possesses, such as cryptographic keys, electronic keycards, smart cards, and physical keys, fall under the category of possession-based authenticators.

Possession-based authenticators are a type of authentication factor that relies on the individual physically possessing an item or device to prove their identity. These authenticators add an extra layer of security by requiring the user to have the physical item in their possession in order to authenticate and gain access to a system, facility, or data. This type of authentication method helps prevent unauthorized access as it requires the combination of something the individual knows (such as a PIN or password) along with something the individual possesses to verify their identity.

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Answer:

  3x -1

Step-by-step explanation:

You want the quotient when (3x² +11x -4) is divided by (x +4).

When the divisor is a linear binomial, the polynomial division is conveniently carried out using synthetic division. The "entry in the left part of the table" referred to in the attachment is the zero of the binomial divisor. Here, that is -4, the value of x that makes (x +4) = 0.

The quotient is 3x -1.

Some graphing calculators are equipped with the capability to manipulate expressions involving variables. The second attachment shows one of those.

  [tex]\boxed{\dfrac{3x^2+11x-4}{x+4}=3x -1}[/tex]

<95141404393>

To determine the proportion of at-home games that were wins, we need to calculate the conditional probability of a win given that the game was played at home. Let's denote the proportion of at-home games that were wins as P(W|H).

We know that 60% of all games were at-home games, which means that 0.60 is the probability of an at-home game (P(H)). We also know that 40% of all games were wins, so the probability of a win (P(W)) is 0.40. Additionally, we are given that 35% of all games were at-home wins, which means P(W∩H) = 0.35.

To find P(W|H), we can use the conditional probability formula:

P(W|H) = P(W∩H) / P(H)

Substituting the given values:

P(W|H) = 0.35 / 0.60

Calculating the result:

P(W|H) ≈ 0.5833

Rounding to two decimal places, the proportion of at-home games that were wins is approximately 0.58 or 58%.

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The  data sets that could most likely be normally distributed is a Algebra test scores.

An example of a continuous probability distribution is the normal distribution, in which the majority of data points cluster in the middle of the range while the remaining ones taper off symmetrically toward either extreme. The distribution's mean is another name for the center of the range.

Algebra test scores can be seen as one that is normal distributed this is because the test scores  can be seen to be around the mean. B Therefore option A

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The inverse function is: f⁻¹(x) = 2x. The domain of the function is x ≥ 0, and the domain of the inverse function is x ∈ R.

Solving for x in terms of y given y = (x - 5)We are to solve for x in terms of y given y = (x - 5).

y = (x - 5)Add 5 to both sides:

y + 5 = xThus, x = y + 5Therefore, x in terms of y is

x = y + 5.The function

f(x) = 2√x can be written as follows:

y = 2√xSquare both sides: y² = (2√x)²y² = 4xSwap x and

y: x = 4y²Take the square root of both sides:

x = 2y.

The domain of the function f(x) = 2√x is x ≥ 0, because we can't have negative numbers under a square root.The domain of the inverse function f⁻¹(x) = 2x is x ∈ R, because we can take any value of x and compute the corresponding value of f⁻¹(x).

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