In Exercises 7-12, complete the square in order to put the equation into standard form. Identify the center and the radius or explain why the equation does not represent a circle.
7. x ^ 2 - 12x + y ^ 2 + 6y = - 9
9. x ^ 2 + y ^ 2 + 14x - 20y - 20 = 0
x ^ 1 - 2x + y ^ 1 + 3/2 * y = - 1
8- 3x ^ 2 - 3y ^ 2 + 27y + 61 = 0
10. x ^ 2 + y ^ 2 - 7x - 3y - 1 = 0
12. 4x ^ 2 - 16x + 4y ^ 2 + 16 = 0

The solution to the initial value problem using the method of separation of variables is y = (x³/3) - (1/2)x + 4/3.

To solve the initial value problem y' = x² - 1/2 with the initial condition y(2) = 3, we can use the method of separation of variables. Here are the steps:

Step 1: Separate the variables

Write the given differential equation in the form:

dy/dx = x² - 1/2

Step 2: Integrate both sides

Integrate both sides of the equation with respect to x:

∫dy = ∫(x² - 1/2) dx

Integration yields:

y = (x³/3) - (1/2)x + C

Step 3: Apply the initial condition

To find the constant C, substitute the initial condition y(2) = 3 into the equation obtained in Step 2:

3 = (2³/3) - (1/2)(2) + C

Simplifying the equation:

3 = 8/3 - 1 + C

3 = 8/3 - 3/3 + C

3 = 5/3 + C

Therefore, C = 3 - 5/3 = 9/3 - 5/3 = 4/3.

Step 4: Write the final solution

Substitute the value of C back into the equation obtained in Step 2:

y = (x³/3) - (1/2)x + 4/3

So, the solution to the initial value problem y' = x² - 1/2, y(2) = 3 is y = (x³/3) - (1/2)x + 4/3.

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The given equation, 9x² + 36y² - 36x + 72y + 36 = 0, represents an ellipse. In standard form, the equation can be written as (x-1)²/4 + (y+1)²/1 = 1. The center of the ellipse is at (1, -1), the vertices are located at (3, -1) and (-1, -1), and the foci are at (2, -1) and (0, -1).

To write the equation 9x² + 36y² - 36x + 72y + 36 = 0 in standard form, we need to complete the square for both the x and y terms. By rearranging the equation, we have 9x² - 36x + 36 + 36y² + 72y + 36 = 0.

Next, we can factor out a 9 from the x terms and a 36 from the y terms: 9(x² - 4x + 4) + 36(y² + 2y + 1) = 0.

Simplifying further, we have 9(x - 2)² + 36(y + 1)² = 36.

Dividing both sides by 36, we get (x - 2)²/4 + (y + 1)²/1 = 1, which is the standard form of an ellipse.

From the standard form, we can determine that the center of the ellipse is located at (1, -1), the vertices are at (3, -1) and (-1, -1), and the foci are at (2, -1) and (0, -1).

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The sequence continues as follows:2, 5, 11, 21, 36, 67, ...

Given sequence2 5 11 21 36...

First differences 3 6 10 15...

Second differences 3 4 5...

Third differences 1 1...

The third differences are constant, which means that we can use a cubic polynomial for the fitting.

The formula for a cubic polynomial is

an³ + b

n² + c

n + d

Let us denote the nth term of the sequence by fn. Then, we have

f1 = 2, f2 = 5, f3 = 11, f4 = 21, f5 = 36...

We can write a system of equations using the first four terms of the sequence.

2 = a + b + c + d

5 = 8a + 4b + 2c + d

11 = 27a + 9b + 3c + d

21 = 64a + 16b + 4c + d

Solving this system, we get a = 1/3, b = 1, c = 11/3, and d = 0.

Thus, the closed-form expression for the nth term of the sequence isf(n) = (1/3)n³ + n² + (11/3)n

The next term in the sequence is f(6) = (1/3)(6)³ + (6)² + (11/3)(6) = 67.

Therefore, the sequence continues as follows:2, 5, 11, 21, 36, 67, ...

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Answer:

1.The two major types of metal electrical cable are:

A.wire and fibre

2. What are the most common forms of wireless connection?

C.Radio news

3. Wi-fi connections have limited range of :

D.300 metres

Answer:

32x

Step-by-step explanation: 4(8x)=32x

Answer:

Other dude is correct

Step-by-step explanation:

The exact value of A in the general solution is 13

Also, the DEQ is separable

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

y = Ax² + C/x

The differential equation is given as

y' + y/x = 39x

When y = Ax² + C/x is differentiated, we have

y' = 2Ax - Cx⁻²

So, we have

2Ax - Cx⁻² + y/x = 39x

Recall that

y = Ax² + C/x

So, we have

2Ax - Cx⁻² + (Ax² + C/x)/x = 39x

Evaluate

2Ax - Cx⁻² + Ax + Cx⁻² = 39x

This gives

2Ax +  Ax  = 39x

So, we have

3Ax = 39x

By comparing both sides of the equation, we have

3A = 39

Divide both sides by 3

A = 13

Hence, the value of A in the general solution is 13

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The solution to the system -8x + 5y = -33 and 8x - 4y = 28 is (x, y) = (7, -1).

To solve the given system of equations using the addition method, let's eliminate one variable by adding the two equations together. The system of equations is:

-8x + 5y = -33 (Equation 1)

8x - 4y = 28 (Equation 2)

When we add Equation 1 and Equation 2, the x terms cancel out:

(-8x + 5y) + (8x - 4y) = -33 + 28

y = -5

Now that we have the value of y, we can substitute it back into either Equation 1 or Equation 2 to solve for x. Let's use Equation 1:

-8x + 5(-5) = -33

-8x - 25 = -33

-8x = -33 + 25

-8x = -8

x = 1

Therefore, the solution to the system is (x, y) = (1, -5).

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Answer:

x+y=5

×-y=1

2x/2=6/2

x=3

while x+y=5

3+y=5

y=5-3

y=2

Answer:

C

Step-by-step explanation:

(3, 2)

The confidence interval using a 90% confidence level is (1.601, 1.819)

Confidence Interval = Sample Mean ± Margin of Error

Given the parameters:

Sample Mean (x) = 1.71

Population Standard Deviation (σ) = 0.86

Sample Size (n) = 170

Confidence Level = 90%

The standard Error(SE) is given by:

SE = σ / √n

SE = 0.86 / √170 ≈ 0.0663

The margin of Error is related to standard error by the relation:

ME = Z × SE

ME = 1.645 × 0.0663 ≈ 0.109

Confidence Interval = Sample Mean ± Margin of Error

Confidence Interval = 1.71 ± 0.109

Confidence Interval= (1.601, 1.819)

A decrease in sample size from 170 to 120 would lead to increase in error margin , since decrease in sample size will increase the standard error.

If confidence interval were raised from 90% to 95% , the error margin would be larger since increase in confidence interval would increase the critical value

Based on the confidence interval , it is very likely that the mean number of personal computers is greater than 1.

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The solution to the differential equation y'' + y = csc(x) using the variation of parameters method is y(x) = cos(x)ln|sin(x)| + Csin(x), where C is a constant.

To solve the differential equation by variation of parameters, we first find the complementary solution (the solution to the homogeneous equation). The homogeneous equation is y'' + y = 0, which has the solution y_c(x) = Acos(x) + Bsin(x), where A and B are constants.

Next, we find the particular solution using the variation of parameters method. We assume the particular solution is of the form y_p(x) = u(x)cos(x) + v(x)sin(x), where u(x) and v(x) are unknown functions.

We differentiate y_p(x) to find y_p' and y_p'' and substitute them into the original differential equation. After simplification, we obtain u'(x)sin(x) - v'(x)cos(x) = csc(x).

To solve this system of equations, we find the derivatives u'(x) and v'(x) and integrate them to obtain u(x) and v(x). Finally, we substitute u(x) and v(x) into the particular solution form y_p(x) = u(x)cos(x) + v(x)sin(x).

The final solution is y(x) = y_c(x) + y_p(x), which simplifies to y(x) = cos(x)ln|sin(x)| + Csin(x), where C is the constant of integration.

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Answer:

The probability of getting a white or green gumball is 35.64%.

Step-by-step explanation:

1. calculate the total number of gumballs in the gumball machine

35 + 22 + 30 + 14 = 101 gumballs

2. calculate the number of white and green gumballs combined

22 + 14 = 36 w/g gumballs

3. divide the number of white and green gumballs combined by the total number of gumballs in the gumball machine

36 / 101 = 0.3564

Answer: the probability of getting a white or green gumball is 35.64% (0.3564).

Probability is the chance of occurring an event from the total possible outcomes.

The probability of getting a white or a green gumball is 57/101.

It is the chance of occurring an event from the total possible outcomes.

We have,

Green gumballs = 35

White gumballs = 22

Purple gumballs = 30

Blue gumballs = 14

Total gumballs = 35 + 22 + 30 + 14 = 101

The combination is given by:

= [tex]^nC_{r}[/tex]

= n! / r! (n-r)!

The probability of choosing a white gumball:

= [tex]\frac{^{22}C_{1}}{^{101}C_{1} }[/tex]

= 22 / 101

The probability of choosing a green gumball:

= [tex]\frac{^{35}C_{1}}{^{101}C_{1} }[/tex]

= 35 / 101

The probability of getting a white or a green gumball:

= 22/101 + 35 / 101

= (22 +35) / 101

= 57 / 101

Thus the probability of getting a white or a green gumball is 57/101.

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The evaluation of ∫30(4f(t) - 6g(t)) dt given that ∫150f(t) dt = -7, ∫30f(t) dt = -8, ∫150g(t) dt = 4, and ∫30g(t) dt = 8 is -80.

Given that ∫150f(t) dt = -7, ∫30f(t) dt = -8, ∫150g(t) dt = 4, and ∫30g(t) dt = 8.

Let us evaluate ∫30(4f(t) - 6g(t)) dt.

Therefore,∫30(4f(t) - 6g(t)) dt = ∫30(4f(t) dt - 6g(t) dt) = 4 ∫30f(t) dt - 6 ∫30g(t) dt

Now, using the given values in the question we can say that,∫30(4f(t) - 6g(t)) dt = 4 ∫30f(t) dt - 6 ∫30g(t) dt = 4 (-8) - 6(8) = -32 - 48 = -80

Therefore, the evaluation of ∫30(4f(t) - 6g(t)) dt given that ∫150f(t) dt = -7, ∫30f(t) dt = -8, ∫150g(t) dt = 4, and ∫30g(t) dt = 8 is -80.

Note: The given integrals ∫150f(t) dt, ∫30f(t) dt, ∫150g(t) dt, and ∫30g(t) dt are only intermediate steps in order to evaluate the final integral.

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The indicated IQ score, corresponding to the shaded area under the curve of 0.5675 in a normal distribution with a mean of 100 and a standard deviation of 15, is approximately 102.55.

To find the indicated IQ score, we need to determine the corresponding z-score using the given information about the normal distribution. Here's how we can calculate it step by step:

Step 1: Identify the shaded area under the curve.

The shaded area under the curve represents the cumulative probability of the IQ scores. In this case, the shaded area is 0.5675 or 56.75%.

Step 2: Convert the cumulative probability to a z-score.

To find the z-score corresponding to the shaded area, we need to find the z-score that gives us a cumulative probability of 56.75%. We can use a standard normal distribution table or a statistical calculator to find the z-score.

Step 3: Find the z-score.

Using a standard normal distribution table, we can search for the closest cumulative probability to 0.5675. The closest cumulative probability we can find in the table is 0.5681, which corresponds to a z-score of approximately 0.17.

Step 4: Convert the z-score to an IQ score.

Now that we have the z-score of 0.17, we can use the formula for transforming z-scores to raw scores:

IQ score = (z-score × standard deviation) + mean

Given that the mean is 100 and the standard deviation is 15, we can calculate the IQ score:

IQ score = (0.17 × 15) + 100

IQ score = 2.55 + 100

IQ score ≈ 102.55

Therefore, the indicated IQ score is approximately 102.55.

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The graph of this function begins at 5 and moves down to 2 and back to 5 again in a repeated pattern over one period of 1.

A cosine function is a periodic function that fluctuates about its midline and follows a predictable pattern. The formula for a cosine function with a midline of y = c, amplitude of a, and period of b is:y = a cos(bx) + cTo shift the graph of a cosine function,

you can add or subtract a value inside the parentheses of the formula, which results in a horizontal shift.

The shift is to the right if you add, and to the left if you subtract.In this instance,

the cosine function has the following characteristics:Midline = y = 5Amplitude = 3Period = 1Horizontal shift to the right = 1/4We'll have to adjust the formula to include all of these parameters.

First and foremost, let's figure out the function's frequency, which is determined by dividing 2π by the period of the cosine function. In this example, the frequency is 2π/1, which equals 2π.

y = a cos(bx) + c is the formula we'll use. We'll substitute the values given into the formula. The resulting formula is:y = 3cos(2π(x - 1/4)) + 5This is the cosine function with a midline of y = 5, an amplitude of 3, a period of 1, and a horizontal shift of 1/4 to the right.

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Answer:

3(k*k)-3(3k)-5(2k)+2(5+5)

Step-by-step explanation:

3k²-19k+20

Just find some number that equal the terms when multiplied. ¯\_(ツ)_/¯

DOUBLE CHECK!

3(k*k)-3(3k)-5(2k)+2(5+5)

Multiply what you need to.

3k²-9k-10k+20

Combine like terms.

3k²-19k+20

---

hope it helps

Answer:

Acute angles are less than 90

obtuse is more than 90

right angles are exactly 90

Step-by-step explanation:

Answer:

First option: 90 degrees

Step-by-step explanation:

Obtuse is greater than 90.

Acute is less than 90.

Right is 90.

Answer:

27

Step-by-step explanation:

You do 3159 ÷ 27

Comes out to 27

I found 27 by just plugging in numbers.

Hope this helps

Answer:

There are 27 students in the class.

Step-by-step explanation:

Since the total weight is 3,159 and the average weight is 117lb, you can just divide 3,159 by 117, which will give you 27.

The probability of a Type I error in rejection region a is 0.01.

We must  find  the area under the t-distribution curve outside the rejection region, assuming a two-tailed test in order to to determine the probability of making a Type I error in each rejection region.

for a. t > 2.718, and df = 11:

Using a t-distribution table,  the probability is 0.01.

b. t < -1.476, df = 5:

Using a t-distribution table the probability is 0.05.

c. t < -2.060 or t > 2.060, df = 25:

Using a t-distribution table, the area in each tail is  0.025.

The combined probability of a Type I error in rejection region c is

0.025 + 0.025 = 0.05.

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240÷30=8

8×30=240

30 height 8 with

Newton has just gone grocery shopping. The mean cost for each item in his bag was $3.38. The price of the 6th item in his bag is $1.64

To determine the price of the 6th item in Newton's bag, we can use the concept of the mean (average). We know that the mean cost for each item in his bag is $3.38, and he bought a total of 6 items.

To find the sum of all 6 item prices, we can multiply the mean cost by the total number of items:

Sum of all item prices = Mean cost * Total number of items

                                    = $3.38 * 6

                                    = $20.28

We also know the prices of 5 of the items, which are $4.09, $4.03, $4.19, $2.24, and $4.09.

To determine the price of the 6th item, we subtract the sum of the known prices from the sum of all item prices:

Price of the 6th item = Sum of all item prices - Sum of known prices

= $20.28 - ($4.09 + $4.03 + $4.19 + $2.24 + $4.09)

= $20.28 - $18.64

= $1.64

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Answer:

Choice B - (1.02)

Step-by-step explanation:

With the teachers starting salary being $40,000, we can rightfully assume that it their salary would not increase with Choice A, $40,000 * 0.2 = $800.

2% otherwise known as 0.02, would make the teachers yearly salary $40,000 * 1 + 0.02, or $40,000 * 1.02. Making the correct answer, Choice B - (1.02). (This would also give the teacher an additional $800 per year!)

Answer:

57.33

Step-by-step explanation:

50+28+3x=180

78+3x=180-78

3x=172/3

x = 57.33

Answer:17

Step-by-step explanation:

0.35•40=14 to find how many they’ve won

14+9=23 is how many they have won or drawn

40-23=17 is how many they’ve lost

Answer:

The measure of the inscribed angle is [tex]40^{\circ}[/tex]

Step-by-step explanation:

Recall that the measure of the inscribed angle is one-half the measure of the central angle subtended by the same arc.

So, if the measure of the central angle is [tex]80^{\circ}[/tex], so the measure of the corresponding inscribed angle is:

[tex]\frac{1}{2}\times 80^{\circ}=40^{\circ}[/tex]

Answer:

Circumference ~ 233.7

Step-by-step explanation:

Use the distance formula to find the radius of the circle with the two coordinates given. The radius of this circle is about 37.2. Plug 37.2 into the formula 2(pi)r to find the Circumference. The answer is 233.7 rounded to the nearest tenth.

The lesser of the two eigenvalues is λ = -1. Its corresponding eigenvector is [-42 29].

The greater of the two eigenvalues is λ = 1. Its corresponding eigenvector is [42 29].

To solve the system of differential equations:

x' = -29x + 42y

y' = -20x + 29y

We can rewrite it in matrix form as follows:

X' = AX

where X = [x y] is a vector, X' represents the derivative of X with respect to time, and A is the coefficient matrix. In this case, A is given by:

A = [ -29 42 ]

[ -20 29 ]

To find the eigenvalues and eigenvectors, we need to solve the characteristic equation:

det(A - λI) = 0

where λ is the eigenvalue and I is the identity matrix. Solving this equation will give us the eigenvalues, and by substituting these eigenvalues back into the equation (A - λI)V = 0, where V is the corresponding eigenvector, we can find the eigenvectors.

Let's calculate the eigenvalues first. We have:

A - λI = [ -29 42 ]

[ -20 29 ] - λ [ 1 0 ]

[ 0 1 ]

Expanding the determinant, we get:

(-29 - λ)(29 - λ) - (42)(-20) = 0

(λ + 29)(λ - 29) + 840 = 0

λ² - 29² + 840 = 0

λ² - 841 + 840 = 0

λ² = 1

λ = ±1

So the eigenvalues are λ = 1 and λ = -1.

To find the corresponding eigenvectors, we substitute each eigenvalue back into the equation (A - λI)V = 0.

For λ = 1, we have:

[ -29 42 ] [ v₁ ] [ 0 ]

[ -20 29 ] [ v₂ ] = [ 0 ]

This gives us the following system of equations:

-29v₁ + 42v₂ = 0

-20v₁ + 29v₂ = 0

Solving this system, we find that v₁ = 42 and v₂ = 29.

Therefore, the eigenvector corresponding to the eigenvalue λ = 1 is [42 29].

For λ = -1, we have:

[ -29 42 ] [ v₁ ] [ 0 ]

[ -20 29 ] [ v₂ ] = [ 0 ]

This gives us the following system of equations:

-29v₁ + 42v₂ = 0

-20v₁ + 29v₂ = 0

Solving this system, we find that v₁ = -42 and v₂ = 29.

Therefore, the eigenvector corresponding to the eigenvalue λ = -1 is [-42 29].

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Complete Question:

Solve the system of differential equations

x' = -29x + 42y

y' = -20x + 29y

x(0) = -15, y(0) = -11

The lesser of the two eigenvalues is Its corresponding eigenvector is

The greater of the two eigenvalues is Its corresponding eigenvector is

Answer:

The coefficient is 6

Step-by-step explanation: