5'000 to the power of 10 3

Using Cramer's rule for first-order condition:

Using Hessian for the second-order condition, critical point (x₁, x₂, x₃) = (-149/444, -69/222, 139/444) is the unique minimum of y.

(a) Using Cramer's rule for the first-order condition:

To optimize the function y, find the critical points where the gradient is equal to zero. The gradient of y is given by:

∇y = [6x₁ - x₂ - 3x₃ - 5, -x₁ + 12x₂ + 2x₃ - 4, 2x₂ + 8x₃ + 2 - 3x₁]

Setting the gradient equal to zero:

6x₁ - x₂ - 3x₃ - 5 = 0 (1)

-x₁ + 12x₂ + 2x₃ - 4 = 0 (2)

2x₂ + 8x₃ + 2 - 3x₁ = 0 (3)

Using Cramer's rule to solve this system of linear equations, the determinant of the coefficient matrix is:

|A| =

| 6 -1 -3 |

|-1 12 2 |

|-3 2 -3|

|A| = 444

The determinant of the matrix obtained by replacing the first column of A with the constants on the right-hand side of the equations is:

|A₁| =

| 5 -1 -3 |

| 0 12 2 |

| 0 2 -3|

|A₁| = -149

The determinant of the matrix obtained by replacing the second column of A with the constants is:

|A₂| =

| 6 5 -3 |

|-1 0 2 |

|-3 0 -3|

|A₂| = -138

The determinant of the matrix obtained by replacing the third column of A with the constants is:

|A₃| =

| 6 -1 5 |

|-1 12 0 |

|-3 2 2|

|A₃| = -278

Therefore, using Cramer's rule:

x₁ = |A₁|/|A| = -149/444

x₂ = |A₂|/|A| = -69/222

x₃ = |A₃|/|A| = 139/444

(b) Using the Hessian for the second-order condition:

To check whether the critical point found in part (a) is a maximum, minimum or saddle point, we need to use the Hessian matrix evaluated at the critical point. The Hessian of y is given by:

(y) =

| 6 0 -3 |

| 0 12 2 |

|-3 2 8 |

Evaluating H(y) at the critical point (x₁, x₂, x₃) = (-149/444, -69/222, 139/444):

H(y) =

| 6 0 -3 |

| 0 12 2 |

|-3 2 8 |

The eigenvalues of H(y) are 2, 6, and 18, which are all positive. Therefore, H(y) is positive definite, and the critical point is a minimum.

Therefore, the critical point (x₁, x₂, x₃) = (-149/444, -69/222, 139/444) is the unique minimum of y.

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Considering the given data-set, the quartiles are given as follows:

There is an even number of elements(26), hence the median is the mean of the 13th and 14th elements, the first half is the 12 elements from 1 to 12, and the second half is the 12 elements from 15 to 26.

The first quartile is the mean of the 6th and 7th elements, which are 67 and 68, hence:

Q1 = (67 + 68)/2 = 67.5.

The third quartile is the mean of the 6th and 7th elements of the second half, hence:

Q3 = (91 + 92)/2 = 91.5.

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The dot isn’t shaded so > or <

From -55, it’s showing small than

From -49, it’s showing greater than

x < -55 or x > -49

Answer:  x < -55 or x > -49

Step-by-step explanation:

x < -55 or x > -49

They both start with an opened circle, and also these two opened circle dots do not connect to each other. On the left side, the number is going from -55 to the smaller side, -57 or less, so we can say that x is smaller than -55. On the right side, the number goes from -49 to the bigger number, we can say that x is greater than -49.

The total amount of money which the Stewart family would have to pay into the annuity each quarter is $242.12.

Mathematically, annuity can be calculated by using this formula:

\(A = P[\frac{(1+\frac{r}{n} )^{nt} - 1)}{\frac{r}{n} }]\)

Given the following data:

Substituting the given parameters into the formula, we have;

\(13000 = P[\frac{(1+\frac{0.036}{4} )^{4 \times 11} - 1)}{\frac{0.036}{4} }]\\\\13000 = P[\frac{(1+0.009 )^{44} - 1)}{0.009 }]\\\\13000 = P[\frac{(1.009 )^{44} - 1)}{0.009 }]\\\\13000 = P[\frac{1.48323960867 - 1}{0.009 }]\\\\13000 = P[\frac{0.48323960867 }{0.009 }]\\\\13000 = 53.6932898522P\)

P = 13000/53.6932898522

P = $242.12.

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Complete Question:

The Stewart family wants to save money to travel the world. They plan to invest in an ordinary annuity that earns 3.6% interest, compounded quarterly. Payments will be made at the end of each quarter. How much money do they need to pay into the annuity each quarter for the annuity to have a total value of $13,000 after 11 years?

Answer:

20

Step-by-step explanation:

The common factors for 60,40,20,80 60 , 40 , 20 , 80 are 1,2,4,5,10,20 1 , 2 , 4 , 5 , 10 , 20 . The GCF for the numerical part is 20 .

The second pair of points representing the solution set of the system of equations is (-6, 29).

To find the second pair of points representing the solution set of the system of equations, we need to substitute the x-coordinate of the second point into one of the equations and solve for y.

Given the system of equations:

y = x^2 - 2x - 19

y + 4x = 5

Substituting the x-coordinate of the second point (-6) into equation 2:

y + 4(-6) = 5

y - 24 = 5

y = 5 + 24

y = 29

Therefore, the second pair of points representing the solution set of the system of equations is (-6, 29).

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Question

Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar.

y = x2 − 2x − 19

y + 4x = 5

The pair of points representing the solution set of this system of equations is (-6, 29) and

_________.

The key features of the function g(x) = 4(x² - 8x - 9) are:

y-intercept: (0, -84)

Zeros: x = -1 and x = 9

Vertex: (4, -100)

What is the polynomial function  

A polynomial function is a function that involves only non-negative integer powers or only positive integer exponents of a variable in an equation like the quadratic equation, cubic equation, etc.

To identify the key features of the function g(x) = 4(x² - 8x - 9), we need to manipulate the function and convert it to vertex form, which is y = a(x - h)² + k.

First, we expand the function:

g(x) = 4x² - 32x - 36

Next, we factor out the 4:

g(x) = 4(x² - 8x - 9)

To complete the square, we add and subtract (8/2)² = 16:

g(x) = 4(x² - 8x + 16 - 16 - 9)

g(x) = 4[(x - 4)² - 25]

g(x) = 4(x - 4)² - 100

Now we can identify the key features of the function:

The y-intercept of g(x) is found by setting x = 0:

g(0) = 4(0 - 4)² - 100 = -84

So the y-intercept is (0, -84).

The zeros of g(x) are found by setting y = 0 and solving for x:

4(x - 4)² - 100 = 0

4(x - 4)² = 100

(x - 4)² = 25

x - 4 = ±5

x = 4 ± 5

So the zeros of g(x) are x = -1 and x = 9.

The vertex of g(x) is (h, k), where h is the x-coordinate of the vertex and k is the y-coordinate of the vertex.

In this case, the vertex is (4, -100), which we found by completing the square.

Therefore, the key features of the function g(x) = 4(x² - 8x - 9) are:

y-intercept: (0, -84)

Zeros: x = -1 and x = 9

Vertex: (4, -100)

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Answer:

12.9x-9

Step-by-step explanation:

5.3x-9+7.6x

5.3x+7.6x = 12.9x

=12.9x-9

Answer: Height = 20 inches, Width = 36 inches

Step-by-step explanation:

Let the height be \(h\) and the width be \(h+16\).

Using the formula for the perimeter of a triangle,

\(2(h+h+16)=112\\\\2(2h+16)=112\\\\2h+16=56\\\\2h=40\\\\h=20 \implies h+16=36\)

The required 95% confidence interval for the mean breaking force of acrylic bone cement is approximately 267.29 to 344.85 newtons.

Find the critical value (t*) for a 95% confidence level and (n - 1) degrees of freedom.

The degrees of freedom (df) is equal to the sample size minus 1 (df = 7 - 1 = 6). From a t-table or statistical software, the critical value for a 95% confidence level and df = 6 is approximately 2.447.

Calculate the standard error of the mean (SE).

The standard error of the mean is given by SE = (sample standard deviation) / sqrt(sample size).

\(SE =\dfrac{41.96}{\sqrt{7}}\)

   ≈ 15.85

Calculate the margin of error (MoE).

The margin of error is MoE = (t*) * SE.

MoE ≈ 2.447 * 15.85
        ≈ 38.78

Calculate the confidence interval.

The confidence interval is given by: Mean ± MoE.

Mean ± MoE ≈ 306.07 ± 38.78

Calculate the lower and upper bounds of the confidence interval.

Lower bound ≈ 306.07 - 38.78
                       = 267.29

Upper bound ≈ 306.07 + 38.78
                       = 344.85

Therefore, the 95% confidence interval for the mean breaking force of acrylic bone cement is approximately 267.29 to 344.85 newtons.

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Therefore, the package with 40 hooks is the better buy in terms of cost efficiency.

To determine which package is the better buy, we need to compare the cost per hook for each package.

For the package of 25 hooks costing $9.95, we divide the total cost by the number of hooks:

Cost per hook = $9.95 / 25 = $0.398

Rounding to the nearest cent, the cost per hook is $0.40.

For the package of 40 hooks costing $13.99, we divide the total cost by the number of hooks:

Cost per hook = $13.99 / 40 = $0.3498

Rounding to the nearest cent, the cost per hook is $0.35.

Comparing the two costs per hook, we can see that the package with 40 hooks for $13.99 offers a better deal, as the cost per hook is lower at $0.35.

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Therefore , the solution of the given problem of probability comes out to be a complete probability model cannot be represented by this list.

The main objective of each considerations technique is to calculate the likelihood that a claim is true or that a particular incident will occur. Anything between 0 and 1, were 0 traditionally denotes a percentage and 1 typically denotes the degree of certainty, can be used to represent chance. A probability illustration shows the likelihood that a particular event will occur.

Here,

The probabilities assigned to the outcomes do not add up to 1, hence the following list cannot be a complete probability model.

=> P(2) + P(3) + P(4)

=> 1/12 + 2/3 + 1/4

=>  (1 + 8 + 3) / 12

=> 12 / 12 = 1

In a complete probability model, the total of the probabilities should always be equal to 1, indicating that there is a probability of 1 that one of the events will occur.

In this instance, there is a missing probability since the sum of the probabilities assigned to the possible possibilities is less than 1.

Therefore, a complete probability model cannot be represented by this list.

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Answer:

To find the probability of selecting a Coke person from this group, we need to divide the number of people who prefer Coke by the total number of people in the group.

The number of people who prefer Coke is 29. The total number of people in the group is 29 + 21 + 37 + 28 + 19 + 16 + 150 + 150 + 2 = 452.

Therefore, the probability of selecting a Coke person from this group is:

P(Coke) = Number of people who prefer Coke / Total number of people in the group = 29 / 452 ≈ 0.064

So the probability of selecting a Coke person from this group is approximately 0.064.

I hope this helps! Let me know if you have any other questions.

Step-by-step explanation:

Answer:

Below

Step-by-step explanation:

Line AB is of the form y = mx + b    where m is the slope = 2

 Line BC is perpendicular to this and will have slope - 1/m = - 1/2   and includes the point C (0,6)

 The point slope form of line BC will then be

     ( y-6) = - 1/2 ( x - 0)    which can be re-arranged to

       y = - 1/2x + 6     < ==== equation for line BC

Answer:

4 15 51

Step-by-step explanation:

if you now what I mean

The formula for the area of a triangle is A = (base · height)/2, but when we look at this problem, there's no base or height!

Looking at the triangle on the right, we can use the Pythagorean Theorem to find the height:

h² + 16² = 20²

h² + 256 = 400

h² = 144

h = ± 12

In this case, the height is 12 because you can't have negatives when you are working with lengths of sides and other things like this in geometry.

So now we've found the height, but we still don't know the full base! So, we can use the height that we just found to use the Pythagorean Theorem for the left triangle!

B² + 12² = 15²

B² + 144 = 225

B² = 81

B = ±9

So once again, our answer is just 9.

But, because the base is CB, we have to add:

b = 9 + 16  

b = 25.

Now we have the base and height, so we can put them into our formula to solve for the area of the triangle:

A = (25 · 12) / 2

A = 300 / 2

A = 150

Answer: y = 4x + 1

Step-by-step explanation: the slope is 4 over 1 (the same as just 4, so we are just putting 4) because the points of the line are going up four and over one (and you always add x to the slope). the y-intercept is 1 because that is where the line meets the y axis. :)

Yes, the solution is correct as x = 4

Algebraic expressions are described as expressions that are made up of terms, variables, constants, factors and coefficients.

These expressions are also composed of operations, such as;

From the information given, we have that;

8 - √x = 10

collect the like terms

- √x = 10 - 8

subtract the values

- √x = 2

Then, we have;

√x = -2

Take the square root of both sides

x = (-2)²

x = 4

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Answer:

Step-by-step explanation:

x+9

Let x, be 4

4+9=13

given condition,

x+9<4x

4+9<4(4)

13<16

The answer is:

Work/explanation:

Our inequality is:

\(\sf{x+9 < 4x}\)

Flip it

\(\sf{4x > x+9}\)

Solve

\(\sf{4x-x > 9}\)

Combine like terms

\(\sf{3x > 9}\)

Divide each side by 3

\(\sf{x > 3}\)

Answer:

1

Step-by-step explanation:

Remember PEMDAS:

P: we have no parenthesis, so skip this step.

E: no exponents either.

MD (multiplication and division): we have 3x5. when we calculate this, it is equal to 15. Therefore our equation is now simplified to 18-15-2.

AS (addition and subtraction): Subtract in order of first to last. The answer is 1.

Answer:

7 schools have fewer than 50 classrooms.

Kamryn would save $159.99 in interest by raising her credit score from 665 to 680 when borrowing $13,250.00 with a simple interest rate loan.

To determine the amount of interest Kamryn will save by raising her credit score from 665 to 680, we need to compare the interest rates associated with each credit score range.

According to the given information, a credit score of 665 falls within the range of 580-669, where the corresponding simple interest rate is 10.358%.

Let's calculate the interest Kamryn would pay on a loan of $13,250.00 at an interest rate of 10.358%.

Interest = Principal * Rate

= $13,250.00 * 0.10358

≈ $1,370.56

Therefore, if Kamryn were to borrow $13,250.00 with a credit score of 665, she would pay approximately $1,370.56 in interest.

Now, let's consider the scenario where Kamryn raises her credit score to 680. A credit score of 680 falls within the range of 670-739, where the corresponding simple interest rate is 9.132%.

Calculating the interest with a credit score of 680:

Interest = Principal * Rate

= $13,250.00 * 0.09132

≈ $1,210.57

Thus, if Kamryn were able to raise her credit score from 665 to 680, she would save approximately $1,370.56 - $1,210.57 = $159.99 in interest.

Therefore, Kamryn would save approximately $159.99 in interest by raising her credit score from 665 to 680 when borrowing $13,250.00 with a simple interest rate loan.

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A) The type of triangles are congruent triangles

B) By the use of SAS Congruency Postulate

C) The distance across for the sink hole is: 52.2 ft

D) The perimeter of triangle ABC is: 172.2 feet.

A) Congruent triangles are defined as the triangles created because of the phrasing "you recreate the same triangle" mentioned in the instructions. Congruent triangles are basically identical carbon copies of each other.

B) If we knew the measure of angle ACB, and then mad use of it to form angle ECD, then we would have enough information to know that triangle ACB was congruent to triangle ECD. Therefore, it would be useful to do the SAS (side angle side) congruence rule.

C) We know that:

AB = ED = 52.2

AB is the distance across the sink hole. Thus, it is 52.2 feet

D) AB = 52.2

BC = 70

AC = 50

Thus:

Perimeter of triangle ABC = AB + BC + AC

Perimeter of triangle ABC = 52.2 + 70 + 50

Perimeter of triangle ABC = 122.2 + 50

Perimeter of triangle ABC = 172.2

The perimeter of triangle ABC is 172.2 feet.

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Answer:

Step-by-step expA) The type of triangles are congruent triangles

B) By the use of SAS Congruency Postulate

C) The distance across for the sink hole is: 52.2 ft

D) The perimeter of triangle ABC is: 172.2 feet.

How to solve congruent triangles?

A) Congruent triangles are defined as the triangles created because of the phrasing "you recreate the same triangle" mentioned in the instructions. Congruent triangles are basically identical carbon copies of each other.

B) If we knew the measure of angle ACB, and then mad use of it to form angle ECD, then we would have enough information to know that triangle ACB was congruent to triangle ECD. Therefore, it would be useful to do the SAS (side angle side) congruence rule.

C) We know that:

AB = ED = 52.2

AB is the distance across the sink hole. Thus, it is 52.2 feet

D) AB = 52.2

BC = 70

AC = 50

Thus:

Perimeter of triangle ABC = AB + BC + AC

Perimeter of triangle ABC = 52.2 + 70 + 50

Perimeter of triangle ABC = 122.2 + 50

Perimeter of triangle ABC = 172.2

The perimeter of triangle ABC is 172.2 feet.

lanation:

Answer:

Here is how to get to the correct answer

Step-by-step explanation:

The radius is half the diameter ; r=3 inches. h= 6 inches.

The volume of 1 candle : 3.14r²h = 3.14*3²*6

Take the volume and multiply it 110

The simplified radical form of the given values would be = 41√3.

A simplified radical form is defined as the expression of the radical form in the simplest way such that there is absence of square roots or cube roots.

The given expression ;

= 4√75 + 7 √ 27

= 4 √ 25 × 3 + 7 √ 9 × 3

= 4 × 5 √3 + 7 × 3 √3

= 20√3 + 21 √ 3

= 41√3

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Answer:

The probability that on a particular week, the hospital will receive on high BP pregnant woman is 0.0068

Step-by-step explanation:

We use the Exponential distribution,

Since we are given that on average, 5 pregnant women with high blood pressure come per week,

So, average = m = 5

Now, on average, 5 people come every week, so,

5 women per week,

so, we get 1 woman per (1/5)th week,

Hence, the mean is m = 1/5 for a woman arriving

and λ = 1/m = 5 = λ

we have to find the probability that it takes higher than a week for a high BP pregnant woman to arrive, i.e,

P(X>1) i.e. the probability that it takes more than a week for a high BP pregnant woman to show up,

Now,

P(X>1) = 1 - P(X<1),

Now, the probability density function is,

\(f(x) = \lambda e^{-\lambda x}\)

And the cumulative distribution function (CDF) is,

\(CDF = 1 - e^{-\lambda x}\)

Now, CDF gives the probability of an event occuring within a given time,

so, for 1 week, we have x = 1, and λ = 5, which gives,

P(X<1) = CDF,

so,

\(P(X < 1)=CDF = 1 - e^{-\lambda x}\\P(X < 1)=1-e^{-5(1)}\\P(X < 1)=1-e^{-5}\\P(X < 1) = 1 - 6.738*10^{-3}\\P(X < 1) = 0.9932\\And,\\P(X > 1) = 1 - 0.9932\\P(X > 1) = 6.8*10^{-3}\\P(X > 1) = 0.0068\)

So, the probability that on a particular week, the hospital will receive on high BP pregnant woman is 0.0068